exercises-definition-graded-algebra

A {\it graded $A$-algebra} is a graded $A$-module $B = \bigoplus_{n \geq 0} B_n$ together with an $A$-bilinear map $$ B \times B \longrightarrow B, \ (b, b') \longmapsto bb' $$ that turns $B$ into an $A$-algebra so that $B_n \cdot B_m \subset B_{n + m}$. Finally, a {\it graded module $M$ over a graded $A$-algebra $B$} is given by a graded $A$-module $M$ together with a (compatible) $B$-module structure such that $B_n \cdot M_d \subset M_{n + d}$. Now you can define {\it homomorphisms of graded modules/rings}, {\it graded submodules}, {\it graded ideals}, {\it exact sequences of graded modules}, etc, etc.


exercises-definition-homogeneous-ideal

Let $R$ be a graded ring. A {\it homogeneous} ideal is simply an ideal $I \subset R$ which is also a graded submodule of $R$. Equivalently, it is an ideal generated by homogeneous elements. Equivalently, if $f \in I$ and $$ f = f_0 + f_1 + \ldots + f_n $$ is the decomposition of $f$ into homogeneous pieces in $R$ then $f_i \in I$ for each $i$.


exercises-definition-Proj-R

We define the {\it homogeneous spectrum $\text{Proj}(R)$} of the graded ring $R$ to be the set of homogeneous, prime ideals ${\mathfrak p}$ of $R$ such that $R_{+} \not \subset {\mathfrak p}$. Note that $\text{Proj}(R)$ is a subset of $\Spec(R)$ and hence has a natural induced topology.


exercises-definition-Dplus-Vplus

Let $R = \oplus_{d \geq 0} R_d$ be a graded ring, let $f\in R_d$ and assume that $d \geq 1$. We define {\it $R_{(f)}$} to be the subring of $R_f$ consisting of elements of the form $r/f^n$ with $r$ homogeneous and $\deg(r) = nd$. Furthermore, we define $$ D_{+}(f) = \{ {\mathfrak p} \in \text{Proj}(R) | f \not\in {\mathfrak p} \}. $$ Finally, for a homogeneous ideal $I \subset R$ we define $V_{+}(I) = V(I) \cap \text{Proj}(R)$.


exercises-definition-CM

A Noetherian local ring $A$ is said to be {\it Cohen-Macaulay} of dimension $d$ if it has dimension $d$ and there exists a system of parameters $x_1, \ldots, x_d$ for $A$ such that $x_i$ is a nonzerodivisor in $A/(x_1, \ldots, x_{i-1})$ for $i = 1, \ldots, d$.


exercises-definition-injective-filtered

Let $\mathcal{A}$ be an abelian category. Let $I$ be a filtered object of $\mathcal{A}$. Assume the filtration on $I$ is finite. We say $I$ is {\it filtered injective} if each $\text{gr}^p(I)$ is an injective object of $\mathcal{A}$.


exercises-definition-finite-filtration-category

Let $\mathcal{A}$ be an abelian category. We denote {\it $\text{Fil}^f(\mathcal{A})$} the full subcategory of $\text{Fil}(\mathcal{A})$ whose objects consist of those $A \in \Ob(\text{Fil}(\mathcal{A}))$ whose filtration is finite.


exercises-definition-filtered-quasi-isomorphism

Let $\mathcal{A}$ be an abelian category. Let $\alpha : K^\bullet \to L^\bullet$ be a morphism of complexes of $\text{Fil}(\mathcal{A})$. We say that $\alpha$ is a {\it filtered quasi-isomorphism} if for each $p \in \mathbf{Z}$ the morphism $\text{gr}^p(K^\bullet) \to \text{gr}^p(L^\bullet)$ is a quasi-isomorphism.


exercises-definition-filtered-acyclic

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex of $\text{Fil}^f(\mathcal{A})$. We say that $K^\bullet$ is {\it filtered acyclic} if for each $p \in \mathbf{Z}$ the complex $\text{gr}^p(K^\bullet)$ is acyclic.


exercises-definition-integral

A scheme $X$ is called {\it integral} if $X$ is nonempty and for every nonempty affine open $U \subset X$ the ring $\Gamma(U, \mathcal{O}_X) = \mathcal{O}_X(U)$ is a domain.


exercises-definition-dual-numbers

For any ring $R$ we denote $R[\epsilon]$ the ring of {\it dual numbers}. As an $R$-module it is free with basis $1$, $\epsilon$. The ring structure comes from setting $\epsilon^2 = 0$.


exercises-definition-tangent-space

Let $f : X \to S$ be a morphism of schemes. Let $x \in X$. We dub the set of dotted arrows of Exercise \ref{exercise-tangent-space-Zariski} the {\it tangent space of $X$ over $S$} and we denote it $T_{X/S, x}$. An element of this space is called a {\it tangent vector} of $X/S$ at $x$.


exercises-definition-quasi-coherent

Let $X$ be a scheme. A sheaf $\mathcal{F}$ of $\mathcal{O}_X$-modules is {\it quasi-coherent} if for every affine open $\Spec(R) = U \subset X$ the restriction $\mathcal{F}|_U$ is of the form $\widetilde M$ for some $R$-module $M$.


exercises-definition-specialization

Let $X$ be a topological space. Let $x, x' \in X$. We say $x$ is a {\it specialization} of $x'$ if and only if $x \in \overline{\{x'\}}$.


exercises-definition-Noetherian-scheme

A scheme $X$ is called {\it locally Noetherian} if and only if for every point $x \in X$ there exists an affine open $\Spec(R) = U \subset X$ such that $R$ is Noetherian. A scheme is {\it Noetherian} if it is locally Noetherian and quasi-compact.


exercises-definition-coherent

Let $X$ be a locally Noetherian scheme. Let $\mathcal{F}$ be a quasi-coherent sheaf of $\mathcal{O}_X$-modules. We say $\mathcal{F}$ is {\it coherent} if for every point $x \in X$ there exists an affine open $\Spec(R) = U \subset X$ such that $\mathcal{F}|_U$ is isomorphic to $\widetilde M$ for some finite $R$-module $M$.


exercises-definition-invertible-sheaf

Let $X$ be a locally ringed space. An {\it invertible ${\mathcal O}_X$-module} on $X$ is a sheaf of ${\mathcal O}_X$-modules ${\mathcal L}$ such that every point has an open neighbourhood $U \subset X$ such that ${\mathcal L}|_U$ is isomorphic to ${\mathcal O}_U$ as ${\mathcal O}_U$-module. We say that ${\mathcal L}$ is trivial if it is isomorphic to ${\mathcal O}_X$ as a ${\mathcal O}_X$-module.


exercises-definition-invertible-module

Let $R$ be a ring. An {\it invertible module $M$} is an $R$-module $M$ such that $\widetilde M$ is an invertible sheaf on the spectrum of $R$. We say $M$ is {\it trivial} if $M \cong R$ as an $R$-module.


exercises-definition-picard-group

Let $X$ be a locally ringed space. The {\it Picard group of $X$} is the set $\Pic(X)$ of isomorphism classes of invertible $\mathcal{O}_X$-modules with addition given by tensor product. See Modules, Definition \ref{modules-definition-pic}. For a ring $R$ we set $\Pic(R) = \Pic(\Spec(R))$.


exercises-definition-delta

(Definition of delta.) Suppose that $$ 0 \to {\mathcal F}_1 \to {\mathcal F}_2 \to {\mathcal F}_3 \to 0 $$ is a short exact sequence of abelian sheaves on any topological space $X$. The boundary map $\delta : H^0(X, {\mathcal F}_3) \to {\check H}^1(X, {\mathcal F}_1)$ is defined as follows. Take an element $\tau \in H^0(X, {\mathcal F}_3)$. Choose an open covering ${\mathcal U} : X = \bigcup_{i\in I} U_i$ such that for each $i$ there exists a section $\tilde \tau_i \in {\mathcal F}_2$ lifting the restriction of $\tau$ to $U_i$. Then consider the assignment $$ (i_0, i_1) \longmapsto \tilde \tau_{i_0}|_{U_{i_0i_1}} - \tilde \tau_{i_1}|_{U_{i_0i_1}}. $$ This is clearly a 1-coboundary in the {\v C}ech complex ${\check C}^\ast({\mathcal U}, {\mathcal F}_2)$. But we observe that (thinking of ${\mathcal F}_1$ as a subsheaf of ${\mathcal F}_2$) the RHS always is a section of ${\mathcal F}_1$ over $U_{i_0i_1}$. Hence we see that the assignment defines a 1-cochain in the complex ${\check C}^\ast({\mathcal U}, {\mathcal F}_2)$. The cohomology class of this 1-cochain is by definition {\it $\delta(\tau)$}.


exercises-definition-divisor

Throughout, let $S$ be any scheme and let $X$ be a Noetherian, integral scheme. \begin{enumerate} \item A {\it Weil divisor} on $X$ is a formal linear combination $\Sigma n_i[Z_i]$ of prime divisors $Z_i$ with integer coefficients. \item A {\it prime divisor} is a closed subscheme $Z \subset X$, which is integral with generic point $\xi \in Z$ such that ${\mathcal O}_{X, \xi}$ has dimension $1$. We will use the notation ${\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi}$ when $\xi \in Z \subset X$ is as above. Note that ${\mathcal O}_{X, Z} \subset K(X)$ is a subring of the function field of $X$. \item The {\it Weil divisor associated to a rational function $f \in K(X)^\ast$} is the sum $\Sigma v_Z(f)[Z]$. Here $v_Z(f)$ is defined as follows \begin{enumerate} \item If $f \in {\mathcal O}_{X, Z}^\ast$ then $v_Z(f) = 0$. \item If $f \in {\mathcal O}_{X, Z}$ then $$ v_Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f)). $$ \item If $f = \frac{a}{b}$ with $a, b \in {\mathcal O}_{X, Z}$ then $$ v_Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)). $$ \end{enumerate} \item An {\it effective Cartier divisor} on a scheme $S$ is a closed subscheme $D \subset S$ such that every point $d\in D$ has an affine open neighbourhood $\Spec(A) = U \subset S$ in $S$ so that $D \cap U = \Spec(A/(f))$ with $f \in A$ a nonzerodivisor. \item The {\it Weil divisor $[D]$ associated to an effective Cartier divisor $D \subset X$} of our Noetherian integral scheme $X$ is defined as the sum $\Sigma v_Z(D)[Z]$ where $v_Z(D)$ is defined as follows \begin{enumerate} \item If the generic point $\xi$ of $Z$ is not in $D$ then $v_Z(D) = 0$. \item If the generic point $\xi$ of $Z$ is in $D$ then $$ v_Z(D) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(f)) $$ where $f \in {\mathcal O}_{X, Z} = {\mathcal O}_{X, \xi}$ is the nonzerodivisor which defines $D$ in an affine neighbourhood of $\xi$ (as in (4) above). \end{enumerate} \item Let $S$ be a scheme. The {\it sheaf of total quotient rings ${\mathcal K}_S$} is the sheaf of ${\mathcal O}_S$-algebras which is the sheafification of the pre-sheaf ${\mathcal K}'$ defined as follows. For $U \subset S$ open we set ${\mathcal K}'(U) = S_U^{-1}{\mathcal O}_S(U)$ where $S_U \subset {\mathcal O}_S(U)$ is the multiplicative subset consisting of sections $f \in {\mathcal O}_S(U)$ such that the germ of $f$ in ${\mathcal O}_{S, u}$ is a nonzerodivisor for every $u\in U$. In particular the elements of $S_U$ are all nonzerodivisors. Thus ${\mathcal O}_S$ is a subsheaf of ${\mathcal K}_S$, and we get a short exact sequence $$ 0 \to {\mathcal O}_S^\ast \to {\mathcal K}_S^\ast \to {\mathcal K}_S^\ast/{\mathcal O}_S^\ast \to 0. $$ \item A {\it Cartier divisor} on a scheme $S$ is a global section of the quotient sheaf ${\mathcal K}_S^\ast/{\mathcal O}_S^\ast$. \item The {\it Weil divisor associated to a Cartier divisor} $\tau \in \Gamma(X, {\mathcal K}_X^\ast/{\mathcal O}_X^\ast)$ over our Noetherian integral scheme $X$ is the sum $\Sigma v_Z(\tau)[Z]$ where $v_Z(\tau)$ is defined as by the following recipe \begin{enumerate} \item If the germ of $\tau$ at the generic point $\xi$ of $Z$ is zero -- in other words the image of $\tau$ in the stalk $({\mathcal K}^\ast/{\mathcal O}^\ast)_\xi$ is ``zero'' -- then $v_Z(\tau) = 0$. \item Find an affine open neighbourhood $\Spec(A) = U \subset X$ so that $\tau|_U$ is the image of a section $f \in {\mathcal K}(U)$ and moreover $f = a/b$ with $a, b \in A$. Then we set $$ v_Z(f) = \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(a)) - \text{length}_{{\mathcal O}_{X, Z}}({\mathcal O}_{X, Z}/(b)). $$ \end{enumerate} \end{enumerate}


spaces-duality-definition-dualizing-scheme

Let $S$ be a scheme. Let $X$ be a locally Noetherian algebraic space over $S$. An object $K$ of $D_\QCoh(\mathcal{O}_X)$ is called a {\it dualizing complex} if $K$ satisfies the equivalent conditions of Lemma \ref{lemma-equivalent-definitions}.


spaces-duality-definition-relative-dualizing-proper-flat

Let $S$ be a scheme. Let $f : X \to Y$ be a proper, flat morphism of algebraic spaces over $S$ which is of finite presentation. A {\it relative dualizing complex} for $X/Y$ is a pair $(\omega_{X/Y}^\bullet, \tau)$ consisting of a $Y$-perfect object $\omega_{X/Y}^\bullet$ of $D(\mathcal{O}_X)$ and a map $$ \tau : Rf_*\omega_{X/Y}^\bullet \longrightarrow \mathcal{O}_Y $$ such that for any cartesian square $$ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^f \\ Y' \ar[r]^g & Y } $$ where $Y'$ is an affine scheme the pair $(L(g')^*\omega_{X/Y}^\bullet, Lg^*\tau)$ is isomorphic to the pair $(a'(\mathcal{O}_{Y'}), \text{Tr}_{f', \mathcal{O}_{Y'}})$ studied in Sections \ref{section-twisted-inverse-image}, \ref{section-base-change-map}, \ref{section-base-change-II}, \ref{section-trace}, \ref{section-compare-with-pullback}, and \ref{section-proper-flat}.


spaces-properties-definition-separated

(Compare Spaces, Definition \ref{spaces-definition-separated}.) Consider a big fppf site $\Sch_{fppf} = (\Sch/\Spec(\mathbf{Z}))_{fppf}$. Let $X$ be an algebraic space over $\Spec(\mathbf{Z})$. Let $\Delta : X \to X \times X$ be the diagonal morphism. \begin{enumerate} \item We say $X$ is {\it separated} if $\Delta$ is a closed immersion. \item We say $X$ is {\it locally separated}\footnote{In the literature this often refers to quasi-separated and locally separated algebraic spaces.} if $\Delta$ is an immersion. \item We say $X$ is {\it quasi-separated} if $\Delta$ is quasi-compact. \item We say $X$ is {\it Zariski locally quasi-separated}\footnote{ This notion was suggested by B.\ Conrad.} if there exists a Zariski covering $X = \bigcup_{i \in I} X_i$ (see Spaces, Definition \ref{spaces-definition-Zariski-open-covering}) such that each $X_i$ is quasi-separated. \end{enumerate} Let $S$ is a scheme contained in $\Sch_{fppf}$, and let $X$ be an algebraic space over $S$. Then we say $X$ is {\it separated}, {\it locally separated}, {\it quasi-separated}, or {\it Zariski locally quasi-separated} if $X$ viewed as an algebraic space over $\Spec(\mathbf{Z})$ (see Spaces, Definition \ref{spaces-definition-base-change}) has the corresponding property.


spaces-properties-definition-points

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A {\it point} of $X$ is an equivalence class of morphisms from spectra of fields into $X$. The set of points of $X$ is denoted $|X|$.


spaces-properties-definition-topological-space

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The underlying {\it topological space} of $X$ is the set of points $|X|$ endowed with the topology constructed in Lemma \ref{lemma-topology-points}.


spaces-properties-definition-quasi-compact

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ is {\it quasi-compact} if there exists a surjective \'etale morphism $U \to X$ with $U$ quasi-compact.


spaces-properties-definition-type-property

Let $\mathcal{P}$ be a property of schemes which is local in the \'etale topology. Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ {\it has property $\mathcal{P}$} if any of the equivalent conditions of Lemma \ref{lemma-type-property} hold.


spaces-properties-definition-property-at-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. Let $\mathcal{P}$ be a property of germs of schemes which is \'etale local. We say $X$ {\it has property $\mathcal{P}$ at $x$} if any of the equivalent conditions of Lemma \ref{lemma-local-source-target-at-point} hold.


spaces-properties-definition-locally-constructible

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $E \subset |X|$ be a subset. We say $E$ is {\it \'etale locally constructible} if the equivalent conditions of Lemma \ref{lemma-locally-constructible} are satisfied.


spaces-properties-definition-dimension-at-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point of $X$. We define the {\it dimension of $X$ at $x$} to be the element $\dim_x(X) \in \{0, 1, 2, \ldots, \infty\}$ such that $\dim_x(X) = \dim_u(U)$ for any (equivalently some) pair $(a : U \to X, u)$ consisting of an \'etale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$. See Definition \ref{definition-property-at-point}, Lemma \ref{lemma-local-source-target-at-point}, and Descent, Lemma \ref{descent-lemma-dimension-at-point-local}.


spaces-properties-definition-dimension

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it dimension} $\dim(X)$ of $X$ is defined by the rule $$ \dim(X) = \sup\nolimits_{x \in |X|} \dim_x(X) $$


spaces-properties-definition-dimension-local-ring

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. The {\it dimension of the local ring of $X$ at $x$} is the element $d \in \{0, 1, 2, \ldots, \infty\}$ satisfying the equivalent conditions of Lemma \ref{lemma-pre-dimension-local-ring}. In this case we will also say {\it $x$ is a point of codimension $d$ on $X$}.


spaces-properties-definition-reduced-induced-space

Let $S$ be a scheme, and let $X$ be an algebraic space over $S$. Let $Z \subset |X|$ be a closed subset. An {\it algebraic space structure on $Z$} is given by a closed subspace $Z'$ of $X$ with $|Z'|$ equal to $Z$. The {\it reduced induced algebraic space structure} on $Z$ is the one constructed in Lemma \ref{lemma-reduced-closed-subspace}. The {\it reduction $X_{red}$ of $X$} is the reduced induced algebraic space structure on $|X|$.


spaces-properties-definition-etale

Let $S$ be a scheme. A morphism $f : X \to Y$ between algebraic spaces over $S$ is called {\it \'etale} if and only if for every \'etale morphism $\varphi : U \to X$ where $U$ is a scheme, the composition $f \circ \varphi$ is \'etale also.


spaces-properties-definition-etale-site

Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$, and let $\Sch_\etale$ be the corresponding big \'etale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The {\it small \'etale site $X_\etale$} of $X$ is defined as follows: \begin{enumerate} \item An object of $X_\etale$ is a morphism $\varphi : U \to X$ where $U \in \Ob((\Sch/S)_\etale)$ is a scheme and $\varphi$ is an \'etale morphism, \item a morphism $(\varphi : U \to X) \to (\varphi' : U' \to X)$ is given by a morphism of schemes $\chi : U \to U'$ such that $\varphi = \varphi' \circ \chi$, and \item a family of morphisms $\{(U_i \to X) \to (U \to X)\}_{i \in I}$ of $X_\etale$ is a covering if and only if $\{U_i \to U\}_{i \in I}$ is a covering of $(\Sch/S)_\etale$. \end{enumerate}


spaces-properties-definition-spaces-etale-site

Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$, and let $\Sch_\etale$ be the corresponding big \'etale site (i.e., having the same underlying category). Let $X$ be an algebraic space over $S$. The site {\it $X_{spaces, \etale}$} of $X$ is defined as follows: \begin{enumerate} \item An object of $X_{spaces, \etale}$ is a morphism $\varphi : U \to X$ where $U$ is an algebraic space over $S$ and $\varphi$ is an \'etale morphism of algebraic spaces over $S$, \item a morphism $(\varphi : U \to X) \to (\varphi' : U' \to X)$ of $X_{spaces, \etale}$ is given by a morphism of algebraic spaces $\chi : U \to U'$ such that $\varphi = \varphi' \circ \chi$, and \item a family of morphisms $\{\varphi_i : (U_i \to X) \to (U \to X)\}_{i \in I}$ of $X_{spaces, \etale}$ is a covering if and only if $|U| = \bigcup \varphi_i(|U_i|)$. \end{enumerate} (As usual we choose a set of coverings of this type, including at least the coverings in $X_\etale$, as in Sets, Lemma \ref{sets-lemma-coverings-site} to turn $X_{spaces, \etale}$ into a site.)


spaces-properties-definition-etale-topos

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it \'etale topos} of $X$, or more precisely the {\it small \'etale topos} of $X$ is the category $\Sh(X_\etale)$ of sheaves of sets on $X_\etale$.


spaces-properties-definition-f-map

Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\mathcal{F}$ be a sheaf of sets on $X_\etale$ and let $\mathcal{G}$ be a sheaf of sets on $Y_\etale$. An {\it $f$-map $\varphi : \mathcal{G} \to \mathcal{F}$} is a collection of maps $\varphi_{(U, V, g)} : \mathcal{G}(V) \to \mathcal{F}(U)$ indexed by commutative diagrams $$ \xymatrix{ U \ar[d]_g \ar[r] & X \ar[d]^f \\ V \ar[r] & Y } $$ where $U \in X_\etale$, $V \in Y_\etale$ such that whenever given an extended diagram $$ \xymatrix{ U' \ar[r] \ar[d]_{g'} & U \ar[d]_g \ar[r] & X \ar[d]^f \\ V' \ar[r] & V \ar[r] & Y } $$ with $V' \to V$ and $U' \to U$ \'etale morphisms of schemes the diagram $$ \xymatrix{ \mathcal{G}(V) \ar[rr]_{\varphi_{(U, V, g)}} \ar[d]_{\text{restriction of }\mathcal{G}} & & \mathcal{F}(U) \ar[d]^{\text{restriction of }\mathcal{F}} \\ \mathcal{G}(V') \ar[rr]^{\varphi_{(U', V', g')}} & & \mathcal{F}(U') } $$ commutes.


spaces-properties-definition-geometric-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. \begin{enumerate} \item A {\it geometric point} of $X$ is a morphism $\overline{x} : \Spec(k) \to X$, where $k$ is an algebraically closed field. We often abuse notation and write $\overline{x} = \Spec(k)$. \item For every geometric point $\overline{x}$ we have the corresponding ``image'' point $x \in |X|$. We say that $\overline{x}$ is a {\it geometric point lying over $x$}. \end{enumerate}


spaces-properties-definition-etale-neighbourhood

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$. \begin{enumerate} \item An {\it \'etale neighborhood} of $\overline{x}$ of $X$ is a commutative diagram $$ \xymatrix{ & U \ar[d]^\varphi \\ {\bar x} \ar[r]^{\bar x} \ar[ur]^{\bar u} & X } $$ where $\varphi$ is an \'etale morphism of algebraic spaces over $S$. We will use the notation $\varphi : (U, \overline{u}) \to (X, \overline{x})$ to indicate this situation. \item A {\it morphism of \'etale neighborhoods} $(U, \overline{u}) \to (U', \overline{u}')$ is an $X$-morphism $h : U \to U'$ such that $\overline{u}' = h \circ \overline{u}$. \end{enumerate}


spaces-properties-definition-stalk

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a presheaf on $X_\etale$. Let $\overline{x}$ be a geometric point of $X$. The {\it stalk} of $\mathcal{F}$ at $\overline{x}$ is $$ \mathcal{F}_{\bar x} = \colim_{(U, \overline{u})} \mathcal{F}(U) $$ where $(U, \overline{u})$ runs over all \'etale neighborhoods of $\overline{x}$ in $X$ with $U \in \Ob(X_\etale)$.


spaces-properties-definition-support

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be an abelian sheaf on $X_\etale$. \begin{enumerate} \item The {\it support of $\mathcal{F}$} is the set of points $x \in |X|$ such that $\mathcal{F}_{\overline{x}} \not = 0$ for any (some) geometric point $\overline{x}$ lying over $x$. \item Let $\sigma \in \mathcal{F}(U)$ be a section. The {\it support of $\sigma$} is the closed subset $U \setminus W$, where $W \subset U$ is the largest open subset of $U$ on which $\sigma$ restricts to zero (see Lemma \ref{lemma-zero-over-image}). \end{enumerate}


spaces-properties-definition-structure-sheaf

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. The {\it structure sheaf} of $X$ is the sheaf of rings $\mathcal{O}_X$ on the small \'etale site $X_\etale$ described in Lemma \ref{lemma-sheaf-condition-holds}.


spaces-properties-definition-etale-local-rings

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\overline{x}$ be a geometric point of $X$ lying over the point $x \in |X|$. \begin{enumerate} \item The {\it \'etale local ring of $X$ at $\overline{x}$} is the stalk of the structure sheaf $\mathcal{O}_X$ on $X_\etale$ at $\overline{x}$. Notation: $\mathcal{O}_{X, \overline{x}}$. \item The {\it strict henselization of $X$ at $\overline{x}$} is the scheme $\Spec(\mathcal{O}_{X, \overline{x}})$. \end{enumerate}


spaces-properties-definition-unibranch

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. We say that $X$ is {\it geometrically unibranch at $x$} if the equivalent conditions of Lemma \ref{lemma-irreducible-local-ring} hold. We say that $X$ is {\it geometrically unibranch} if $X$ is geometrically unibranch at every $x \in |X|$.


spaces-properties-definition-number-of-geometric-branches

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$. The {\it number of geometric branches of $X$ at $x$} is either $n \in \mathbf{N}$ if the equivalent conditions of Lemma \ref{lemma-nr-branches-local-ring} hold, or else $\infty$.


spaces-properties-definition-noetherian

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. We say $X$ is {\it Noetherian} if $X$ is quasi-compact, quasi-separated and locally Noetherian.


spaces-properties-definition-regular-at-point

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $x \in |X|$ be a point. We say {\it $X$ is regular at $x$} if $\mathcal{O}_{U, u}$ is a regular local ring for any (equivalently some) pair $(a : U \to X, u)$ consisting of an \'etale morphism $a : U \to X$ from a scheme to $X$ and a point $u \in U$ with $a(u) = x$.


spaces-properties-definition-quasi-coherent

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. A {\it quasi-coherent} $\mathcal{O}_X$-module is a quasi-coherent module on the ringed site $(X_\etale, \mathcal{O}_X)$ in the sense of Modules on Sites, Definition \ref{sites-modules-definition-site-local}. The category of quasi-coherent sheaves on $X$ is denoted $\QCoh(\mathcal{O}_X)$.


spaces-properties-definition-locally-projective

Let $S$ be a scheme. Let $X$ be an algebraic space over $S$. Let $\mathcal{F}$ be a quasi-coherent $\mathcal{O}_X$-module. We say $\mathcal{F}$ is {\it locally projective} if the equivalent conditions of Lemma \ref{lemma-locally-projective} are satisfied.


intersection-definition-proper-intersection

Let $X$ be a nonsingular variety. \begin{enumerate} \item Let $W,V \subset X$ be closed subvarieties with $\dim(W) = s$ and $\dim(V) = r$. We say that $W$ and $V$ {\it intersect properly} if $\dim(V \cap W) \leq r + s - \dim(X)$. \item Let $\alpha = \sum n_i [W_i]$ be an $s$-cycle, and $\beta = \sum_j m_j [V_j]$ be an $r$-cycle on $X$. We say that $\alpha$ and $\beta$ {\it intersect properly} if $W_i$ and $V_j$ intersect properly for all $i$ and $j$. \end{enumerate}


intersection-definition-multiplicity

In the situation above, if $d \geq \dim(\text{Supp}(M))$, then we set $e_I(M, d)$ equal to $0$ if $d > \dim(\text{Supp}(M))$ and equal to $d!$ times the leading coefficient of the numerical polynomial $\chi_{I, M}$ so that $$ \chi_{I, M}(n) \sim e_I(M, d) \frac{n^d}{d!} + \text{lower order terms} $$ The {\it multiplicity of $M$ for the ideal of definition $I$} is $e_I(M) = e_I(M, \dim(\text{Supp}(M)))$.


homology-definition-preadditive

A category $\mathcal{A}$ is called {\it preadditive} if each morphism set $\Mor_\mathcal{A}(x, y)$ is endowed with the structure of an abelian group such that the compositions $$ \Mor(x, y) \times \Mor(y, z) \longrightarrow \Mor(x, z) $$ are bilinear. A functor $F : \mathcal{A} \to \mathcal{B}$ of preadditive categories is called {\it additive} if and only if $F : \Mor(x, y) \to \Mor(F(x), F(y))$ is a homomorphism of abelian groups for all $x, y \in \Ob(\mathcal{A})$.


homology-definition-zero-object

In a preadditive category $\mathcal{A}$ we call {\it zero object}, and we denote it $0$ any final and initial object as in Lemma \ref{lemma-preadditive-zero} above.


homology-definition-direct-sum

Given a pair of objects $x, y$ in a preadditive category $\mathcal{A}$, the {\it direct sum} $x \oplus y$ of $x$ and $y$ is the direct product $x \times y$ endowed with the morphisms $i, j, p, q$ as in Lemma \ref{lemma-preadditive-direct-sum} above.


homology-definition-additive-category

A category $\mathcal{A}$ is called {\it additive} if it is preadditive and finite products exist, in other words it has a zero object and direct sums.


homology-definition-kernel

Let $\mathcal{A}$ be a preadditive category. Let $f : x \to y$ be a morphism. \begin{enumerate} \item A {\it kernel} of $f$ is a morphism $i : z \to x$ such that (a) $f \circ i = 0$ and (b) for any $i' : z' \to x$ such that $f \circ i' = 0$ there exists a unique morphism $g : z' \to z$ such that $i' = i \circ g$. \item If the kernel of $f$ exists, then we denote this $\Ker(f) \to x$. \item A {\it cokernel} of $f$ is a morphism $p : y \to z$ such that (a) $p \circ f = 0$ and (b) for any $p' : y \to z'$ such that $p' \circ f = 0$ there exists a unique morphism $g : z \to z'$ such that $p' = g \circ p$. \item If a cokernel of $f$ exists we denote this $y \to \Coker(f)$. \item If a kernel of $f$ exists, then a {\it coimage of $f$} is a cokernel for the morphism $\Ker(f) \to x$. \item If a kernel and coimage exist then we denote this $x \to \Coim(f)$. \item If a cokernel of $f$ exists, then the {\it image of $f$} is a kernel of the morphism $y \to \Coker(f)$. \item If a cokernel and image of $f$ exist then we denote this $\Im(f) \to y$. \end{enumerate}


homology-definition-karoubian

Let $\mathcal{C}$ be a preadditive category. We say $\mathcal{C}$ is {\it Karoubian} if every idempotent endomorphism of an object of $\mathcal{C}$ has a kernel.


homology-definition-abelian-category

A category $\mathcal{A}$ is {\it abelian} if it is additive, if all kernels and cokernels exist, and if the natural map $\Coim(f) \to \Im(f)$ is an isomorphism for all morphisms $f$ of $\mathcal{A}$.


homology-definition-injective-surjective

Let $f : x \to y$ be a morphism in an abelian category. \begin{enumerate} \item We say $f$ is {\it injective} if $\Ker(f) = 0$. \item We say $f$ is {\it surjective} if $\Coker(f) = 0$. \end{enumerate} If $x \to y$ is injective, then we say that $x$ is a {\it subobject} of $y$ and we use the notation $x \subset y$. If $x \to y$ is surjective, then we say that $y$ is a {\it quotient} of $x$.


homology-definition-exact

Let $\mathcal{A}$ be an additive category. We say a sequence of morphisms $$ \ldots \to x \to y \to z \to \ldots $$ in $\mathcal{A}$ is a {\it complex} if the composition of any two (drawn) arrows is zero. If $\mathcal{A}$ is abelian then we say a sequence as above is {\it exact at $y$} if $\Im(x \to y) = \Ker(y \to z)$. We say it is {\it exact} if it is exact at every object. A {\it short exact sequence} is an exact complex of the form $$ 0 \to A  \to B \to C \to 0. $$


homology-definition-ses-split

Let $\mathcal{A}$ be an abelian category. Let $i : A \to B$ and $q : B \to C$ be morphisms of $\mathcal{A}$ such that $0 \to A \to B \to C \to 0$ is a short exact sequence. We say the short exact sequence is {\it split} if there exist morphisms $j : C \to B$ and $p : B \to A$ such that $(B, i, j, p, q)$ is the direct sum of $A$ and $C$.


homology-definition-extension

Let $\mathcal{A}$ be an abelian category. Let $A, B \in \Ob(\mathcal{A})$. An {\it extension $E$ of $B$ by $A$} is a short exact sequence $$ 0 \to A \to E \to B \to 0. $$ An {\it morphism of extensions} between two extensions $0 \to A \to E \to B \to 0$ and $0 \to A \to F \to B \to 0$ means a morphism $f : E \to F$ in $\mathcal{A}$ making the diagram $$ \xymatrix{ 0 \ar[r] & A \ar[r] \ar[d]^{\text{id}} & E \ar[r] \ar[d]^f & B \ar[r] \ar[d]^{\text{id}} & 0 \\ 0 \ar[r] & A \ar[r] & F \ar[r] & B \ar[r] & 0 } $$ commutative. Thus, the extensions of $B$ by $A$ form a category.


homology-definition-ext-group

Let $\mathcal{A}$ be an abelian category. Let $A, B \in \Ob(\mathcal{A})$. The set of isomorphism classes of extensions of $B$ by $A$ is denoted $$ \Ext_\mathcal{A}(B, A). $$ This is called the {\it $\Ext$-group}.


homology-definition-simple

Let $\mathcal{A}$ be an abelian category. An object $A$ of $\mathcal{A}$ is said to be {\it simple} if it is nonzero and the only subobjects of $A$ are $0$ and $A$.


homology-definition-Artinian

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item We say an object $A$ of $\mathcal{A}$ is {\it Artinian} if and only if it satisfies the descending chain condition for subobjects. \item We say $\mathcal{A}$ is {\it Artinian} if every object of $\mathcal{A}$ is Artinian. \end{enumerate}


homology-definition-Noetherian

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item We say an object $A$ of $\mathcal{A}$ is {\it Noetherian} if and only if it satisfies the ascending chain condition for subobjects. \item We say $\mathcal{A}$ is {\it Noetherian} if every object of $\mathcal{A}$ is Noetherian. \end{enumerate}


homology-definition-serre-subcategory

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item A {\it Serre subcategory} of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence $$ A \to B \to C $$ with $A, C \in \Ob(\mathcal{C})$, then also $B \in \Ob(\mathcal{C})$. \item A {\it weak Serre subcategory} of $\mathcal{A}$ is a nonempty full subcategory $\mathcal{C}$ of $\mathcal{A}$ such that given an exact sequence $$ A_0 \to A_1 \to A_2 \to A_3 \to A_4 $$ with $A_0, A_1, A_3, A_4$ in $\mathcal{C}$, then also $A_2$ in $\mathcal{C}$. \end{enumerate}


homology-definition-kernel-category

Let $\mathcal{A}$, $\mathcal{B}$ be abelian categories. Let $F : \mathcal{A} \to \mathcal{B}$ be an exact functor. Then the full subcategory of objects $C$ of $\mathcal{A}$ such that $F(C) = 0$ is called the {\it kernel of the functor $F$}, and is sometimes denoted $\Ker(F)$.


homology-definition-K-zero

Let $\mathcal{A}$ be an abelian category. We denote $K_0(\mathcal{A})$ the {\it zeroth $K$-group of $\mathcal{A}$}. It is the abelian group constructed as follows. Take the free abelian group on the objects on $\mathcal{A}$ and for every short exact sequence $0 \to A \to B \to C \to 0$ impose the relation $[B] - [A] - [C] = 0$.


homology-definition-cohomological-delta-functor

Let $\mathcal{A}, \mathcal{B}$ be abelian categories. A {\it cohomological $\delta$-functor} or simply a {\it $\delta$-functor} from $\mathcal{A}$ to $\mathcal{B}$ is given by the following data: \begin{enumerate} \item a collection $F^n : \mathcal{A} \to \mathcal{B}$, $n \geq 0$ of additive functors, and \item for every short exact sequence $0 \to A \to B \to C \to 0$ of $\mathcal{A}$ a collection $\delta_{A \to B \to C} : F^n(C) \to F^{n + 1}(A)$, $n \geq 0$ of morphisms of $\mathcal{B}$. \end{enumerate} These data are assumed to satisfy the following axioms \begin{enumerate} \item for every short exact sequence as above the sequence $$ \xymatrix{ 0 \ar[r] & F^0(A) \ar[r] & F^0(B) \ar[r] & F^0(C) \ar[lld]^{\delta_{A \to B \to C}} \\  & F^1(A) \ar[r] & F^1(B) \ar[r] & F^1(C) \ar[lld]^{\delta_{A \to B \to C}} \\  & F^2(A) \ar[r] & F^2(B) \ar[r] & \ldots } $$ is exact, and \item for every morphism $(A \to B \to C) \to (A' \to B' \to C')$ of short exact sequences of $\mathcal{A}$ the diagrams $$ \xymatrix{ F^n(C) \ar[d] \ar[rr]_{\delta_{A \to B \to C}} & & F^{n + 1}(A) \ar[d] \\ F^n(C') \ar[rr]^{\delta_{A' \to B' \to C'}} & & F^{n + 1}(A') } $$ are commutative. \end{enumerate}


homology-definition-morphism-delta-functors

Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $(F^n, \delta_F)$ and $(G^n, \delta_G)$ be $\delta$-functors from $\mathcal{A}$ to $\mathcal{B}$. A {\it morphism of $\delta$-functors from $F$ to $G$} is a collection of transformation of functors $t^n : F^n \to G^n$, $n \geq 0$ such that for every short exact sequence $0 \to A \to B \to C \to 0$ of $\mathcal{A}$ the diagrams $$ \xymatrix{ F^n(C) \ar[d]_{t^n} \ar[rr]_{\delta_{F, A \to B \to C}} & & F^{n + 1}(A) \ar[d]^{t^{n + 1}} \\ G^n(C) \ar[rr]^{\delta_{G, A \to B \to C}} & & G^{n + 1}(A) } $$ are commutative.


homology-definition-universal-delta-functor

Let $\mathcal{A}, \mathcal{B}$ be abelian categories. Let $F = (F^n, \delta_F)$ be a $\delta$-functor from $\mathcal{A}$ to $\mathcal{B}$. We say $F$ is a {\it universal $\delta$-functor} if and only if for every $\delta$-functor $G = (G^n, \delta_G)$ and any morphism of functors $t : F^0 \to G^0$ there exists a unique morphism of $\delta$-functors $\{t^n\}_{n \geq 0} : F \to G$ such that $t = t^0$.


homology-definition-homotopy-equivalent

Let $\mathcal{A}$ be an additive category. We say a morphism $a : A_\bullet \to B_\bullet$ is a {\it homotopy equivalence} if there exists a morphism $b : B_\bullet \to A_\bullet$ such that there exists a homotopy between $a \circ b$ and $\text{id}_A$ and there exists a homotopy between $b \circ a$ and $\text{id}_B$. If there exists such a morphism between $A_\bullet$ and $B_\bullet$, then we say that $A_\bullet$ and $B_\bullet$ are {\it homotopy equivalent}.


homology-definition-quasi-isomorphism

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item A morphism of chain complexes $f : A_\bullet \to B_\bullet$ is called a {\it quasi-isomorphism} if the induced map $H_i(f) : H_i(A_\bullet) \to H_i(B_\bullet)$ is an isomorphism for all $i \in \mathbf{Z}$. \item A chain complex $A_\bullet$ is called {\it acyclic} if all of its homology objects $H_i(A_\bullet)$ are zero. \end{enumerate}


homology-definition-homotopy-equivalent-cochain

Let $\mathcal{A}$ be an additive category. We say a morphism $a : A^\bullet \to B^\bullet$ is a {\it homotopy equivalence} if there exists a morphism $b : B^\bullet \to A^\bullet$ such that there exists a homotopy between $a \circ b$ and $\text{id}_A$ and there exists a homotopy between $b \circ a$ and $\text{id}_B$. If there exists such a morphism between $A^\bullet$ and $B^\bullet$, then we say that $A^\bullet$ and $B^\bullet$ are {\it homotopy equivalent}.


homology-definition-quasi-isomorphism-cochain

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item A morphism of cochain complexes $f : A^\bullet \to B^\bullet$ of $\mathcal{A}$ is called a {\it quasi-isomorphism} if the induced maps $H^i(f) : H^i(A^\bullet) \to H^i(B^\bullet)$ is an isomorphism for all $i \in \mathbf{Z}$. \item A cochain complex $A^\bullet$ is called {\it acyclic} if all of its cohomology objects $H^i(A^\bullet)$ are zero. \end{enumerate}


homology-definition-shift

Let $\mathcal{A}$ be an additive category. Let $A_\bullet$ be a chain complex with boundary maps $d_{A, n} : A_n \to A_{n - 1}$. For any $k \in \mathbf{Z}$ we define the {\it $k$-shifted chain complex $A[k]_\bullet$} as follows: \begin{enumerate} \item we set $A[k]_n = A_{n + k}$, and \item we set $d_{A[k], n} : A[k]_n \to A[k]_{n - 1}$ equal to $d_{A[k], n} = (-1)^k d_{A, n + k}$. \end{enumerate} If $f : A_\bullet \to B_\bullet$ is a morphism of chain complexes, then we let $f[k] : A[k]_\bullet \to B[k]_\bullet$ be the morphism of chain complexes with $f[k]_n = f_{k + n}$.


homology-definition-homology-shift

Let $\mathcal{A}$ be an abelian category. Let $A_\bullet$ be a chain complex with boundary maps $d_{A, n} : A_n \to A_{n - 1}$. For any $k \in \mathbf{Z}$ we identify {\it $H_{i + k}(A_\bullet) \rightarrow H_i(A[k]_\bullet)$} via the identification $A_{i + k} = A[k]_i$.


homology-definition-shift-cochain

Let $\mathcal{A}$ be an additive category. Let $A^\bullet$ be a cochain complex with boundary maps $d_A^n : A^n \to A^{n + 1}$. For any $k \in \mathbf{Z}$ we define the {\it $k$-shifted cochain complex $A[k]^\bullet$} as follows: \begin{enumerate} \item we set $A[k]^n = A^{n + k}$, and \item we set $d_{A[k]}^n : A[k]^n \to A[k]^{n + 1}$ equal to $d_{A[k]}^n = (-1)^k d_A^{n + k}$. \end{enumerate} If $f : A^\bullet \to B^\bullet$ is a morphism of cochain complexes, then we let $f[k] : A[k]^\bullet \to B[k]^\bullet$ be the morphism of cochain complexes with $f[k]^n = f^{k + n}$.


homology-definition-cohomology-shift

Let $\mathcal{A}$ be an abelian category. Let $A^\bullet$ be a cochain complex with boundary maps $d_A^n : A^n \to A^{n + 1}$. For any $k \in \mathbf{Z}$ we identify {\it $H^{i + k}(A^\bullet) \longrightarrow H^i(A[k]^\bullet)$} via the identification $A^{i + k} = A[k]^i$.


homology-definition-graded

Let $\mathcal{A}$ be an additive category. The {\it category of graded objects of $\mathcal{A}$}, denoted $\text{Gr}(\mathcal{A})$, is the category with \begin{enumerate} \item objects $A = (A^i)$ are families of objects $A^i$, $i \in \mathbf{Z}$ of objects of $\mathcal{A}$, and \item morphisms $f : A = (A^i) \to B = (B^i)$ are families of morphisms $f^i : A^i \to B^i$ of $\mathcal{A}$. \end{enumerate}


homology-definition-graded-shift

Let $\mathcal{A}$ be an additive category. If $A = (A^i)$ is a graded object, then the $k$th {\it shift} $A[k]$ is the graded object with $A[k]^i = A^{k + i}$.


homology-definition-additive-monoidal

An {\it additive monoidal category} is an additive category $\mathcal{A}$ endowed with a monoidal structure $\otimes, \phi$ (Categories, Definition \ref{categories-definition-monoidal-category}) such that $\otimes$ is an additive functor in each variable.


homology-definition-double-complex

Let $\mathcal{A}$ be an additive category. A {\it double complex} in $\mathcal{A}$ is given by a system $(\{A^{p, q}, d_1^{p, q}, d_2^{p, q}\}_{p, q\in \mathbf{Z}})$, where each $A^{p, q}$ is an object of $\mathcal{A}$ and $d_1^{p, q} : A^{p, q} \to A^{p + 1, q}$ and $d_2^{p, q} : A^{p, q} \to A^{p, q + 1}$ are morphisms of $\mathcal{A}$ such that the following rules hold: \begin{enumerate} \item $d_1^{p + 1, q} \circ d_1^{p, q} = 0$ \item $d_2^{p, q + 1} \circ d_2^{p, q} = 0$ \item $d_1^{p, q + 1} \circ d_2^{p, q} = d_2^{p + 1, q} \circ d_1^{p, q}$ \end{enumerate} for all $p, q \in \mathbf{Z}$.


homology-definition-associated-simple-complex

Let $\mathcal{A}$ be an additive category. Let $A^{\bullet, \bullet}$ be a double complex. The {\it associated simple complex}, denoted $sA^\bullet$, also often called the {\it associated total complex}, denoted $\text{Tot}(A^{\bullet, \bullet})$, is given by $$ sA^n = \text{Tot}^n(A^{\bullet, \bullet}) = \bigoplus\nolimits_{n = p + q} A^{p, q} $$ (if it exists) with differential $$ d_{sA^\bullet}^n = d_{\text{Tot}(A^{\bullet, \bullet})}^n = \sum\nolimits_{n = p + q} (d_1^{p, q} + (-1)^p d_2^{p, q}) $$


homology-definition-filtered

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item A {\it decreasing filtration} $F$ on an object $A$ is a family $(F^nA)_{n \in \mathbf{Z}}$ of subobjects of $A$ such that $$ A \supset \ldots \supset F^nA \supset F^{n + 1}A \supset \ldots \supset 0 $$ \item A {\it filtered object of $\mathcal{A}$} is pair $(A, F)$ consisting of an object $A$ of $\mathcal{A}$ and a decreasing filtration $F$ on $A$. \item A {\it morphism $(A, F) \to (B, F)$ of filtered objects} is given by a morphism $\varphi : A \to B$ of $\mathcal{A}$ such that $\varphi(F^iA) \subset F^iB$ for all $i \in \mathbf{Z}$. \item The category of filtered objects is denoted $\text{Fil}(\mathcal{A})$. \item Given a filtered object $(A, F)$ and a subobject $X \subset A$ the {\it induced filtration} on $X$ is the filtration with $F^nX = X \cap F^nA$. \item Given a filtered object $(A, F)$ and a surjection $\pi : A \to Y$ the {\it quotient filtration} is the filtration with $F^nY = \pi(F^nA)$. \item A filtration $F$ on an object $A$ is said to be {\it finite} if there exist $n, m$ such that $F^nA = A$ and $F^mA = 0$. \item Given a filtered object $(A, F)$ we say $\bigcap F^iA$ exists if there exists a biggest subobject of $A$ contained in all $F^iA$. We say $\bigcup F^iA$ exists if there exists a smallest subobject of $A$ containing all $F^iA$. \item The filtration on a filtered object $(A, F)$ is said to be {\it separated} if $\bigcap F^iA = 0$ and {\it exhaustive} if $\bigcup F^iA = A$. \end{enumerate}


homology-definition-strict

Let $\mathcal{A}$ be an abelian category. A morphism $f : A \to B$ of filtered objects of $\mathcal{A}$ is said to be {\it strict} if $f(F^iA) = f(A) \cap F^iB$ for all $i \in \mathbf{Z}$.


homology-definition-spectral-sequence

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item A {\it spectral sequence in $\mathcal{A}$} is given by a system $(E_r, d_r)_{r \geq 1}$ where each $E_r$ is an object of $\mathcal{A}$, each $d_r : E_r \to E_r$ is a morphism such that $d_r \circ d_r = 0$ and $E_{r + 1} = \Ker(d_r)/\Im(d_r)$ for $r \geq 1$. \item A {\it morphism of spectral sequences} $f : (E_r, d_r)_{r \geq 1} \to (E'_r, d'_r)_{r \geq 1}$ is given by a family of morphisms $f_r : E_r \to E'_r$ such that $f_r \circ d_r = d'_r \circ f_r$ and such that $f_{r + 1}$ is the morphism induced by $f_r$ via the identifications $E_{r + 1} = \Ker(d_r)/\Im(d_r)$ and $E'_{r + 1} = \Ker(d'_r)/\Im(d'_r)$. \end{enumerate}


homology-definition-limit-spectral-sequence

Let $\mathcal{A}$ be an abelian category. Let $(E_r, d_r)_{r \geq 1}$ be a spectral sequence. \begin{enumerate} \item If the subobjects $Z_{\infty} = \bigcap Z_r$ and $B_{\infty} = \bigcup B_r$ of $E_1$ exist then we define the {\it limit}\footnote{This notation is not universally accepted. In some references an additional pair of subobjects $Z_\infty$ and $B_\infty$ of $E_1$ such that $0 = B_1 \subset B_2 \subset \ldots \subset B_\infty \subset Z_\infty \subset \ldots \subset Z_2 \subset Z_1 = E_1$ is part of the data comprising a spectral sequence!} of the spectral sequence to be the object $E_{\infty} = Z_{\infty}/B_{\infty}$. \item We say that the spectral sequence {\it degenerates at $E_r$} if the differentials $d_r, d_{r + 1}, \ldots$ are all zero. \end{enumerate}


homology-definition-exact-couple

Let $\mathcal{A}$ be an abelian category. \begin{enumerate} \item An {\it exact couple} is a datum $(A, E, \alpha, f, g)$ where $A$, $E$ are objects of $\mathcal{A}$ and $\alpha$, $f$, $g$ are morphisms as in the following diagram $$ \xymatrix{ A \ar[rr]_{\alpha} & & A \ar[ld]^g \\ & E \ar[lu]^f & } $$ with the property that the kernel of each arrow is the image of its predecessor. So $\Ker(\alpha) = \Im(f)$, $\Ker(f) = \Im(g)$, and $\Ker(g) = \Im(\alpha)$. \item A {\it morphism of exact couples} $t : (A, E, \alpha, f, g) \to (A', E', \alpha', f', g')$ is given by morphisms $t_A : A \to A'$ and $t_E : E \to E'$ such that $\alpha' \circ t_A = t_A \circ \alpha$, $f' \circ t_E = t_A \circ f$, and $g' \circ t_A = t_E \circ g$. \end{enumerate}


homology-definition-spectral-sequence-associated-exact-couple

Let $\mathcal{A}$ be an abelian category. Let $(A, E, \alpha, f, g)$ be an exact couple. The {\it spectral sequence associated to the exact couple} is the spectral sequence $(E_r, d_r)_{r \geq 1}$ with $E_1 = E$, $d_1 = d$, $E_2 = E'$, $d_2 = d' = g' \circ f'$, $E_3 = E''$, $d_3 = d'' = g'' \circ f''$, and so on.


homology-definition-differential-object

Let $\mathcal{A}$ be an abelian category. A {\it differential object} of $\mathcal{A}$ is a pair $(A, d)$ consisting of an object $A$ of $\mathcal{A}$ endowed with a selfmap $d$ such that $d \circ d = 0$. A {\it morphism of differential objects} $(A, d) \to (B, d)$ is given by a morphism $\alpha : A \to B$ such that $d \circ \alpha = \alpha \circ d$.


homology-definition-differential-object-homology

For a differential object $(A, d)$ we denote $$ H(A, d) = \Ker(d)/\Im(d) $$ its {\it homology}.


homology-definition-differential-object-selfmap

Let $\mathcal{A}$ be an abelian category. Let $(A, d)$ be a differential object of $\mathcal{A}$. Let $\alpha : A \to A$ be an injective selfmap of $A$ which commutes with $d$. The {\it spectral sequence associated to $(A, d, \alpha)$} is the spectral sequence $(E_r, d_r)_{r \geq 0}$ described above.


homology-definition-filtered-differential

Let $\mathcal{A}$ be an abelian category. A {\it filtered differential object} $(K, F, d)$ is a filtered object $(K, F)$ of $\mathcal{A}$ endowed with an endomorphism $d : (K, F) \to (K, F)$ whose square is zero: $d \circ d = 0$.


homology-definition-filtration-cohomology-filtered-differential

Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. The {\it induced filtration} on $H(K, d)$ is the filtration defined by $F^pH(K, d) = \Im(H(F^pK, d) \to H(K, d))$.


homology-definition-filtered-differential-ss-converges

Let $\mathcal{A}$ be an abelian category. Let $(K, F, d)$ be a filtered differential object of $\mathcal{A}$. We say the spectral sequence associated to $(K, F, d)$ \begin{enumerate} \item {\it weakly converges to $H(K)$} if $\text{gr}H(K) = E_{\infty}$ via Lemma \ref{lemma-compute-filtered-cohomology}, \item {\it abuts to $H(K)$} if it weakly converges to $H(K)$ and we have $\bigcap F^pH(K) = 0$ and $\bigcup F^pH(K) = H(K)$, \end{enumerate}


homology-definition-filtered-complex

Let $\mathcal{A}$ be an abelian category. A {\it filtered complex $K^\bullet$ of $\mathcal{A}$} is a complex of $\text{Fil}(\mathcal{A})$ (see Definition \ref{definition-filtered}).


homology-definition-filtration-cohomology-filtered-complex

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. The {\it induced filtration} on $H^n(K^\bullet)$ is the filtration defined by $F^pH^n(K^\bullet) = \Im(H^n(F^pK^\bullet) \to H^n(K^\bullet))$.


homology-definition-bounded-ss

Let $\mathcal{A}$ be an abelian category. Let $(E_r, d_r)_{r \geq r_0}$ be a spectral sequence of bigraded objects of $\mathcal{A}$ with $d_r$ of bidegree $(r, -r + 1)$. We say such a spectral sequence is \begin{enumerate} \item {\it regular} if for all $p, q \in \mathbf{Z}$ there is a $b = b(p, q)$ such that the maps $d_r^{p, q} : E_r^{p, q} \to E_r^{p + r, q - r + 1}$ are zero for $r \geq b$, \item {\it coregular} if for all $p, q \in \mathbf{Z}$ there is a $b = b(p, q)$ such that the maps $d_r^{p - r, q + r - 1} : E_r^{p - r, q + r - 1} \to E_r^{p, q}$ are zero for $r \geq b$, \item {\it bounded} if for all $n$ there are only a finite number of nonzero $E_{r_0}^{p, n - p}$, \item {\it bounded below} if for all $n$ there is a $b = b(n)$ such that $E_{r_0}^{p, n - p} = 0$ for $p \geq b$. \item {\it bounded above} if for all $n$ there is a $b = b(n)$ such that $E_{r_0}^{p, n - p} = 0$ for $p \leq b$. \end{enumerate}


homology-definition-filtered-complex-ss-converges

Let $\mathcal{A}$ be an abelian category. Let $(K^\bullet, F)$ be a filtered complex of $\mathcal{A}$. We say the spectral sequence associated to $(K^\bullet, F)$ \begin{enumerate} \item {\it weakly converges to $H^*(K^\bullet)$} if $\text{gr}^pH^n(K^\bullet) = E_{\infty}^{p, n - p}$ via Lemma \ref{lemma-compute-cohomology-filtered-complex} for all $p, n \in \mathbf{Z}$, \item {\it abuts to $H^*(K^\bullet)$} if it weakly converges to $H^*(K^\bullet)$ and $\bigcap_p F^pH^n(K^\bullet) = 0$ and $\bigcup_p F^p H^n(K^\bullet) = H^n(K^\bullet)$ for all $n$, \item {\it converges to $H^*(K^\bullet)$} if it is regular, abuts to $H^*(K^\bullet)$, and $H^n(K^\bullet) = \lim_p H^n(K^\bullet)/F^pH^n(K^\bullet)$. \end{enumerate}


homology-definition-ss-double-complex-converge

Let $\mathcal{A}$ be an abelian category. Let $K^{\bullet, \bullet}$ be a double complex. We say the spectral sequence $({}'E_r, {}'d_r)_{r \geq 0}$ {\it weakly converges to $H^n(\text{Tot}(K^{\bullet, \bullet}))$}, {\it abuts to $H^n(\text{Tot}(K^{\bullet, \bullet}))$}, or {\it converges to $H^n(\text{Tot}(K^{\bullet, \bullet}))$} if Definition \ref{definition-filtered-complex-ss-converges} applies. Similarly we say the spectral sequence $({}''E_r, {}''d_r)_{r \geq 0}$ {\it weakly converges to $H^n(\text{Tot}(K^{\bullet, \bullet}))$}, {\it abuts to $H^n(\text{Tot}(K^{\bullet, \bullet}))$}, or {\it converges to $H^n(\text{Tot}(K^{\bullet, \bullet}))$} if Definition \ref{definition-filtered-complex-ss-converges} applies.


homology-definition-injective

Let $\mathcal{A}$ be an abelian category. An object $J \in \Ob(\mathcal{A})$ is called {\it injective} if for every injection $A \hookrightarrow B$ and every morphism $A \to J$ there exists a morphism $B \to J$ making the following diagram commute $$ \xymatrix{ A \ar[r] \ar[d] & B \ar@{-->}[ld] \\ J & } $$


homology-definition-enough-injectives

Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has {\it enough injectives} if every object $A$ has an injective morphism $A \to J$ into an injective object $J$.


homology-definition-functorial-injective-embedding

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has {\it functorial injective embeddings} if there exists a functor $$ J : \mathcal{A} \longrightarrow \text{Arrows}(\mathcal{A}) $$ such that \begin{enumerate} \item $s \circ J = \text{id}_\mathcal{A}$, \item for any object $A \in \Ob(\mathcal{A})$ the morphism $J(A)$ is injective, and \item for any object $A \in \Ob(\mathcal{A})$ the object $t(J(A))$ is an injective object of $\mathcal{A}$. \end{enumerate} We will denote such a functor by $A \mapsto (A \to J(A))$.


homology-definition-projective

Let $\mathcal{A}$ be an abelian category. An object $P \in \Ob(\mathcal{A})$ is called {\it projective} if for every surjection $A \rightarrow B$ and every morphism $P \to B$ there exists a morphism $P \to A$ making the following diagram commute $$ \xymatrix{ A \ar[r] & B \\ P \ar@{-->}[u] \ar[ru] & } $$


homology-definition-enough-projectives

Let $\mathcal{A}$ be an abelian category. We say $\mathcal{A}$ has {\it enough projectives} if every object $A$ has an surjective morphism $P \to A$ from an projective object $P$ onto it.


homology-definition-functorial-projective-surjections

Let $\mathcal{A}$ be an abelian category. We say that $\mathcal{A}$ has {\it functorial projective surjections} if there exists a functor $$ P : \mathcal{A} \longrightarrow \text{Arrows}(\mathcal{A}) $$ such that \begin{enumerate} \item $t \circ J = \text{id}_\mathcal{A}$, \item for any object $A \in \Ob(\mathcal{A})$ the morphism $P(A)$ is surjective, and \item for any object $A \in \Ob(\mathcal{A})$ the object $s(P(A))$ is an projective object of $\mathcal{A}$. \end{enumerate} We will denote such a functor by $A \mapsto (P(A) \to A)$.


homology-definition-Mittag-Leffler

Let $\mathcal{C}$ be an abelian category. We say the inverse system $(A_i)$ satisfies the {\it Mittag-Leffler condition}, or for short is {\it ML}, if for every $i$ there exists a $c = c(i) \geq i$ such that $$ \Im(A_k \to A_i) = \Im(A_c \to A_i) $$ for all $k \geq c$.


categories-definition-category

A {\it category} $\mathcal{C}$ consists of the following data: \begin{enumerate} \item A set of objects $\Ob(\mathcal{C})$. \item For each pair $x, y \in \Ob(\mathcal{C})$ a set of morphisms $\Mor_\mathcal{C}(x, y)$. \item For each triple $x, y, z\in \Ob(\mathcal{C})$ a composition map $ \Mor_\mathcal{C}(y, z) \times \Mor_\mathcal{C}(x, y) \to \Mor_\mathcal{C}(x, z) $, denoted $(\phi, \psi) \mapsto \phi \circ \psi$. \end{enumerate} These data are to satisfy the following rules: \begin{enumerate} \item For every element $x\in \Ob(\mathcal{C})$ there exists a morphism $\text{id}_x\in \Mor_\mathcal{C}(x, x)$ such that $\text{id}_x \circ \phi = \phi$ and $\psi \circ \text{id}_x = \psi $ whenever these compositions make sense. \item Composition is associative, i.e., $(\phi \circ \psi) \circ \chi = \phi \circ ( \psi \circ \chi)$ whenever these compositions make sense. \end{enumerate}


categories-definition-isomorphism

A morphism $\phi : x \to y$ is an {\it isomorphism} of the category $\mathcal{C}$ if there exists a morphism $\psi : y \to x$ such that $\phi \circ \psi = \text{id}_y$ and $\psi \circ \phi = \text{id}_x$.


categories-definition-groupoid

A {\it groupoid} is a category where every morphism is an isomorphism.


categories-definition-functor

A {\it functor} $F : \mathcal{A} \to \mathcal{B}$ between two categories $\mathcal{A}, \mathcal{B}$ is given by the following data: \begin{enumerate} \item A map $F : \Ob(\mathcal{A}) \to \Ob(\mathcal{B})$. \item For every $x, y \in \Ob(\mathcal{A})$ a map $F : \Mor_\mathcal{A}(x, y) \to \Mor_\mathcal{B}(F(x), F(y))$, denoted $\phi \mapsto F(\phi)$. \end{enumerate} These data should be compatible with composition and identity morphisms in the following manner: $F(\phi \circ \psi) = F(\phi) \circ F(\psi)$ for a composable pair $(\phi, \psi)$ of morphisms of $\mathcal{A}$ and $F(\text{id}_x) = \text{id}_{F(x)}$.


categories-definition-faithful

Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. \begin{enumerate} \item We say $F$ is {\it faithful} if for any objects $x, y$ of $\Ob(\mathcal{A})$ the map $$ F : \Mor_\mathcal{A}(x, y) \to \Mor_\mathcal{B}(F(x), F(y)) $$ is injective. \item If these maps are all bijective then $F$ is called {\it fully faithful}. \item The functor $F$ is called {\it essentially surjective} if for any object $y \in \Ob(\mathcal{B})$ there exists an object $x \in \Ob(\mathcal{A})$ such that $F(x)$ is isomorphic to $y$ in $\mathcal{B}$. \end{enumerate}


categories-definition-subcategory

A {\it subcategory} of a category $\mathcal{B}$ is a category $\mathcal{A}$ whose objects and arrows form subsets of the objects and arrows of $\mathcal{B}$ and such that source, target and composition in $\mathcal{A}$ agree with those of $\mathcal{B}$. We say $\mathcal{A}$ is a {\it full subcategory} of $\mathcal{B}$ if $\Mor_\mathcal{A}(x, y) = \Mor_\mathcal{B}(x, y)$ for all $x, y \in \Ob(\mathcal{A})$. We say $\mathcal{A}$ is a {\it strictly full} subcategory of $\mathcal{B}$ if it is a full subcategory and given $x \in \Ob(\mathcal{A})$ any object of $\mathcal{B}$ which is isomorphic to $x$ is also in $\mathcal{A}$.


categories-definition-transformation-functors

Let $F, G : \mathcal{A} \to \mathcal{B}$ be functors. A {\it natural transformation}, or a {\it morphism of functors} $t : F \to G$, is a collection $\{t_x\}_{x\in \Ob(\mathcal{A})}$ such that \begin{enumerate} \item $t_x : F(x) \to G(x)$ is a morphism in the category $\mathcal{B}$, and \item for every morphism $\phi : x \to y$ of $\mathcal{A}$ the following diagram is commutative $$ \xymatrix{ F(x) \ar[r]^{t_x} \ar[d]_{F(\phi)} & G(x) \ar[d]^{G(\phi)} \\ F(y) \ar[r]^{t_y} & G(y) } $$ \end{enumerate}


categories-definition-equivalence-categories

An {\it equivalence of categories} $F : \mathcal{A} \to \mathcal{B}$ is a functor such that there exists a functor $G : \mathcal{B} \to \mathcal{A}$ such that the compositions $F \circ G$ and $G \circ F$ are isomorphic to the identity functors $\text{id}_\mathcal{B}$, respectively $\text{id}_\mathcal{A}$. In this case we say that $G$ is a {\it quasi-inverse} to $F$.


categories-definition-product-category

Let $\mathcal{A}$, $\mathcal{B}$ be categories. We define the {\it product category} $\mathcal{A} \times \mathcal{B}$ to be the category with objects $\Ob(\mathcal{A} \times \mathcal{B}) = \Ob(\mathcal{A}) \times \Ob(\mathcal{B})$ and $$ \Mor_{\mathcal{A} \times \mathcal{B}}((x, y), (x', y')) := \Mor_\mathcal{A}(x, x')\times \Mor_\mathcal{B}(y, y'). $$ Composition is defined componentwise.


categories-definition-opposite

Given a category $\mathcal{C}$ the {\it opposite category} $\mathcal{C}^{opp}$ is the category with the same objects as $\mathcal{C}$ but all morphisms reversed.


categories-definition-contravariant

Let $\mathcal{C}$, $\mathcal{S}$ be categories. A {\it contravariant} functor $F$ from $\mathcal{C}$ to $\mathcal{S}$ is a functor $\mathcal{C}^{opp}\to \mathcal{S}$.


categories-definition-presheaf

Let $\mathcal{C}$ be a category. \begin{enumerate} \item A {\it presheaf of sets on $\mathcal{C}$} or simply a {\it presheaf} is a contravariant functor $F$ from $\mathcal{C}$ to $\textit{Sets}$. \item The category of presheaves is denoted $\textit{PSh}(\mathcal{C})$. \end{enumerate}


categories-definition-representable-functor

A contravariant functor $F : \mathcal{C}\to \textit{Sets}$ is said to be {\it representable} if it is isomorphic to the functor of points $h_U$ for some object $U$ of $\mathcal{C}$.


categories-definition-products

Let $x, y\in \Ob(\mathcal{C})$. A {\it product} of $x$ and $y$ is an object $x \times y \in \Ob(\mathcal{C})$ together with morphisms $p\in \Mor_{\mathcal C}(x \times y, x)$ and $q\in\Mor_{\mathcal C}(x \times y, y)$ such that the following universal property holds: for any $w\in \Ob(\mathcal{C})$ and morphisms $\alpha \in \Mor_{\mathcal C}(w, x)$ and $\beta \in \Mor_\mathcal{C}(w, y)$ there is a unique $\gamma\in \Mor_{\mathcal C}(w, x \times y)$ making the diagram $$ \xymatrix{ w \ar[rrrd]^\beta \ar@{-->}[rrd]_\gamma \ar[rrdd]_\alpha & & \\ & & x \times y \ar[d]_p \ar[r]_q & y \\ & & x & } $$ commute.


categories-definition-has-products-of-pairs

We say the category $\mathcal{C}$ {\it has products of pairs of objects} if a product $x \times y$ exists for any $x, y \in \Ob(\mathcal{C})$.


categories-definition-coproducts

Let $x, y \in \Ob(\mathcal{C})$. A {\it coproduct}, or {\it amalgamated sum} of $x$ and $y$ is an object $x \amalg y \in \Ob(\mathcal{C})$ together with morphisms $i \in \Mor_{\mathcal C}(x, x \amalg y)$ and $j \in \Mor_{\mathcal C}(y, x \amalg y)$ such that the following universal property holds: for any $w \in \Ob(\mathcal{C})$ and morphisms $\alpha \in \Mor_{\mathcal C}(x, w)$ and $\beta \in \Mor_\mathcal{C}(y, w)$ there is a unique $\gamma \in \Mor_{\mathcal C}(x \amalg y, w)$ making the diagram $$ \xymatrix{ & y \ar[d]^j \ar[rrdd]^\beta \\ x \ar[r]^i \ar[rrrd]_\alpha & x \amalg y \ar@{-->}[rrd]^\gamma \\ & & & w } $$ commute.


categories-definition-has-coproducts-of-pairs

We say the category $\mathcal{C}$ {\it has coproducts of pairs of objects} if a coproduct $x \amalg y$ exists for any $x, y \in \Ob(\mathcal{C})$.


categories-definition-fibre-products

Let $x, y, z\in \Ob(\mathcal{C})$, $f\in \Mor_\mathcal{C}(x, y)$ and $g\in \Mor_{\mathcal C}(z, y)$. A {\it fibre product} of $f$ and $g$ is an object $x \times_y z\in \Ob(\mathcal{C})$ together with morphisms $p \in \Mor_{\mathcal C}(x \times_y z, x)$ and $q \in \Mor_{\mathcal C}(x \times_y z, z)$ making the diagram $$ \xymatrix{ x \times_y z \ar[r]_q \ar[d]_p & z \ar[d]^g \\ x \ar[r]^f & y } $$ commute, and such that the following universal property holds: for any $w\in \Ob(\mathcal{C})$ and morphisms $\alpha \in \Mor_{\mathcal C}(w, x)$ and $\beta \in \Mor_\mathcal{C}(w, z)$ with $f \circ \alpha = g \circ \beta$ there is a unique $\gamma \in \Mor_{\mathcal C}(w, x \times_y z)$ making the diagram $$ \xymatrix{ w \ar[rrrd]^\beta \ar@{-->}[rrd]_\gamma \ar[rrdd]_\alpha & & \\ & & x \times_y z \ar[d]^p \ar[r]_q & z \ar[d]^g \\ & & x \ar[r]^f & y } $$ commute.


categories-definition-cartesian

We say a commutative diagram $$ \xymatrix{ w \ar[r] \ar[d] & z \ar[d] \\ x \ar[r] & y } $$ in a category is {\it cartesian} if $w$ and the morphisms $w \to x$ and $w \to z$ form a fibre product of the morphisms $x \to y$ and $z \to y$.


categories-definition-has-fibre-products

We say the category $\mathcal{C}$ {\it has fibre products} if the fibre product exists for any $f\in \Mor_{\mathcal C}(x, y)$ and $g\in \Mor_{\mathcal C}(z, y)$.


categories-definition-representable-morphism

A morphism $f : x \to y$ of a category $\mathcal{C}$ is said to be {\it representable} if for every morphism $z \to y$ in $\mathcal{C}$ the fibre product $x \times_y z$ exists.


categories-definition-representable-map-presheaves

Let $\mathcal{C}$ be a category. Let $F, G : \mathcal{C}^{opp} \to \textit{Sets}$ be functors. We say a morphism $a : F \to G$ is {\it representable}, or that {\it $F$ is relatively representable over $G$}, if for every $U \in \Ob(\mathcal{C})$ and any $\xi \in G(U)$ the functor $h_U \times_G F$ is representable.


categories-definition-pushouts

Let $x, y, z\in \Ob(\mathcal{C})$, $f\in \Mor_\mathcal{C}(y, x)$ and $g\in \Mor_{\mathcal C}(y, z)$. A {\it pushout} of $f$ and $g$ is an object $x\amalg_y z\in \Ob(\mathcal{C})$ together with morphisms $p\in \Mor_{\mathcal C}(x, x\amalg_y z)$ and $q\in\Mor_{\mathcal C}(z, x\amalg_y z)$ making the diagram $$ \xymatrix{ y \ar[r]_g \ar[d]_f & z \ar[d]^q \\ x \ar[r]^p & x\amalg_y z } $$ commute, and such that the following universal property holds: For any $w\in \Ob(\mathcal{C})$ and morphisms $\alpha \in \Mor_{\mathcal C}(x, w)$ and $\beta \in \Mor_\mathcal{C}(z, w)$ with $\alpha \circ f = \beta \circ g$ there is a unique $\gamma\in \Mor_{\mathcal C}(x\amalg_y z, w)$ making the diagram $$ \xymatrix{ y \ar[r]_g \ar[d]_f & z \ar[d]^q \ar[rrdd]^\beta & & \\ x \ar[r]^p \ar[rrrd]^\alpha & x \amalg_y z  \ar@{-->}[rrd]^\gamma & & \\ & & & w } $$ commute.


categories-definition-cocartesian

We say a commutative diagram $$ \xymatrix{ y \ar[r] \ar[d] & z \ar[d] \\ x \ar[r] & w } $$ in a category is {\it cocartesian} if $w$ and the morphisms $x \to w$ and $z \to w$ form a pushout of the morphisms $y \to x$ and $y \to z$.


categories-definition-equalizers

Suppose that $X$, $Y$ are objects of a category $\mathcal{C}$ and that $a, b : X \to Y$ are morphisms. We say a morphism $e : Z \to X$ is an {\it equalizer} for the pair $(a, b)$ if $a \circ e = b \circ e$ and if $(Z, e)$ satisfies the following universal property: For every morphism $t : W \to X$ in $\mathcal{C}$ such that $a \circ t = b \circ t$ there exists a unique morphism $s : W \to Z$ such that $t = e \circ s$.


categories-definition-coequalizers

Suppose that $X$, $Y$ are objects of a category $\mathcal{C}$ and that $a, b : X \to Y$ are morphisms. We say a morphism $c : Y \to Z$ is a {\it coequalizer} for the pair $(a, b)$ if $c \circ a = c \circ b$ and if $(Z, c)$ satisfies the following universal property: For every morphism $t : Y \to W$ in $\mathcal{C}$ such that $t \circ a = t \circ b$ there exists a unique morphism $s : Z \to W$ such that $t = s \circ c$.


categories-definition-initial-final

Let $\mathcal{C}$ be a category. \begin{enumerate} \item An object $x$ of the category $\mathcal{C}$ is called an {\it initial} object if for every object $y$ of $\mathcal{C}$ there is exactly one morphism $x \to y$. \item An object $x$ of the category $\mathcal{C}$ is called a {\it final} object if for every object $y$ of $\mathcal{C}$ there is exactly one morphism $y \to x$. \end{enumerate}


categories-definition-mono-epi

Let $\mathcal{C}$ be a category and let $f : X \to Y$ be a morphism of $\mathcal{C}$. \begin{enumerate} \item We say that $f$ is a {\it monomorphism} if for every object $W$ and every pair of morphisms $a, b : W \to X$ such that $f \circ a = f \circ b$ we have $a = b$. \item We say that $f$ is an {\it epimorphism} if for every object $W$ and every pair of morphisms $a, b : Y \to W$ such that $a \circ f = b \circ f$ we have $a = b$. \end{enumerate}


categories-definition-limit

A {\it limit} of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\lim_\mathcal{I} M$ in $\mathcal{C}$ together with morphisms $p_i : \lim_\mathcal{I} M \to M_i$ such that \begin{enumerate} \item for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $p_{i'} =  M(\phi) \circ p_i$, and \item for any object $W$ in $\mathcal{C}$ and any family of morphisms $q_i : W \to M_i$ (indexed by $i \in \mathcal{I}$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $q_{i'} = M(\phi) \circ q_i$ there exists a unique morphism $q : W \to \lim_\mathcal{I} M$ such that $q_i = p_i \circ q$ for every object $i$ of $\mathcal{I}$. \end{enumerate}


categories-definition-colimit

A {\it colimit} of the $\mathcal{I}$-diagram $M$ in the category $\mathcal{C}$ is given by an object $\colim_\mathcal{I} M$ in $\mathcal{C}$ together with morphisms $s_i : M_i \to \colim_\mathcal{I} M$ such that \begin{enumerate} \item for $\phi : i \to i'$ a morphism in $\mathcal{I}$ we have $s_i = s_{i'} \circ M(\phi)$, and \item for any object $W$ in $\mathcal{C}$ and any family of morphisms $t_i : M_i \to W$ (indexed by $i \in \mathcal{I}$) such that for all $\phi : i \to i'$ in $\mathcal{I}$ we have $t_i = t_{i'} \circ M(\phi)$ there exists a unique morphism $t : \colim_\mathcal{I} M \to W$ such that $t_i = t \circ s_i$ for every object $i$ of $\mathcal{I}$. \end{enumerate}


categories-definition-product

Suppose that $I$ is a set, and suppose given for every $i \in I$ an object $M_i$ of the category $\mathcal{C}$. A {\it product} $\prod_{i\in I} M_i$ is by definition $\lim_\mathcal{I} M$ (if it exists) where $\mathcal{I}$ is the category having only identities as morphisms and having the elements of $I$ as objects.


categories-definition-coproduct

Suppose that $I$ is a set, and suppose given for every $i \in I$ an object $M_i$ of the category $\mathcal{C}$. A {\it coproduct} $\coprod_{i\in I} M_i$ is by definition $\colim_\mathcal{I} M$ (if it exists) where $\mathcal{I}$ is the category having only identities as morphisms and having the elements of $I$ as objects.


categories-definition-category-connected

We say that a category $\mathcal{I}$ is {\it connected} if the equivalence relation generated by $x \sim y \Leftrightarrow \Mor_\mathcal{I}(x, y) \not = \emptyset$ has exactly one equivalence class.


categories-definition-cofinal

Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say {\it $\mathcal{I}$ is cofinal in $\mathcal{J}$} or that $H$ is {\it cofinal} if \begin{enumerate} \item for all $y \in \Ob(\mathcal{J})$ there exists a $x \in \Ob(\mathcal{I})$ and a morphism $y \to H(x)$, and \item given $y \in \Ob(\mathcal{J})$, $x, x' \in \Ob(\mathcal{I})$ and morphisms $y \to H(x)$ and $y \to H(x')$ there exists a sequence of morphisms $$ x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x' $$ in $\mathcal{I}$ and morphisms $y \to H(x_i)$ in $\mathcal{J}$ such that the diagrams $$ \xymatrix{ & y \ar[ld] \ar[d] \ar[rd] \\ H(x_{2k}) & H(x_{2k + 1}) \ar[l] \ar[r] & H(x_{2k + 2}) } $$ commute for $k = 0, \ldots, n - 1$. \end{enumerate}


categories-definition-initial

Let $H : \mathcal{I} \to \mathcal{J}$ be a functor between categories. We say {\it $\mathcal{I}$ is initial in $\mathcal{J}$} or that $H$ is {\it initial} if \begin{enumerate} \item for all $y \in \Ob(\mathcal{J})$ there exists a $x \in \Ob(\mathcal{I})$ and a morphism $H(x) \to y$, \item for any $y \in \Ob(\mathcal{J})$, $x , x' \in \Ob(\mathcal{I})$ and morphisms $H(x) \to y$, $H(x') \to y$ in $\mathcal{J}$ there exists a sequence of morphisms $$ x = x_0 \leftarrow x_1 \rightarrow x_2 \leftarrow x_3 \rightarrow \ldots \rightarrow x_{2n} = x' $$ in $\mathcal{I}$ and morphisms $H(x_i) \to y$ in $\mathcal{J}$ such that the diagrams $$ \xymatrix{ H(x_{2k}) \ar[rd] & H(x_{2k + 1}) \ar[l] \ar[r] \ar[d] & H(x_{2k + 2}) \ar[ld] \\ & y } $$ commute for $k = 0, \ldots, n - 1$. \end{enumerate}


categories-definition-directed

We say that a diagram $M : \mathcal{I} \to \mathcal{C}$ is {\it directed}, or {\it filtered} if the following conditions hold: \begin{enumerate} \item the category $\mathcal{I}$ has at least one object, \item for every pair of objects $x, y$ of $\mathcal{I}$ there exists an object $z$ and morphisms $x \to z$, $y \to z$, and \item for every pair of objects $x, y$ of $\mathcal{I}$ and every pair of morphisms $a, b : x \to y$ of $\mathcal{I}$ there exists a morphism $c : y \to z$ of $\mathcal{I}$ such that $M(c \circ a) = M(c \circ b)$ as morphisms in $\mathcal{C}$. \end{enumerate} We say that an index category $\mathcal{I}$ is {\it directed}, or {\it filtered} if $\text{id} : \mathcal{I} \to \mathcal{I}$ is filtered (in other words you erase the $M$ in part (3) above).


categories-definition-codirected

We say that a diagram $M : \mathcal{I} \to \mathcal{C}$ is {\it codirected} or {\it cofiltered} if the following conditions hold: \begin{enumerate} \item the category $\mathcal{I}$ has at least one object, \item for every pair of objects $x, y$ of $\mathcal{I}$ there exists an object $z$ and morphisms $z \to x$, $z \to y$, and \item for every pair of objects $x, y$ of $\mathcal{I}$ and every pair of morphisms $a, b : x \to y$ of $\mathcal{I}$ there exists a morphism $c : w \to x$ of $\mathcal{I}$ such that $M(a \circ c) = M(b \circ c)$ as morphisms in $\mathcal{C}$. \end{enumerate} We say that an index category $\mathcal{I}$ is {\it codirected}, or {\it cofiltered} if $\text{id} : \mathcal{I} \to \mathcal{I}$ is cofiltered (in other words you erase the $M$ in part (3) above).


categories-definition-directed-set

Let $I$ be a set and let $\leq$ be a binary relation on $I$. \begin{enumerate} \item We say $\leq$ is a {\it preorder} if it is transitive (if $i \leq j$ and $j \leq k$ then $i \leq k$) and reflexive ($i \leq i$ for all $i \in I$). \item A {\it preordered set} is a set endowed with a preorder. \item A {\it directed set} is a preordered set $(I, \leq)$ such that $I$ is not empty and such that $\forall i, j \in I$, there exists $k \in I$ with $i \leq k, j \leq k$. \item We say $\leq$ is a {\it partial order} if it is a preorder which is antisymmetric (if $i \leq j$ and $j \leq i$, then $i = j$). \item A {\it partially ordered set} is a set endowed with a partial order. \item A {\it directed partially ordered set} is a directed set whose ordering is a partial order. \end{enumerate}


categories-definition-system-over-poset

Let $(I, \leq)$ be a preordered set. Let $\mathcal{C}$ be a category. \begin{enumerate} \item A {\it system over $I$ in $\mathcal{C}$}, sometimes called a {\it inductive system over $I$ in $\mathcal{C}$} is given by objects $M_i$ of $\mathcal{C}$ and for every $i \leq i'$ a morphism $f_{ii'} : M_i \to M_{i'}$ such that $f_{ii} = \text{id}$ and such that $f_{ii''} = f_{i'i''} \circ f_{i i'}$ whenever $i \leq i' \leq i''$. \item An {\it inverse system over $I$ in $\mathcal{C}$}, sometimes called a {\it projective system over $I$ in $\mathcal{C}$} is given by objects $M_i$ of $\mathcal{C}$ and for every $i' \leq i$ a morphism $f_{ii'} : M_i \to M_{i'}$ such that $f_{ii} = \text{id}$ and such that $f_{ii''} = f_{i'i''} \circ f_{i i'}$ whenever $i'' \leq i' \leq i$. (Note reversal of inequalities.) \end{enumerate} We will say $(M_i, f_{ii'})$ is a (inverse) system over $I$ to denote this. The maps $f_{ii'}$ are sometimes called the {\it transition maps}.


categories-definition-directed-system

Let $I$ be a preordered set. We say a system (resp.\ inverse system) $(M_i, f_{ii'})$ is a {\it directed system} (resp.\ {\it directed inverse system}) if $I$ is a directed set (Definition \ref{definition-directed-set}): $I$ is nonempty and for all $i_1, i_2 \in I$ there exists $i\in I$ such that $i_1 \leq i$ and $i_2 \leq i$.


categories-definition-essentially-constant-diagram

Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram in a category $\mathcal{C}$. \begin{enumerate} \item Assume the index category $\mathcal{I}$ is filtered and let $(X, \{M_i \to X\}_i)$ be a cocone for $M$, see Remark \ref{remark-cones-and-cocones}. We say $M$ is {\it essentially constant} with {\it value} $X$ if there exists an $i \in \mathcal{I}$ and a morphism $X \to M_i$ such that \begin{enumerate} \item $X \to M_i \to X$ is $\text{id}_X$, and \item for all $j$ there exist $k$ and morphisms $i \to k$ and $j \to k$ such that the morphism $M_j \to M_k$ equals the composition $M_j \to X \to M_i \to M_k$. \end{enumerate} \item Assume the index category $\mathcal{I}$ is cofiltered and let $(X, \{X \to M_i\}_i)$ be a cone for $M$, see Remark \ref{remark-cones-and-cocones}. We say $M$ is {\it essentially constant} with {\it value} $X$ if there exists an $i \in \mathcal{I}$ and a morphism $M_i \to X$ such that \begin{enumerate} \item $X \to M_i \to X$ is $\text{id}_X$, and \item for all $j$ there exist $k$ and morphisms $k \to i$ and $k \to j$ such that the morphism $M_k \to M_j$ equals the composition $M_k \to M_i \to X \to M_j$. \end{enumerate} \end{enumerate} Please keep in mind Lemma \ref{lemma-essentially-constant-is-limit-colimit} when using this definition.


categories-definition-essentially-constant-system

Let $\mathcal{C}$ be a category. A directed system $(M_i, f_{ii'})$ is an {\it essentially constant system} if $M$ viewed as a functor $I \to \mathcal{C}$ defines an essentially constant diagram. A directed inverse system $(M_i, f_{ii'})$ is an {\it essentially constant inverse system} if $M$ viewed as a functor $I^{opp} \to \mathcal{C}$ defines an essentially constant inverse diagram.


categories-definition-exact

Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. \begin{enumerate} \item Suppose all finite limits exist in $\mathcal{A}$. We say $F$ is {\it left exact} if it commutes with all finite limits. \item Suppose all finite colimits exist in $\mathcal{A}$. We say $F$ is {\it right exact} if it commutes with all finite colimits. \item We say $F$ is {\it exact} if it is both left and right exact. \end{enumerate}


categories-definition-adjoint

Let $\mathcal{C}$, $\mathcal{D}$ be categories. Let $u : \mathcal{C} \to \mathcal{D}$ and $v : \mathcal{D} \to \mathcal{C}$ be functors. We say that $u$ is a {\it left adjoint} of $v$, or that $v$ is a {\it right adjoint} to $u$ if there are bijections $$ \Mor_\mathcal{D}(u(X), Y) \longrightarrow \Mor_\mathcal{C}(X, v(Y)) $$ functorial in $X \in \Ob(\mathcal{C})$, and $Y \in \Ob(\mathcal{D})$.


categories-definition-compact-object

Let $\mathcal{C}$ be a big\footnote{See Remark \ref{remark-big-categories}.} category. An object $X$ of $\mathcal{C}$ is called a {\it categorically compact} if we have $$ \Mor_\mathcal{C}(X, \colim_i M_i) = \colim_i \Mor_\mathcal{C}(X, M_i) $$ for every filtered diagram $M : \mathcal{I} \to \mathcal{C}$ such that $\colim_i M_i$ exists.


categories-definition-multiplicative-system

Let $\mathcal{C}$ be a category. A set of arrows $S$ of $\mathcal{C}$ is called a {\it left multiplicative system} if it has the following properties: \begin{enumerate} \item[LMS1] The identity of every object of $\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$. \item[LMS2] Every solid diagram $$ \xymatrix{ X \ar[d]_t \ar[r]_g & Y \ar@{..>}[d]^s \\ Z \ar@{..>}[r]^f & W } $$ with $t \in S$ can be completed to a commutative dotted square with $s \in S$. \item[LMS3] For every pair of morphisms $f, g : X \to Y$ and $t \in S$ with target $X$ such that $f \circ t = g \circ t$ there exists a $s \in S$ with source $Y$ such that $s \circ f = s \circ g$. \end{enumerate} A set of arrows $S$ of $\mathcal{C}$ is called a {\it right multiplicative system} if it has the following properties: \begin{enumerate} \item[RMS1] The identity of every object of $\mathcal{C}$ is in $S$ and the composition of two composable elements of $S$ is in $S$. \item[RMS2] Every solid diagram $$ \xymatrix{ X \ar@{..>}[d]_t \ar@{..>}[r]_g & Y \ar[d]^s \\ Z \ar[r]^f & W } $$ with $s \in S$ can be completed to a commutative dotted square with $t \in S$. \item[RMS3] For every pair of morphisms $f, g : X \to Y$ and $s \in S$ with source $Y$ such that $s \circ f = s \circ g$ there exists a $t \in S$ with target $X$ such that $f \circ t = g \circ t$. \end{enumerate} A set of arrows $S$ of $\mathcal{C}$ is called a {\it multiplicative system} if it is both a left multiplicative system and a right multiplicative system. In other words, this means that MS1, MS2, MS3 hold, where MS1 $=$ LMS1 $+$ RMS1, MS2 $=$ LMS2 $+$ RMS2, and MS3 $=$ LMS3 $+$ RMS3. (That said, of course LMS1 $=$ RMS1 $=$ MS1.)


categories-definition-left-localization-as-fraction

Let $\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\mathcal{C}$. Given any morphism $f : X \to Y'$ in $\mathcal{C}$ and any morphism $s : Y \to Y'$ in $S$, we denote by {\it $s^{-1} f$} the equivalence class of the pair $(f : X \to Y', s : Y \to Y')$. This is a morphism from $X$ to $Y$ in $S^{-1} \mathcal{C}$.


categories-definition-right-localization-as-fraction

Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\mathcal{C}$. Given any morphism $f : X' \to Y$ in $\mathcal{C}$ and any morphism $s : X' \to X$ in $S$, we denote by {\it $f s^{-1}$} the equivalence class of the pair $(f : X' \to Y, s : X' \to X)$. This is a morphism from $X$ to $Y$ in $S^{-1} \mathcal{C}$.


categories-definition-saturated-multiplicative-system

Let $\mathcal{C}$ be a category and let $S$ be a multiplicative system. We say $S$ is {\it saturated} if, in addition to MS1, MS2, MS3, we also have \begin{enumerate} \item[MS4] Given three composable morphisms $f, g, h$, if $fg, gh \in S$, then $g \in S$. \end{enumerate}


categories-definition-horizontal-composition

Given a diagram as in the left hand side of: $$ \xymatrix{ \mathcal{A} \rtwocell^F_{F'}{t} & \mathcal{B} \rtwocell^G_{G'}{s} & \mathcal{C} } \text{ gives } \xymatrix{ \mathcal{A} \rrtwocell^{G \circ F} _{G' \circ F'}{\ \ s \star t} & & \mathcal{C} } $$ we define the {\it horizontal} composition $s \star t$ to be the transformation of functors ${}_{G'}t \circ s_F =  s_{F'}\circ {}_Gt$.


categories-definition-2-category

A (strict) {\it $2$-category} $\mathcal{C}$ consists of the following data \begin{enumerate} \item A set of objects $\Ob(\mathcal{C})$. \item For each pair $x, y \in \Ob(\mathcal{C})$ a category $\Mor_\mathcal{C}(x, y)$. The objects of $\Mor_\mathcal{C}(x, y)$ will be called {\it $1$-morphisms} and denoted $F : x \to y$. The morphisms between these $1$-morphisms will be called {\it $2$-morphisms} and denoted $t : F' \to F$. The composition of $2$-morphisms in $\Mor_\mathcal{C}(x, y)$ will be called {\it vertical} composition and will be denoted $t \circ t'$ for $t : F' \to F$ and $t' : F'' \to F'$. \item For each triple $x, y, z\in \Ob(\mathcal{C})$ a functor $$ (\circ, \star) : \Mor_\mathcal{C}(y, z) \times \Mor_\mathcal{C}(x, y) \longrightarrow \Mor_\mathcal{C}(x, z). $$ The image of the pair of $1$-morphisms $(F, G)$ on the left hand side will be called the {\it composition} of $F$ and $G$, and denoted $F\circ G$. The image of the pair of $2$-morphisms $(t, s)$ will be called the {\it horizontal} composition and denoted $t \star s$. \end{enumerate} These data are to satisfy the following rules: \begin{enumerate} \item The set of objects together with the set of $1$-morphisms endowed with composition of $1$-morphisms forms a category. \item Horizontal composition of $2$-morphisms is associative. \item The identity $2$-morphism $\text{id}_{\text{id}_x}$ of the identity $1$-morphism $\text{id}_x$ is a unit for horizontal composition. \end{enumerate}


categories-definition-sub-2-category

Let $\mathcal{C}$ be a $2$-category. A {\it sub $2$-category} $\mathcal{C}'$ of $\mathcal{C}$, is given by a subset $\Ob(\mathcal{C}')$ of $\Ob(\mathcal{C})$ and sub categories $\Mor_{\mathcal{C}'}(x, y)$ of the categories $\Mor_\mathcal{C}(x, y)$ for all $x, y \in \Ob(\mathcal{C}')$ such that these, together with the operations $\circ$ (composition $1$-morphisms), $\circ$ (vertical composition $2$-morphisms), and $\star$ (horizontal composition) form a $2$-category.


categories-definition-equivalence

Two objects $x, y$ of a $2$-category are {\it equivalent} if there exist $1$-morphisms $F : x \to y$ and $G : y \to x$ such that $F \circ G$ is $2$-isomorphic to $\text{id}_y$ and $G \circ F$ is $2$-isomorphic to $\text{id}_x$.


categories-definition-functor-into-2-category

Let $\mathcal{A}$ be a category and let $\mathcal{C}$ be a $2$-category. \begin{enumerate} \item A {\it functor} from an ordinary category into a $2$-category will ignore the $2$-morphisms unless mentioned otherwise. In other words, it will be a ``usual'' functor into the category formed out of 2-category by forgetting all the 2-morphisms. \item A {\it weak functor}, or a {\it pseudo functor} $\varphi$ from $\mathcal{A}$ into the 2-category $\mathcal{C}$ is given by the following data \begin{enumerate} \item a map $\varphi : \Ob(\mathcal{A}) \to \Ob(\mathcal{C})$, \item for every pair $x, y\in \Ob(\mathcal{A})$, and every morphism $f : x \to y$ a $1$-morphism $\varphi(f) : \varphi(x) \to \varphi(y)$, \item for every $x\in \Ob(A)$ a $2$-morphism $\alpha_x : \text{id}_{\varphi(x)} \to \varphi(\text{id}_x)$, and \item for every pair of composable morphisms $f : x \to y$, $g : y \to z$ of $\mathcal{A}$ a $2$-morphism $\alpha_{g, f} : \varphi(g \circ f) \to \varphi(g) \circ \varphi(f)$. \end{enumerate} These data are subject to the following conditions: \begin{enumerate} \item the $2$-morphisms $\alpha_x$ and $\alpha_{g, f}$ are all isomorphisms, \item for any morphism $f : x \to y$ in $\mathcal{A}$ we have $\alpha_{\text{id}_y, f} = \alpha_y \star \text{id}_{\varphi(f)}$: $$ \xymatrix{ \varphi(x) \rrtwocell^{\varphi(f)}_{\varphi(f)}{\ \ \ \ \text{id}_{\varphi(f)}} & & \varphi(y) \rrtwocell^{\text{id}_{\varphi(y)}}_{\varphi(\text{id}_y)}{\alpha_y} & & \varphi(y) } = \xymatrix{ \varphi(x) \rrtwocell^{\varphi(f)}_{\varphi(\text{id}_y) \circ \varphi(f)}{\ \ \ \ \alpha_{\text{id}_y, f}} & & \varphi(y) } $$ \item for any morphism $f : x \to y$ in $\mathcal{A}$ we have $\alpha_{f, \text{id}_x} = \text{id}_{\varphi(f)} \star \alpha_x$, \item for any triple of composable morphisms $f : w \to x$, $g : x \to y$, and $h : y \to z$ of $\mathcal{A}$ we have $$ (\text{id}_{\varphi(h)} \star \alpha_{g, f}) \circ \alpha_{h, g \circ f} = (\alpha_{h, g} \star \text{id}_{\varphi(f)}) \circ \alpha_{h \circ g, f} $$ in other words the following diagram with objects $1$-morphisms and arrows $2$-morphisms commutes $$ \xymatrix{ \varphi(h \circ g \circ f) \ar[d]_{\alpha_{h, g \circ f}} \ar[rr]_{\alpha_{h \circ g, f}} & & \varphi(h \circ g) \circ \varphi(f) \ar[d]^{\alpha_{h, g} \star \text{id}_{\varphi(f)}} \\ \varphi(h) \circ \varphi(g \circ f) \ar[rr]^{\text{id}_{\varphi(h)} \star \alpha_{g, f}} & & \varphi(h) \circ \varphi(g) \circ \varphi(f) } $$ \end{enumerate} \end{enumerate}


categories-definition-2-1-category

A (strict) {\it $(2, 1)$-category} is a $2$-category in which all $2$-morphisms are isomorphisms.


categories-definition-final-object-2-category

A {\it final object} of a $(2, 1)$-category $\mathcal{C}$ is an object $x$ such that \begin{enumerate} \item for every $y \in \Ob(\mathcal{C})$ there is a morphism $y \to x$, and \item every two morphisms $y \to x$ are isomorphic by a unique 2-morphism. \end{enumerate}


categories-definition-2-fibre-products

Let $\mathcal{C}$ be a $(2, 1)$-category. Let $x, y, z\in \Ob(\mathcal{C})$ and $f\in \Mor_\mathcal{C}(x, z)$ and $g\in \Mor_{\mathcal C}(y, z)$. A {\it 2-fibre product of $f$ and $g$} is a final object in the category of 2-commutative diagrams described above. If a 2-fibre product exists we will denote it $x \times_z y\in \Ob(\mathcal{C})$, and denote the required morphisms $p\in \Mor_{\mathcal C}(x \times_z y, x)$ and $q\in \Mor_{\mathcal C}(x \times_z y, y)$ making the diagram $$ \xymatrix{ & x \times_z y \ar[r]^{p} \ar[d]_q & x \ar[d]^{f} \\ & y \ar[r]^{g} & z } $$ 2-commute and we will denote the given invertible 2-morphism exhibiting this by $\psi : f \circ p \to g \circ q$.


categories-definition-categories-over-C

Let $\mathcal{C}$ be a category. The {\it $2$-category of categories over $\mathcal{C}$} is the $2$-category defined as follows: \begin{enumerate} \item Its objects will be functors $p : \mathcal{S} \to \mathcal{C}$. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$. \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate} In this situation we will denote $$ \Mor_{\textit{Cat}/\mathcal{C}}(\mathcal{S}, \mathcal{S}') $$ the category of $1$-morphisms between $(\mathcal{S}, p)$ and $(\mathcal{S}', p')$


categories-definition-fibre-category

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. \begin{enumerate} \item The {\it fibre category} over an object $U\in \Ob(\mathcal{C})$ is the category $\mathcal{S}_U$ with objects $$ \Ob(\mathcal{S}_U) = \{x\in \Ob(\mathcal{S}) : p(x) = U\} $$ and morphisms $$ \Mor_{\mathcal{S}_U}(x, y) = \{ \phi \in \Mor_\mathcal{S}(x, y) : p(\phi) = \text{id}_U\}. $$ \item A {\it lift} of an object $U \in \Ob(\mathcal{C})$ is an object $x\in \Ob(\mathcal{S})$ such that $p(x) = U$, i.e., $x\in \Ob(\mathcal{S}_U)$. We will also sometime say that {\it $x$ lies over $U$}. \item Similarly, a {\it lift} of a morphism $f : V \to U$ in $\mathcal{C}$ is a morphism $\phi : y \to x$ in $\mathcal{S}$ such that $p(\phi) = f$. We sometimes say that {\it $\phi$ lies over $f$}. \end{enumerate}


categories-definition-cartesian-over-C

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. A {\it strongly cartesian morphism}, or more precisely a {\it strongly $\mathcal{C}$-cartesian morphism} is a morphism $\varphi : y \to x$ of $\mathcal{S}$ such that for every $z \in \Ob(\mathcal{S})$ the map $$ \Mor_\mathcal{S}(z, y) \longrightarrow \Mor_\mathcal{S}(z, x) \times_{\Mor_\mathcal{C}(p(z), p(x))} \Mor_\mathcal{C}(p(z), p(y)), $$ given by $\psi \longmapsto (\varphi \circ \psi, p(\psi))$ is bijective.


categories-definition-fibred-category

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. We say $\mathcal{S}$ is a {\it fibred category over $\mathcal{C}$} if given any $x \in \Ob(\mathcal{S})$ lying over $U \in \Ob(\mathcal{C})$ and any morphism $f : V \to U$ of $\mathcal{C}$, there exists a strongly cartesian morphism $f^*x \to x$ lying over $f$.


categories-definition-pullback-functor-fibred-category

Assume $p : \mathcal{S} \to \mathcal{C}$ is a fibred category. \begin{enumerate} \item A {\it choice of pullbacks}\footnote{This is probably nonstandard terminology. In some texts this is called a ``cleavage''  but it conjures up the wrong image. Maybe a ``cleaving'' would be a better word. A related notion is that of a ``splitting'', but in many texts a ``splitting'' means a choice of pullbacks such that $g^*f^* = (f \circ g)^*$ for any composable pair of morphisms. Compare also with Definition \ref{definition-split-fibred-category}.} for $p : \mathcal{S} \to \mathcal{C}$ is given by a choice of a strongly cartesian morphism $f^\ast x \to x$ lying over $f$ for any morphism $f: V \to U$ of $\mathcal{C}$ and any $x \in \Ob(\mathcal{S}_U)$. \item Given a choice of pullbacks, for any morphism $f : V \to U$ of $\mathcal{C}$ the functor $f^* : \mathcal{S}_U \to \mathcal{S}_V$ described above is called a {\it pullback functor} (associated to the choices $f^*x \to x$ made above). \end{enumerate}


categories-definition-fibred-categories-over-C

Let $\mathcal{C}$ be a category. The {\it $2$-category of fibred categories over $\mathcal{C}$} is the sub $2$-category of the $2$-category of categories over $\mathcal{C}$ (see Definition \ref{definition-categories-over-C}) defined as follows: \begin{enumerate} \item Its objects will be fibred categories $p : \mathcal{S} \to \mathcal{C}$. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ and such that $G$ maps strongly cartesian morphisms to strongly cartesian morphisms. \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate} In this situation we will denote $$ \Mor_{\textit{Fib}/\mathcal{C}}(\mathcal{S}, \mathcal{S}') $$ the category of $1$-morphisms between $(\mathcal{S}, p)$ and $(\mathcal{S}', p')$


categories-definition-inertia-fibred-category

Let $\mathcal{C}$ be a category. \begin{enumerate} \item Let $F : \mathcal{S} \to \mathcal{S}'$ be a $1$-morphism of fibred categories over $\mathcal{C}$. The {\it relative inertia of $\mathcal{S}$ over $\mathcal{S}'$} is the fibred category $\mathcal{I}_{\mathcal{S}/\mathcal{S}'} \to \mathcal{C}$ of Lemma \ref{lemma-inertia-fibred-category}. \item By the {\it inertia fibred category $\mathcal{I}_\mathcal{S}$ of $\mathcal{S}$} we mean $\mathcal{I}_\mathcal{S} = \mathcal{I}_{\mathcal{S}/\mathcal{C}}$. \end{enumerate}


categories-definition-fibred-groupoids

Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. We say that $\mathcal{S}$ is {\it fibred in groupoids} over $\mathcal{C}$ if the following two conditions hold: \begin{enumerate} \item For every morphism $f : V \to U$ in $\mathcal{C}$ and every lift $x$ of $U$ there is a lift $\phi : y \to x$ of $f$ with target $x$. \item For every pair of morphisms $\phi : y \to x$ and $ \psi : z \to x$ and any morphism $f : p(z) \to p(y)$ such that $p(\phi) \circ f = p(\psi)$ there exists a unique lift $\chi : z \to y$ of $f$ such that $\phi \circ \chi = \psi$. \end{enumerate}


categories-definition-categories-fibred-in-groupoids-over-C

Let $\mathcal{C}$ be a category. The {\it $2$-category of categories fibred in groupoids over $\mathcal{C}$} is the sub $2$-category of the $2$-category of fibred categories over $\mathcal{C}$ (see Definition \ref{definition-fibred-categories-over-C}) defined as follows: \begin{enumerate} \item Its objects will be categories $p : \mathcal{S} \to \mathcal{C}$ fibred in groupoids. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them). \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate}


categories-definition-split-fibred-category

Let $\mathcal{C}$ be a category. Suppose that $F : \mathcal{C}^{opp} \to \textit{Cat}$ is a functor to the $2$-category of categories. We will write $p_F : \mathcal{S}_F \to \mathcal{C}$ for the fibred category constructed in Example \ref{example-functor-categories}. A {\it split fibred category} is a fibred category isomorphic (!) over $\mathcal{C}$ to one of these categories {\it $\mathcal{S}_F$}.


categories-definition-split-category-fibred-in-groupoids

Let $\mathcal{C}$ be a category. Suppose that $F : \mathcal{C}^{opp} \to \textit{Groupoids}$ is a functor to the $2$-category of groupoids. We will write $p_F : \mathcal{S}_F \to \mathcal{C}$ for the category fibred in groupoids constructed in Example \ref{example-functor-groupoids}. A {\it split category fibred in groupoids} is a category fibred in groupoids isomorphic (!) over $\mathcal{C}$ to one of these categories {\it $\mathcal{S}_F$}.


categories-definition-discrete

A category is called {\it discrete} if the only morphisms are the identity morphisms.


categories-definition-category-fibred-sets

Let $\mathcal{C}$ be a category. A {\it category fibred in sets}, or a {\it category fibred in discrete categories} is a category fibred in groupoids all of whose fibre categories are discrete.


categories-definition-categories-fibred-in-sets-over-C

Let $\mathcal{C}$ be a category. The {\it $2$-category of categories fibred in sets over $\mathcal{C}$} is the sub $2$-category of the category of categories fibred in groupoids over $\mathcal{C}$ (see Definition \ref{definition-categories-fibred-in-groupoids-over-C}) defined as follows: \begin{enumerate} \item Its objects will be categories $p : \mathcal{S} \to \mathcal{C}$ fibred in sets. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them). \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate}


categories-definition-setoid

Let us call a category a {\it setoid}\footnote{A set on steroids!?} if it is a groupoid where every object has exactly one automorphism: the identity.


categories-definition-category-fibred-setoids

Let $\mathcal{C}$ be a category. A {\it category fibred in setoids} is a category fibred in groupoids all of whose fibre categories are setoids.


categories-definition-categories-fibred-in-setoids-over-C

Let $\mathcal{C}$ be a category. The {\it $2$-category of categories fibred in setoids over $\mathcal{C}$} is the sub $2$-category of the category of categories fibred in groupoids over $\mathcal{C}$ (see Definition \ref{definition-categories-fibred-in-groupoids-over-C}) defined as follows: \begin{enumerate} \item Its objects will be categories $p : \mathcal{S} \to \mathcal{C}$ fibred in setoids. \item Its $1$-morphisms $(\mathcal{S}, p) \to (\mathcal{S}', p')$ will be functors $G : \mathcal{S} \to \mathcal{S}'$ such that $p' \circ G = p$ (since every morphism is strongly cartesian $G$ automatically preserves them). \item Its $2$-morphisms $t : G \to H$ for $G, H : (\mathcal{S}, p) \to (\mathcal{S}', p')$ will be morphisms of functors such that $p'(t_x) = \text{id}_{p(x)}$ for all $x \in \Ob(\mathcal{S})$. \end{enumerate}


categories-definition-representable-fibred-category

Let $\mathcal{C}$ be a category. A category fibred in groupoids $p : \mathcal{S} \to \mathcal{C}$ is called {\it representable} if there exists an object $X$ of $\mathcal{C}$ and an equivalence $j : \mathcal{S} \to \mathcal{C}/X$ (in the $2$-category of groupoids over $\mathcal{C}$).


categories-definition-representable-map-categories-fibred-in-groupoids

Let $\mathcal{C}$ be a category. Let $\mathcal{X}$, $\mathcal{Y}$ be categories fibred in groupoids over $\mathcal{C}$. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism. We say $F$ is {\it representable}, or that {\it $\mathcal{X}$ is relatively representable over $\mathcal{Y}$}, if for every $U \in \Ob(\mathcal{C})$ and any $G : \mathcal{C}/U \to \mathcal{Y}$ the category fibred in groupoids $$ (\mathcal{C}/U) \times_\mathcal{Y} \mathcal{X} \longrightarrow \mathcal{C}/U $$ is representable.


categories-definition-monoidal-category

A triple $(\mathcal{C}, \otimes, \phi)$ where $\mathcal{C}$ is a category, $\otimes : \mathcal{C} \times \mathcal{C} \to \mathcal{C}$ is a functor, and $\phi$ is an associativity constraint is called a {\it monoidal category} if there exists a unit $\mathbf{1}$.


categories-definition-functor-monoidal-categories

Let $\mathcal{C}$ and $\mathcal{C}'$ be monoidal categories. A {\it functor of monoidal categories} $F : \mathcal{C} \to \mathcal{C}'$ is given by a functor $F$ as indicated and a natural transformation $$ F(X) \otimes F(Y) \to F(X \otimes Y) $$ such that for all objects $X, Y, Z$ the diagram $$ \xymatrix{ F(X) \otimes (F(Y) \otimes F(Z)) \ar[r] \ar[d] & F(X) \otimes F(Y \otimes Z) \ar[r] & F(X \otimes (Y \otimes Z)) \ar[d] \\ (F(X) \otimes F(Y)) \otimes F(Z) \ar[r] & F(X \otimes Y) \otimes F(Z) \ar[r] & F((X \otimes Y) \otimes Z) } $$ commutes and such that $F(\mathbf{1})$ is a unit in $\mathcal{C}'$.


categories-definition-invertible

Let $\mathcal{C}$ be a monoidal category. An object $X$ of $\mathcal{C}$ is called {\it invertible} if any (or all) of the equivalent conditions of Lemma \ref{lemma-invertible} hold.


categories-definition-dual

Given a monoidal category $(\mathcal{C}, \otimes, \phi)$ and an object $X$ a {\it left dual} is an object $Y$ together with morphisms $\eta : \mathbf{1} \to X \otimes Y$ and $\epsilon : Y \otimes X \to \mathbf{1}$ such that the diagrams $$ \vcenter{ \xymatrix{ X \ar[rd]_1 \ar[r]_-{\eta \otimes 1} & X \otimes Y \otimes X  \ar[d]^{1 \otimes \epsilon} \\ & X } } \quad\text{and}\quad \vcenter{ \xymatrix{ Y \ar[rd]_1 \ar[r]_-{1 \otimes \eta} & Y \otimes X \otimes Y  \ar[d]^{\epsilon \otimes 1} \\ & Y } } $$ commute. In this situation we say that $X$ is a {\it right dual} of $Y$.


categories-definition-symmetric-monoidal-category

A quadruple $(\mathcal{C}, \otimes, \phi, \psi)$ where $\mathcal{C}$ is a category, $\otimes : \mathcal{C} \otimes \mathcal{C} \to \mathcal{C}$ is a functor, $\phi$ is an associativity constraint, and $\psi$ is a commutativity constraint compatible with $\phi$ is called a {\it symmetric monoidal category} if there exists a unit.