homology-lemma-double-complex-gives-resolution

Let $\mathcal{A}$ be an abelian category. Let $K^\bullet$ be a complex. Let $A^{\bullet, \bullet}$ be a double complex. Let $\alpha^p : K^p \to A^{p, 0}$ be morphisms. Assume that \begin{enumerate} \item For every $n \in \mathbf{Z}$ there are only finitely many nonzero $A^{p, q}$ with $p + q = n$. \item We have $A^{p, q} = 0$ if $q < 0$. \item The morphisms $\alpha^p$ give rise to a morphism of complexes $\alpha : K^\bullet \to A^{\bullet, 0}$. \item The complex $A^{p, \bullet}$ is exact in all degrees $q \not = 0$ and the morphism $K^p \to A^{p, 0}$ induces an isomorphism $K^p \to \Ker(d_2^{p, 0})$. \end{enumerate} Then $\alpha$ induces a quasi-isomorphism $$ K^\bullet \longrightarrow \text{Tot}(A^{\bullet, \bullet}) $$ of complexes. Moreover, there is a variant of this lemma involving the second variable $q$ instead of $p$.


homology-lemma-homotopy-complex-complexes

Let $\mathcal{A}$ be an abelian category. Let $M^\bullet$ be a complex of $\mathcal{A}$. Let $$ a : M^\bullet[0] \longrightarrow \left(A^{0, \bullet} \to A^{1, \bullet} \to A^{2, \bullet} \to \ldots \right) $$ be a homotopy equivalence in the category of complexes of complexes of $\mathcal{A}$. Then the map $\alpha : M^\bullet \to \text{Tot}(A^{\bullet, \bullet})$ induced by $M^\bullet \to A^{0, \bullet}$ is a homotopy equivalence.


homology-lemma-good-resolution-gives-qis

Let $M^\bullet$ be a complex of abelian groups. Let $$ \ldots \to A_2^\bullet \to A_1^\bullet \to A_0^\bullet \to M^\bullet \to 0 $$ be an exact complex of complexes of abelian groups such that for all $p \in \mathbf{Z}$ the complexes $$ \ldots \to \Ker(d_{A_2^\bullet}^p) \to \Ker(d_{A_1^\bullet}^p) \to \Ker(d_{A_0^\bullet}^p) \to \Ker(d_{M^\bullet}^p) \to 0 $$ are exact as well. Set $A^{p, q} = A_{-p}^q$ to obtain a double complex. Then $\text{Tot}(A^{\bullet, \bullet}) \to M^\bullet$ induced by $A_0^\bullet \to M^\bullet$ is a quasi-isomorphism.


homology-lemma-good-right-resolution-gives-qis

Let $M^\bullet$ be a complex of abelian groups. Let $$ 0 \to M^\bullet \to A_0^\bullet \to A_1^\bullet \to A_2^\bullet \to \ldots $$ be an exact complex of complexes of abelian groups such that for all $p \in \mathbf{Z}$ the complexes $$ 0 \to \Coker(d_{M^\bullet}^p) \to \Coker(d_{A_0^\bullet}^p) \to \Coker(d_{A_1^\bullet}^p) \to \Coker(d_{A_2^\bullet}^p) \to \ldots $$ are exact as well. Set $A^{p, q} = A_p^q$ to obtain a double complex. Let $\text{Tot}_\pi(A^{\bullet, \bullet})$ be the product total complex associated to the double complex (see proof). Then the map $M^\bullet \to \text{Tot}_\pi(A^{\bullet, \bullet})$ induced by $M^\bullet \to A_0^\bullet$ is a quasi-isomorphism.


homology-lemma-resolution-gives-qis

Let $M^\bullet$ be a complex of abelian groups. Let $$ \ldots \to A_2^\bullet \to A_1^\bullet \to A_0^\bullet \to M^\bullet \to 0 $$ be an exact complex of complexes of abelian groups. Set $A^{p, q} = A_{-p}^q$ to obtain a double complex. Let $\text{Tot}_\pi(A^{\bullet, \bullet})$ be the product total complex associated to the double complex (see proof). Then the map $\text{Tot}_\pi(A^{\bullet, \bullet}) \to M^\bullet$ induced by $A_0^\bullet \to M^\bullet$ is a quasi-isomorphism.


homology-lemma-characterize-injectives

Let $\mathcal{A}$ be an abelian category. Let $I$ be an object of $\mathcal{A}$. The following are equivalent: \begin{enumerate} \item The object $I$ is injective. \item The functor $B \mapsto \Hom_\mathcal{A}(B, I)$ is exact. \item Any short exact sequence $$ 0 \to I \to A \to B \to 0 $$ in $\mathcal{A}$ is split. \item We have $\Ext_\mathcal{A}(B, I) = 0$ for all $B \in \Ob(\mathcal{A})$. \end{enumerate}


homology-lemma-product-injectives

Let $\mathcal{A}$ be an abelian category. Suppose $I_\omega$, $\omega \in \Omega$ is a set of injective objects of $\mathcal{A}$. If $\prod_{\omega \in \Omega} I_\omega$ exists then it is injective.


homology-lemma-characterize-projectives

Let $\mathcal{A}$ be an abelian category. Let $P$ be an object of $\mathcal{A}$. The following are equivalent: \begin{enumerate} \item The object $P$ is projective. \item The functor $B \mapsto \Hom_\mathcal{A}(P, B)$ is exact. \item Any short exact sequence $$ 0 \to A \to B \to P \to 0 $$ in $\mathcal{A}$ is split. \item We have $\Ext_\mathcal{A}(P, A) = 0$ for all $A \in \Ob(\mathcal{A})$. \end{enumerate}


homology-lemma-coproduct-projectives

Let $\mathcal{A}$ be an abelian category. Suppose $P_\omega$, $\omega \in \Omega$ is a set of projective objects of $\mathcal{A}$. If $\coprod_{\omega \in \Omega} P_\omega$ exists then it is projective.


homology-lemma-adjoint-preserve-injectives

\begin{slogan} A functor with an exact left adjoint preserves injectives \end{slogan} Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume \begin{enumerate} \item $u$ is right adjoint to $v$, and \item $v$ transforms injective maps into injective maps. \end{enumerate} Then $u$ transforms injectives into injectives.


homology-lemma-adjoint-enough-injectives

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume \begin{enumerate} \item $u$ is right adjoint to $v$, \item $v$ transforms injective maps into injective maps, \item $\mathcal{A}$ has enough injectives, and \item $vB = 0$ implies $B = 0$ for any $B \in \Ob(\mathcal{B})$. \end{enumerate} Then $\mathcal{B}$ has enough injectives.


homology-lemma-adjoint-functorial-injectives

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ and $v : \mathcal{B} \to \mathcal{A}$ be additive functors. Assume \begin{enumerate} \item $u$ is right adjoint to $v$, \item $v$ transforms injective maps into injective maps, \item $\mathcal{A}$ has enough injectives, \item $vB = 0$ implies $B = 0$ for any $B \in \Ob(\mathcal{B})$, and \item $\mathcal{A}$ has functorial injective hulls. \end{enumerate} Then $\mathcal{B}$ has functorial injective hulls.


homology-lemma-partially-defined-adjoint

Let $\mathcal{A}$ and $\mathcal{B}$ be abelian categories. Let $u : \mathcal{A} \to \mathcal{B}$ be a functor. If there exists a subset $\mathcal{P} \subset \Ob(\mathcal{B})$ such that \begin{enumerate} \item every object of $\mathcal{B}$ is a quotient of an element of $\mathcal{P}$, and \item for every $P \in \mathcal{P}$ there exists an object $Q$ of $\mathcal{A}$ such that $\Hom_\mathcal{A}(Q, A) = \Hom_\mathcal{B}(P, u(A))$ functorially in $A$, \end{enumerate} then there exists a left adjoint $v$ of $u$.


homology-lemma-direct-sum-from-product-colimit

Let $\mathcal{I}$ be a category. Let $\mathcal{A}$ be an additive, Karoubian category. Let $F : \mathcal{I} \to \mathcal{A}$ and $G : \mathcal{I} \to \mathcal{A}$ be functors. The following are equivalent \begin{enumerate} \item $\colim_\mathcal{I} F \oplus G$ exists, and \item $\colim_\mathcal{I} F$ and $\colim_\mathcal{I} G$ exist. \end{enumerate} In this case $\colim_\mathcal{I} F \oplus G = \colim_\mathcal{I} F \oplus \colim_\mathcal{I} G$.


homology-lemma-direct-sum-from-product-essentially-constant

Let $\mathcal{I}$ be a filtered category. Let $\mathcal{A}$ be an additive, Karoubian category. Let $F : \mathcal{I} \to \mathcal{A}$ and $G : \mathcal{I} \to \mathcal{A}$ be functors. The following are equivalent \begin{enumerate} \item $F \oplus G : \mathcal{I} \to \mathcal{A}$ is essentially constant, and \item $F$ and $G$ are essentially constant. \end{enumerate}


homology-lemma-inverse-systems-abelian

Let $\mathcal{C}$ be a category. \begin{enumerate} \item If $\mathcal{C}$ is an additive category, then the category of inverse systems with values in $\mathcal{C}$ is an additive category. \item If $\mathcal{C}$ is an abelian category, then the category of inverse systems with values in $\mathcal{C}$ is an abelian category. A sequence $(K_i) \to (L_i) \to (M_i)$ of inverse systems is exact if and only if each $K_i \to L_i \to N_i$ is exact. \end{enumerate}


homology-lemma-Mittag-Leffler

Let $$ 0 \to (A_i) \to (B_i) \to (C_i) \to 0 $$ be a short exact sequence of inverse systems of abelian groups. \begin{enumerate} \item In any case the sequence $$ 0 \to \lim_i A_i \to \lim_i B_i \to \lim_i C_i $$ is exact. \item If $(B_i)$ is ML, then also $(C_i)$ is ML. \item If $(A_i)$ is ML, then $$ 0 \to \lim_i A_i \to \lim_i B_i \to \lim_i C_i \to 0 $$ is exact. \end{enumerate}


homology-lemma-apply-Mittag-Leffler

Let $$ (A_i) \to (B_i) \to (C_i) \to (D_i) $$ be an exact sequence of inverse systems of abelian groups. If the system $(A_i)$ is ML, then the sequence $$ \lim_i B_i \to \lim_i C_i \to \lim_i D_i $$ is exact.


homology-lemma-essentially-constant

Let $\mathcal{A}$ be an abelian category. Let $(A_i)$ be an inverse system in $\mathcal{A}$ with limit $A = \lim A_i$. Then $(A_i)$ is essentially constant (see Categories, Definition \ref{categories-definition-essentially-constant-diagram}) if and only if there exists an $i$ and for all $j \geq i$ a direct sum decomposition $A_j = A \oplus Z_j$ such that (a) the maps $A_{j'} \to A_j$ are compatible with the direct sum decompositions, (b) for all $j$ there exists some $j' \geq j$ such that $Z_{j'} \to Z_j$ is zero.


homology-lemma-exact-sequence-ML

Let $$ 0 \to (A_i) \to (B_i) \to (C_i) \to 0 $$ be an exact sequence of inverse systems of abelian groups. If $(A_i)$ has ML and $(C_i)$ is essentially constant, then $(B_i)$ has ML.


homology-lemma-apply-Mittag-Leffler-again

Let $$ (A^{-2}_i \to A^{-1}_i \to A^0_i \to A^1_i) $$ be an inverse system of complexes of abelian groups and denote $A^{-2} \to A^{-1} \to A^0 \to A^1$ its limit. Denote $(H_i^{-1})$, $(H_i^0)$ the inverse systems of cohomologies, and denote $H^{-1}$, $H^0$ the cohomologies of $A^{-2} \to A^{-1} \to A^0 \to A^1$. If $(A^{-2}_i)$ and $(A^{-1}_i)$ are ML and $(H^{-1}_i)$ is essentially constant, then $H^0 = \lim H_i^0$.


homology-lemma-ML-over-ordinals

Let $\alpha$ be an ordinal. Let $K_\beta^\bullet$, $\beta < \alpha$ be an inverse system of complexes of abelian groups over $\alpha$. If for all $\beta < \alpha$ the complex $K_\beta^\bullet$ is acyclic and the map $$ K^n_\beta \longrightarrow \lim_{\gamma < \beta} K^n_\gamma $$ is surjective, then the complex $\lim_{\beta < \alpha} K_\beta^\bullet$ is acyclic.


homology-lemma-product-abelian-groups-exact

Let $I$ be a set. For $i \in I$ let $L_i \to M_i \to N_i$ be a complex of abelian groups. Let $H_i = \Ker(M_i \to N_i)/\Im(L_i \to M_i)$ be the cohomology. Then $$ \prod L_i \to \prod M_i \to \prod N_i $$ is a complex of abelian groups with homology $\prod H_i$.


categories-theorem-adjoint-functor

Let $G : \mathcal{C} \to \mathcal{D}$ be a functor of big categories. Assume $\mathcal{C}$ has limits, $G$ commutes with them, and for every object $y$ of $\mathcal{D}$ there exists a set of pairs $(x_i, f_i)_{i \in I}$ with $x_i \in \Ob(\mathcal{C})$, $f_i \in \Mor_\mathcal{D}(y, G(x_i))$ such that for any pair $(x, f)$ with $x \in \Ob(\mathcal{C})$, $f \in \Mor_\mathcal{C}(y, G(x))$ there is an $i$ and a morphism $h : x_i \to x$ such that $f = G(h) \circ f_i$. Then $G$ has a left adjoint $F$.


categories-lemma-construct-quasi-inverse

Let $F : \mathcal{A} \to \mathcal{B}$ be a fully faithful functor. Suppose for every $X \in \Ob(\mathcal{B})$ given an object $j(X)$ of $\mathcal{A}$ and an isomorphism $i_X : X \to F(j(X))$. Then there is a unique functor $j : \mathcal{B} \to \mathcal{A}$ such that $j$ extends the rule on objects, and the isomorphisms $i_X$ define an isomorphism of functors $\text{id}_\mathcal{B} \to F \circ j$. Moreover, $j$ and $F$ are quasi-inverse equivalences of categories.


categories-lemma-equivalence-categories

A functor is an equivalence of categories if and only if it is both fully faithful and essentially surjective.


categories-lemma-yoneda

\begin{reference} Appeared in some form in \cite{Yoneda-homology}. Used by Grothendieck in a generalized form in \cite{Gr-II}. \end{reference} Let $U, V \in \Ob(\mathcal{C})$. Given any morphism of functors $s : h_U \to h_V$ there is a unique morphism $\phi : U \to V$ such that $h(\phi) = s$. In other words the functor $h$ is fully faithful. More generally, given any contravariant functor $F$ and any object $U$ of $\mathcal{C}$ we have a natural bijection $$ \Mor_{\textit{PSh}(\mathcal{C})}(h_U, F) \longrightarrow F(U), \quad s \longmapsto s_U(\text{id}_U). $$


categories-lemma-composition-representable

Let $\mathcal{C}$ be a category. Let $f : x \to y$, and $g : y \to z$ be representable. Then $g \circ f : x \to z$ is representable.


categories-lemma-base-change-representable

Let $\mathcal{C}$ be a category. Let $f : x \to y$ be representable. Let $y' \to y$ be a morphism of $\mathcal{C}$. Then the morphism $x' := x \times_y y' \to y'$ is representable also.


categories-lemma-fibre-product-presheaves

Let $\mathcal{C}$ be a category. Let $F, G, H : \mathcal{C}^{opp} \to \textit{Sets}$ be functors. Let $a : F \to G$ and $b : H \to G$ be transformations of functors. Then the fibre product $F \times_{a, G, b} H$ in the category $\textit{PSh}(\mathcal{C})$ exists and is given by the formula $$ (F \times_{a, G, b} H)(X) = F(X) \times_{a_X, G(X), b_X} H(X) $$ for any object $X$ of $\mathcal{C}$.


categories-lemma-representable-over-representable

Let $\mathcal{C}$ be a category. Let $a : F \to G$ be a morphism of contravariant functors from $\mathcal{C}$ to $\textit{Sets}$. If $a$ is representable, and $G$ is a representable functor, then $F$ is representable.


categories-lemma-representable-diagonal

Let $\mathcal{C}$ be a category. Let $F : \mathcal{C}^{opp} \to \textit{Sets}$ be a functor. Assume $\mathcal{C}$ has products of pairs of objects and fibre products. The following are equivalent: \begin{enumerate} \item the diagonal $\Delta : F \to F \times F$ is representable, \item for every $U$ in $\mathcal{C}$, and any $\xi \in F(U)$ the map $\xi : h_U \to F$ is representable, \item for every pair $U, V$ in $\mathcal{C}$ and any $\xi \in F(U)$, $\xi' \in F(V)$ the fibre product $h_U \times_{\xi, F, \xi'} h_V$ is representable. \end{enumerate}


categories-lemma-characterize-mono-epi

Let $\mathcal{C}$ be a category, and let $f : X \to Y$ be a morphism of $\mathcal{C}$. Then \begin{enumerate} \item $f$ is a monomorphism if and only if $X$ is the fibre product $X \times_Y X$, and \item $f$ is an epimorphism if and only if $Y$ is the pushout $Y \amalg_X Y$. \end{enumerate}


categories-lemma-functorial-colimit

Suppose that $M : \mathcal{I} \to \mathcal{C}$, and $N : \mathcal{J} \to \mathcal{C}$ are diagrams whose colimits exist. Suppose $H : \mathcal{I} \to \mathcal{J}$ is a functor, and suppose $t : M \to N \circ H$ is a transformation of functors. Then there is a unique morphism $$ \theta : \colim_\mathcal{I} M \longrightarrow \colim_\mathcal{J} N $$ such that all the diagrams $$ \xymatrix{ M_i \ar[d]_{t_i} \ar[r] & \colim_\mathcal{I} M \ar[d]^{\theta} \\ N_{H(i)} \ar[r] & \colim_\mathcal{J} N } $$ commute.


categories-lemma-functorial-limit

Suppose that $M : \mathcal{I} \to \mathcal{C}$, and $N : \mathcal{J} \to \mathcal{C}$ are diagrams whose limits exist. Suppose $H : \mathcal{I} \to \mathcal{J}$ is a functor, and suppose $t : N \circ H \to M$ is a transformation of functors. Then there is a unique morphism $$ \theta : \lim_\mathcal{J} N \longrightarrow \lim_\mathcal{I} M $$ such that all the diagrams $$ \xymatrix{ \lim_\mathcal{J} N \ar[d]^{\theta} \ar[r] & N_{H(i)} \ar[d]_{t_i} \\ \lim_\mathcal{I} M \ar[r] & M_i } $$ commute.


categories-lemma-colimits-commute

Let $\mathcal{I}$, $\mathcal{J}$ be index categories. Let $M : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$ be a functor. We have $$ \colim_i \colim_j M_{i, j} = \colim_{i, j} M_{i, j} = \colim_j \colim_i M_{i, j} $$ provided all the indicated colimits exist. Similar for limits.


categories-lemma-limits-products-equalizers

Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. Write $I = \Ob(\mathcal{I})$ and $A = \text{Arrows}(\mathcal{I})$. Denote $s, t : A \to I$ the source and target maps. Suppose that $\prod_{i \in I} M_i$ and $\prod_{a \in A} M_{t(a)}$ exist. Suppose that the equalizer of $$ \xymatrix{ \prod_{i \in I} M_i \ar@<1ex>[r]^\phi \ar@<-1ex>[r]_\psi & \prod_{a \in A} M_{t(a)} } $$ exists, where the morphisms are determined by their components as follows: $p_a \circ \psi = M(a) \circ p_{s(a)}$ and $p_a \circ \phi = p_{t(a)}$. Then this equalizer is the limit of the diagram.


categories-lemma-colimits-coproducts-coequalizers

\begin{slogan} If all coproducts and coequalizers exist, all colimits exist. \end{slogan} Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. Write $I = \Ob(\mathcal{I})$ and $A = \text{Arrows}(\mathcal{I})$. Denote $s, t : A \to I$ the source and target maps. Suppose that $\coprod_{i \in I} M_i$ and $\coprod_{a \in A} M_{s(a)}$ exist. Suppose that the coequalizer of $$ \xymatrix{ \coprod_{a \in A} M_{s(a)} \ar@<1ex>[r]^\phi \ar@<-1ex>[r]_\psi & \coprod_{i \in I} M_i } $$ exists, where the morphisms are determined by their components as follows: The component $M_{s(a)}$ maps via $\psi$ to the component $M_{t(a)}$ via the morphism $a$. The component $M_{s(a)}$ maps via $\phi$ to the component $M_{s(a)}$ by the identity morphism. Then this coequalizer is the colimit of the diagram.


categories-lemma-connected-limit-over-X

Let $\mathcal{C}$ be a category. Let $X$ be an object of $\mathcal{C}$. Let $M : \mathcal{I} \to \mathcal{C}/X$ be a diagram in the category of objects over $X$. If the index category $\mathcal{I}$ is connected and the limit of $M$ exists in $\mathcal{C}/X$, then the limit of the composition $\mathcal{I} \to \mathcal{C}/X \to \mathcal{C}$ exists and is the same.


categories-lemma-cofinal

Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is cofinal in $\mathcal{J}$. Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism $$ \colim_\mathcal{I} M \circ H = \colim_\mathcal{J} M $$ if either side exists.


categories-lemma-initial

Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of categories. Assume $\mathcal{I}$ is initial in $\mathcal{J}$. Then for every diagram $M : \mathcal{J} \to \mathcal{C}$ we have a canonical isomorphism $$ \lim_\mathcal{I} M \circ H = \lim_\mathcal{J} M $$ if either side exists.


categories-lemma-colimit-constant-connected-fibers

Let $F : \mathcal{I} \to \mathcal{I}'$ be a functor. Assume \begin{enumerate} \item the fibre categories (see Definition \ref{definition-fibre-category}) of $\mathcal{I}$ over $\mathcal{I}'$ are all connected, and \item for every morphism $\alpha' : x' \to y'$ in $\mathcal{I}'$ there exist a morphism $\alpha : x \to y$ in $\mathcal{I}$ such that $F(\alpha) = \alpha'$. \end{enumerate} Then for every diagram $M : \mathcal{I}' \to \mathcal{C}$ the colimit $\colim_\mathcal{I} M \circ F$ exists if and only if $\colim_{\mathcal{I}'} M$ exists and if so these colimits agree.


categories-lemma-finite-diagram-category

Let $\mathcal{I}$ be a category with \begin{enumerate} \item $\Ob(\mathcal{I})$ is finite, and \item there exist finitely many morphisms $f_1, \ldots, f_m \in \text{Arrows}(\mathcal{I})$ such that every morphism of $\mathcal{I}$ is a composition $f_{j_1} \circ f_{j_2} \circ \ldots \circ f_{j_k}$. \end{enumerate} Then there exists a functor $F : \mathcal{J} \to \mathcal{I}$ such that \begin{enumerate} \item[(a)] $\mathcal{J}$ is a finite category, and \item[(b)] for any diagram $M : \mathcal{I} \to \mathcal{C}$ the (co)limit of $M$ over $\mathcal{I}$ exists if and only if the (co)limit of $M \circ F$ over $\mathcal{J}$ exists and in this case the (co)limits are canonically isomorphic. \end{enumerate} Moreover, $\mathcal{J}$ is connected (resp.\ nonempty) if and only if $\mathcal{I}$ is so.


categories-lemma-fibre-products-equalizers-exist

Let $\mathcal{C}$ be a category. The following are equivalent: \begin{enumerate} \item Connected finite limits exist in $\mathcal{C}$. \item Equalizers and fibre products exist in $\mathcal{C}$. \end{enumerate}


categories-lemma-almost-finite-limits-exist

Let $\mathcal{C}$ be a category. The following are equivalent: \begin{enumerate} \item Nonempty finite limits exist in $\mathcal{C}$. \item Products of pairs and equalizers exist in $\mathcal{C}$. \item Products of pairs and fibre products exist in $\mathcal{C}$. \end{enumerate}


categories-lemma-finite-limits-exist

Let $\mathcal{C}$ be a category. The following are equivalent: \begin{enumerate} \item Finite limits exist in $\mathcal{C}$. \item Finite products and equalizers exist. \item The category has a final object and fibred products exist. \end{enumerate}


categories-lemma-almost-finite-colimits-exist

Let $\mathcal{C}$ be a category. The following are equivalent: \begin{enumerate} \item Nonempty finite colimits exist in $\mathcal{C}$. \item Coproducts of pairs and coequalizers exist in $\mathcal{C}$. \item Coproducts of pairs and pushouts exist in $\mathcal{C}$. \end{enumerate}


categories-lemma-colimits-exist

Let $\mathcal{C}$ be a category. The following are equivalent: \begin{enumerate} \item Finite colimits exist in $\mathcal{C}$, \item Finite coproducts and coequalizers exist in $\mathcal{C}$, and \item The category has an initial object and pushouts exist. \end{enumerate}


categories-lemma-directed-commutes

Let $\mathcal{I}$ and $\mathcal{J}$ be index categories. Assume that $\mathcal{I}$ is filtered and $\mathcal{J}$ is finite. Let $M : \mathcal{I} \times \mathcal{J} \to \textit{Sets}$, $(i, j) \mapsto M_{i, j}$ be a diagram of diagrams of sets. In this case $$ \colim_i \lim_j M_{i, j} = \lim_j \colim_i M_{i, j}. $$ In particular, colimits over $\mathcal{I}$ commute with finite products, fibre products, and equalizers of sets.


categories-lemma-cofinal-in-filtered

Let $\mathcal{I}$ be a category. Let $\mathcal{J}$ be a full subcategory. Assume that $\mathcal{I}$ is filtered. Assume also that for any object $i$ of $\mathcal{I}$, there exists a morphism $i \to j$ to some object $j$ of $\mathcal{J}$. Then $\mathcal{J}$ is filtered and cofinal in $\mathcal{I}$.


categories-lemma-colimits-abelian-as-sets

Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every pair of objects $x, y$ of $\mathcal{I}$ there exists an object $z$ and morphisms $x \to z$ and $y \to z$. Let $M : \mathcal{I} \to \textit{Ab}$ be a diagram of abelian groups over $\mathcal{I}$. Then the colimit of $M$ in the category of sets surjects onto the colimit of $M$ in the category of abelian groups.


categories-lemma-split-into-connected

Let $\mathcal{I}$ be an index category, i.e., a category. Assume that for every solid diagram $$ \xymatrix{ x \ar[d] \ar[r] & y \ar@{..>}[d] \\ z \ar@{..>}[r] & w } $$ in $\mathcal{I}$ there exists an object $w$ and dotted arrows making the diagram commute. Then $\mathcal{I}$ is either empty or a nonempty disjoint union of connected categories having the same property.


categories-lemma-split-into-directed

Let $\mathcal{I}$ be an index category, i.e., a category. Assume \begin{enumerate} \item for every pair of morphisms $a : w \to x$ and $b : w \to y$ in $\mathcal{I}$ there exists an object $z$ and morphisms $c : x \to z$ and $d : y \to z$ such that $c \circ a = d \circ b$, and \item for every pair of morphisms $a, b : x \to y$ there exists a morphism $c : y \to z$ such that $c \circ a = c \circ b$. \end{enumerate} Then $\mathcal{I}$ is a (possibly empty) union of disjoint filtered index categories $\mathcal{I}_j$.


categories-lemma-almost-directed-commutes-equalizers

Let $\mathcal{I}$ be an index category satisfying the hypotheses of Lemma \ref{lemma-split-into-directed} above. Then colimits over $\mathcal{I}$ commute with fibre products and equalizers in sets (and more generally with finite connected limits).


categories-lemma-directed-category-system

Let $\mathcal{I}$ be a filtered index category. There exists a directed set $I$ and a system $(x_i, \varphi_{ii'})$ over $I$ in $\mathcal{I}$ with the following properties: \begin{enumerate} \item For every category $\mathcal{C}$ and every diagram $M : \mathcal{I} \to \mathcal{C}$ with values in $\mathcal{C}$, denote $(M(x_i), M(\varphi_{ii'}))$ the corresponding system over $I$. If $\colim_{i \in I} M(x_i)$ exists then so does $\colim_\mathcal{I} M$ and the transformation $$ \theta : \colim_{i \in I} M(x_i) \longrightarrow \colim_\mathcal{I} M $$ of Lemma \ref{lemma-functorial-colimit} is an isomorphism. \item For every category $\mathcal{C}$ and every diagram $M : \mathcal{I}^{opp} \to \mathcal{C}$ in $\mathcal{C}$, denote $(M(x_i), M(\varphi_{ii'}))$ the corresponding inverse system over $I$. If $\lim_{i \in I} M(x_i)$ exists then so does $\lim_\mathcal{I} M$ and the transformation $$ \theta : \lim_{\mathcal{I}^{opp}} M \longrightarrow \lim_{i \in I} M(x_i) $$ of Lemma \ref{lemma-functorial-limit} is an isomorphism. \end{enumerate}


categories-lemma-nonempty-limit

If $S : \mathcal{I} \to \textit{Sets}$ is a cofiltered diagram of sets and all the $S_i$ are finite nonempty, then $\lim_i S_i$ is nonempty. In other words, the limit of a directed inverse system of finite nonempty sets is nonempty.


categories-lemma-essentially-constant-is-limit-colimit

Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram. If $\mathcal{I}$ is filtered and $M$ is essentially constant as an ind-object, then $X = \colim M_i$ exists and $M$ is essentially constant with value $X$. If $\mathcal{I}$ is cofiltered and $M$ is essentially constant as a pro-object, then $X = \lim M_i$ exists and $M$ is essentially constant with value $X$.


categories-lemma-image-essentially-constant

Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with filtered (resp.\ cofiltered) index category $\mathcal{I}$. Let $F : \mathcal{C} \to \mathcal{D}$ be a functor. If $M$ is essentially constant as an ind-object (resp.\ pro-object), then so is $F \circ M : \mathcal{I} \to \mathcal{D}$.


categories-lemma-characterize-essentially-constant-ind

Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with filtered index category $\mathcal{I}$. The following are equivalent \begin{enumerate} \item $M$ is an essentially constant ind-object, and \item $X = \colim_i M_i$ exists and for any $W$ in $\mathcal{C}$ the map $$ \colim_i \Mor_\mathcal{C}(W, M_i) \longrightarrow \Mor_\mathcal{C}(W, X) $$ is bijective. \end{enumerate}


categories-lemma-characterize-essentially-constant-pro

Let $\mathcal{C}$ be a category. Let $M : \mathcal{I} \to \mathcal{C}$ be a diagram with cofiltered index category $\mathcal{I}$. The following are equivalent \begin{enumerate} \item $M$ is an essentially constant pro-object, and \item $X = \lim_i M_i$ exists and for any $W$ in $\mathcal{C}$ the map $$ \colim_{i \in \mathcal{I}^{opp}} \Mor_\mathcal{C}(M_i, W) \longrightarrow \Mor_\mathcal{C}(X, W) $$ is bijective. \end{enumerate}


categories-lemma-cofinal-essentially-constant

Let $\mathcal{C}$ be a category. Let $H : \mathcal{I} \to \mathcal{J}$ be a functor of filtered index categories. If $H$ is cofinal, then any diagram $M : \mathcal{J} \to \mathcal{C}$ is essentially constant if and only if $M \circ H$ is essentially constant.


categories-lemma-essentially-constant-over-product

Let $\mathcal{I}$ and $\mathcal{J}$ be filtered categories and denote $p : \mathcal{I} \times \mathcal{J} \to \mathcal{J}$ the projection. Then $\mathcal{I} \times \mathcal{J}$ is filtered and a diagram $M : \mathcal{J} \to \mathcal{C}$ is essentially constant if and only if $M \circ p : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$ is essentially constant.


categories-lemma-characterize-left-exact

Let $F : \mathcal{A} \to \mathcal{B}$ be a functor. Suppose all finite limits exist in $\mathcal{A}$, see Lemma \ref{lemma-finite-limits-exist}. The following are equivalent: \begin{enumerate} \item $F$ is left exact, \item $F$ commutes with finite products and equalizers, and \item $F$ transforms a final object of $\mathcal{A}$ into a final object of $\mathcal{B}$, and commutes with fibre products. \end{enumerate}


categories-lemma-adjoint-exists

Let $u : \mathcal{C} \to \mathcal{D}$ be a functor between categories. If for each $y \in \Ob(\mathcal{D})$ the functor $x \mapsto \Mor_\mathcal{D}(u(x), y)$ is representable, then $u$ has a right adjoint.


categories-lemma-left-adjoint-composed-fully-faithful

\begin{reference} Bhargav Bhatt, private communication. \end{reference} Let $u$ be a left adjoint to $v$ as in Definition \ref{definition-adjoint}. \begin{enumerate} \item If $v \circ u$ is fully faithful, then $u$ is fully faithful. \item If $u \circ v$ is fully faithful, then $v$ is fully faithful. \end{enumerate}


categories-lemma-adjoint-fully-faithful

Let $u$ be a left adjoint to $v$ as in Definition \ref{definition-adjoint}. Then \begin{enumerate} \item $u$ is fully faithful $\Leftrightarrow$ $\text{id} \cong v \circ u$ $\Leftrightarrow$ $\eta : \textit{id} \to v \circ u$ is an isomorphism, \item $v$ is fully faithful $\Leftrightarrow$ $u \circ v \cong \text{id}$ $\Leftrightarrow$ $\epsilon : u \circ v \to \text{id}$ is an isomorphism. \end{enumerate}


categories-lemma-adjoint-exact

Let $u$ be a left adjoint to $v$ as in Definition \ref{definition-adjoint}. \begin{enumerate} \item Suppose that $M : \mathcal{I} \to \mathcal{C}$ is a diagram, and suppose that $\colim_\mathcal{I} M$ exists in $\mathcal{C}$. Then $u(\colim_\mathcal{I} M) = \colim_\mathcal{I} u \circ M$. In other words, $u$ commutes with (representable) colimits. \item Suppose that $M : \mathcal{I} \to \mathcal{D}$ is a diagram, and suppose that $\lim_\mathcal{I} M$ exists in $\mathcal{D}$. Then $v(\lim_\mathcal{I} M) = \lim_\mathcal{I} v \circ M$. In other words $v$ commutes with representable limits. \end{enumerate}


categories-lemma-exact-adjoint

Let $u$ be a left adjoint of $v$ as in Definition \ref{definition-adjoint}. \begin{enumerate} \item If $\mathcal{C}$ has finite colimits, then $u$ is right exact. \item If $\mathcal{D}$ has finite limits, then $v$ is left exact. \end{enumerate}


categories-lemma-transformation-between-functors-and-adjoints

Let $u_1, u_2 : \mathcal{C} \to \mathcal{D}$ be functors with right adjoints $v_1, v_2 : \mathcal{D} \to \mathcal{C}$. Let $\beta : u_2 \to u_1$ be a transformation of functors. Let $\beta^\vee : v_1 \to v_2$ be the corresponding transformation of adjoint functors. Then $$ \xymatrix{ u_2 \circ v_1 \ar[r]_\beta \ar[d]_{\beta^\vee} & u_1 \circ v_1 \ar[d] \\ u_2 \circ v_2 \ar[r] & \text{id} } $$ is commutative where the unlabeled arrows are the counit transformations.


categories-lemma-compose-counits

Let $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ be categories. Let $v : \mathcal{A} \to \mathcal{B}$ and $v' : \mathcal{B} \to \mathcal{C}$ be functors with left adjoints $u$ and $u'$ respectively. Then \begin{enumerate} \item The functor $v'' = v' \circ v$ has a left adjoint equal to $u'' = u \circ u'$. \item Given $X$ in $\mathcal{A}$ we have \begin{equation} \label{equation-compose-counits} \epsilon_X^v \circ u(\epsilon^{v'}_{v(X)}) = \epsilon^{v''}_X : u''(v''(X)) \to X \end{equation} Where $\epsilon$ is the counit of the adjunctions. \end{enumerate}


categories-lemma-a-version-of-brown

Let $\mathcal{C}$ be a big\footnote{See Remark \ref{remark-big-categories}.} category which has limits. Let $F : \mathcal{C} \to \textit{Sets}$ be a functor. Assume that \begin{enumerate} \item $F$ commutes with limits, \item there exists a family $\{x_i\}_{i \in I}$ of objects of $\mathcal{C}$ and for each $i \in I$ an element $f_i \in F(x_i)$ such that for $y \in \Ob(\mathcal{C})$ and $g \in F(y)$ there exists an $i$ and a morphism $\varphi : x_i \to y$ with $F(\varphi)(f_i) = g$. \end{enumerate} Then $F$ is representable, i.e., there exists an object $x$ of $\mathcal{C}$ such that $$ F(y) = \Mor_\mathcal{C}(x, y) $$ functorially in $y$.


categories-lemma-extend-functor-by-colim

Let $\mathcal{C}$ and $\mathcal{D}$ be big categories having filtered colimits. Let $\mathcal{C}' \subset \mathcal{C}$ be a small full subcategory consisting of categorically compact objects of $\mathcal{C}$ such that every object of $\mathcal{C}$ is a filtered colimit of objects of $\mathcal{C}'$. Then every functor $F' : \mathcal{C}' \to \mathcal{D}$ has a unique extension $F : \mathcal{C} \to \mathcal{D}$ commuting with filtered colimits.


categories-lemma-left-localization

Let $\mathcal{C}$ be a category and let $S$ be a left multiplicative system. \begin{enumerate} \item The relation on pairs defined above is an equivalence relation. \item The composition rule given above is well defined on equivalence classes. \item Composition is associative (and the identity morphisms satisfy the identity axioms), and hence $S^{-1}\mathcal{C}$ is a category. \end{enumerate}


categories-lemma-morphisms-left-localization

Let $\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\mathcal{C}$. Given any finite collection $g_i : X_i \to Y$ of morphisms of $S^{-1}\mathcal{C}$ (indexed by $i$), we can find an element $s : Y \to Y'$ of $S$ and a family of morphisms $f_i : X_i \to Y'$ of $\mathcal{C}$ such that each $g_i$ is the equivalence class of the pair $(f_i : X_i \to Y', s : Y \to Y')$.


categories-lemma-equality-morphisms-left-localization

Let $\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\mathcal{C}$. Let $A, B : X \to Y$ be morphisms of $S^{-1}\mathcal{C}$ which are the equivalence classes of $(f : X \to Y', s : Y \to Y')$ and $(g : X \to Y', s : Y \to Y')$. Then $A = B$ if and only if there exists a morphism $a : Y' \to Y''$ with $a \circ s \in S$ and such that $a \circ f = a \circ g$.


categories-lemma-properties-left-localization

Let $\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\mathcal{C}$. \begin{enumerate} \item The rules $X \mapsto X$ and $(f : X \to Y) \mapsto (f : X \to Y, \text{id}_Y : Y \to Y)$ define a functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$. \item For any $s \in S$ the morphism $Q(s)$ is an isomorphism in $S^{-1}\mathcal{C}$. \item If $G : \mathcal{C} \to \mathcal{D}$ is any functor such that $G(s)$ is invertible for every $s \in S$, then there exists a unique functor $H : S^{-1}\mathcal{C} \to \mathcal{D}$ such that $H \circ Q = G$. \end{enumerate}


categories-lemma-left-localization-limits

Let $\mathcal{C}$ be a category and let $S$ be a left multiplicative system of morphisms of $\mathcal{C}$. The localization functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ commutes with finite colimits.


categories-lemma-left-localization-diagram

Let $\mathcal{C}$ be a category. Let $S$ be a left multiplicative system. If $f : X \to Y$, $f' : X' \to Y'$ are two morphisms of $\mathcal{C}$ and if $$ \xymatrix{ Q(X) \ar[d]_{Q(f)} \ar[r]_a & Q(X') \ar[d]^{Q(f')} \\ Q(Y) \ar[r]^b & Q(Y') } $$ is a commutative diagram in $S^{-1}\mathcal{C}$, then there exists a morphism $f'' : X'' \to Y''$ in $\mathcal{C}$ and a commutative diagram $$ \xymatrix{ X \ar[d]_f \ar[r]_g & X'' \ar[d]^{f''} & X' \ar[d]^{f'} \ar[l]^s \\ Y \ar[r]^h & Y'' & Y' \ar[l]_t } $$ in $\mathcal{C}$ with $s, t \in S$ and $a = s^{-1}g$, $b = t^{-1}h$.


categories-lemma-right-localization

Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system. \begin{enumerate} \item The relation on pairs defined above is an equivalence relation. \item The composition rule given above is well defined on equivalence classes. \item Composition is associative (and the identity morphisms satisfy the identity axioms), and hence $S^{-1}\mathcal{C}$ is a category. \end{enumerate}


categories-lemma-morphisms-right-localization

Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\mathcal{C}$. Given any finite collection $g_i : X \to Y_i$ of morphisms of $S^{-1}\mathcal{C}$ (indexed by $i$), we can find an element $s : X' \to X$ of $S$ and a family of morphisms $f_i : X' \to Y_i$ of $\mathcal{C}$ such that $g_i$ is the equivalence class of the pair $(f_i : X' \to Y_i, s : X' \to X)$.


categories-lemma-equality-morphisms-right-localization

Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\mathcal{C}$. Let $A, B : X \to Y$ be morphisms of $S^{-1}\mathcal{C}$ which are the equivalence classes of $(f : X' \to Y, s : X' \to X)$ and $(g : X' \to Y, s : X' \to X)$. Then $A = B$ if and only if there exists a morphism $a : X'' \to X'$ with $s \circ a \in S$ and such that $f \circ a = g \circ a$.


categories-lemma-properties-right-localization

Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\mathcal{C}$. \begin{enumerate} \item The rules $X \mapsto X$ and $(f : X \to Y) \mapsto (f : X \to Y, \text{id}_X : X \to X)$ define a functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$. \item For any $s \in S$ the morphism $Q(s)$ is an isomorphism in $S^{-1}\mathcal{C}$. \item If $G : \mathcal{C} \to \mathcal{D}$ is any functor such that $G(s)$ is invertible for every $s \in S$, then there exists a unique functor $H : S^{-1}\mathcal{C} \to \mathcal{D}$ such that $H \circ Q = G$. \end{enumerate}


categories-lemma-right-localization-limits

Let $\mathcal{C}$ be a category and let $S$ be a right multiplicative system of morphisms of $\mathcal{C}$. The localization functor $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ commutes with finite limits.


categories-lemma-multiplicative-system

Let $\mathcal{C}$ be a category and let $S$ be a multiplicative system. The category of left fractions and the category of right fractions $S^{-1}\mathcal{C}$ are canonically isomorphic.


categories-lemma-what-gets-inverted

Let $\mathcal{C}$ be a category and let $S$ be a multiplicative system. Denote $Q : \mathcal{C} \to S^{-1}\mathcal{C}$ the localization functor. The set $$ \hat S = \{f \in \text{Arrows}(\mathcal{C}) \mid Q(f) \text{ is an isomorphism}\} $$ is equal to $$ S' = \{f \in \text{Arrows}(\mathcal{C}) \mid \text{there exist }g, h\text{ such that }gf, fh \in S\} $$ and is the smallest saturated multiplicative system containing $S$. In particular, if $S$ is saturated, then $\hat S = S$.


categories-lemma-properties-2-cat-cats

The horizontal and vertical compositions have the following properties \begin{enumerate} \item $\circ$ and $\star$ are associative, \item the identity transformations $\text{id}_F$ are units for $\circ$, \item the identity transformations of the identity functors $\text{id}_{\text{id}_\mathcal{A}}$ are units for $\star$ and $\circ$, and \item given a diagram $$ \xymatrix{ \mathcal{A} \rruppertwocell^F{t} \ar[rr]_(.3){F'} \rrlowertwocell_{F''}{t'} & & \mathcal{B} \rruppertwocell^G{s} \ar[rr]_(.3){G'} \rrlowertwocell_{G''}{s'} & & \mathcal{C} } $$ we have $ (s' \circ s) \star (t' \circ t) = (s' \star t') \circ (s \star t)$. \end{enumerate}


categories-lemma-2-fibre-product-categories

In the $(2, 1)$-category of categories $2$-fibre products exist and are given by the construction of Example \ref{example-2-fibre-product-categories}.


categories-lemma-functoriality-2-fibre-product

Let $$ \xymatrix{ & \mathcal{Y} \ar[d]_I \ar[rd]^K & \\ \mathcal{X} \ar[r]^H \ar[rd]^L & \mathcal{Z} \ar[rd]^M & \mathcal{B} \ar[d]^G \\ & \mathcal{A} \ar[r]^F & \mathcal{C} } $$ be a $2$-commutative diagram of categories. A choice of isomorphisms $\alpha : G \circ K \to M \circ I$ and $\beta : M \circ H \to F \circ L$ determines a morphism $$ \mathcal{X} \times_\mathcal{Z} \mathcal{Y} \longrightarrow \mathcal{A} \times_\mathcal{C} \mathcal{B} $$ of $2$-fibre products associated to this situation.


categories-lemma-equivalence-2-fibre-product

Assumptions as in Lemma \ref{lemma-functoriality-2-fibre-product}. \begin{enumerate} \item If $K$ and $L$ are faithful then the morphism $\mathcal{X} \times_\mathcal{Z} \mathcal{Y} \to \mathcal{A} \times_\mathcal{C} \mathcal{B}$ is faithful. \item If $K$ and $L$ are fully faithful and $M$ is faithful then the morphism $\mathcal{X} \times_\mathcal{Z} \mathcal{Y} \to \mathcal{A} \times_\mathcal{C} \mathcal{B}$ is fully faithful. \item If $K$ and $L$ are equivalences and $M$ is fully faithful then the morphism $\mathcal{X} \times_\mathcal{Z} \mathcal{Y} \to \mathcal{A} \times_\mathcal{C} \mathcal{B}$ is an equivalence. \end{enumerate}


categories-lemma-associativity-2-fibre-product

Let $$ \xymatrix{ \mathcal{A} \ar[rd] & & \mathcal{C} \ar[ld] \ar[rd] & & \mathcal{E} \ar[ld] \\ & \mathcal{B} & & \mathcal{D} } $$ be a diagram of categories and functors. Then there is a canonical isomorphism $$ (\mathcal{A} \times_\mathcal{B} \mathcal{C}) \times_\mathcal{D} \mathcal{E} \cong \mathcal{A} \times_\mathcal{B} (\mathcal{C} \times_\mathcal{D} \mathcal{E}) $$ of categories.


categories-lemma-triple-2-fibre-product-pr02

Let $$ \xymatrix{ \mathcal{A} \ar[rd] & & \mathcal{C} \ar[ld] \ar[rd] & & \mathcal{E} \ar[ld] \\ & \mathcal{B} \ar[rd]_F & & \mathcal{D} \ar[ld]^G \\ & & \mathcal{F} & } $$ be a commutative diagram of categories and functors. Then there is a canonical functor $$ \text{pr}_{02} : \mathcal{A} \times_\mathcal{B} \mathcal{C} \times_\mathcal{D} \mathcal{E} \longrightarrow \mathcal{A} \times_\mathcal{F} \mathcal{E} $$ of categories.


categories-lemma-2-fibre-product-erase-factor

Let $$ \mathcal{A} \to \mathcal{B} \leftarrow \mathcal{C} \leftarrow \mathcal{D} $$ be a diagram of categories and functors. Then there is a canonical isomorphism $$ \mathcal{A} \times_\mathcal{B} \mathcal{C} \times_\mathcal{C} \mathcal{D} \cong \mathcal{A} \times_\mathcal{B} \mathcal{D} $$ of categories.


categories-lemma-diagonal-1

Let $$ \xymatrix{ \mathcal{C}_3 \ar[r] \ar[d] & \mathcal{S} \ar[d]^\Delta \\ \mathcal{C}_1 \times \mathcal{C}_2 \ar[r]^{G_1 \times G_2} & \mathcal{S} \times \mathcal{S} } $$ be a $2$-fibre product of categories. Then there is a canonical isomorphism $\mathcal{C}_3 \cong \mathcal{C}_1 \times_{G_1, \mathcal{S}, G_2} \mathcal{C}_2$.


categories-lemma-diagonal-2

Let $$ \xymatrix{ \mathcal{C}' \ar[r] \ar[d] & \mathcal{S} \ar[d]^\Delta \\ \mathcal{C} \ar[r]^{G_1 \times G_2} & \mathcal{S} \times \mathcal{S} } $$ be a $2$-fibre product of categories. Then there is a canonical isomorphism $$ \mathcal{C}' \cong (\mathcal{C} \times_{G_1, \mathcal{S}, G_2} \mathcal{C}) \times_{(p, q), \mathcal{C} \times \mathcal{C}, \Delta} \mathcal{C}. $$


categories-lemma-fibre-product-after-map

Let $\mathcal{A} \to \mathcal{C}$, $\mathcal{B} \to \mathcal{C}$ and $\mathcal{C} \to \mathcal{D}$ be functors between categories. Then the diagram $$ \xymatrix{ \mathcal{A} \times_\mathcal{C} \mathcal{B} \ar[d] \ar[r] & \mathcal{A} \times_\mathcal{D} \mathcal{B} \ar[d] \\ \mathcal{C} \ar[r]^-{\Delta_{\mathcal{C}/\mathcal{D}}} \ar[r] & \mathcal{C} \times_\mathcal{D} \mathcal{C} } $$ is a $2$-fibre product diagram.


categories-lemma-base-change-diagonal

Let $$ \xymatrix{ \mathcal{U} \ar[d] \ar[r] & \mathcal{V} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{Y} } $$ be a $2$-fibre product of categories. Then the diagram $$ \xymatrix{ \mathcal{U} \ar[d] \ar[r] & \mathcal{U} \times_\mathcal{V} \mathcal{U} \ar[d] \\ \mathcal{X} \ar[r] & \mathcal{X} \times_\mathcal{Y} \mathcal{X} } $$ is $2$-cartesian.


categories-lemma-2-product-categories-over-C

Let $\mathcal{C}$ be a category. The $(2, 1)$-category of categories over $\mathcal{C}$ has 2-fibre products. Suppose that $F : \mathcal{X} \to \mathcal{S}$ and $G : \mathcal{Y} \to \mathcal{S}$ are morphisms of categories over $\mathcal{C}$. An explicit 2-fibre product $\mathcal{X} \times_\mathcal{S}\mathcal{Y}$ is given by the following description \begin{enumerate} \item an object of $\mathcal{X} \times_\mathcal{S} \mathcal{Y}$ is a quadruple $(U, x, y, f)$, where $U \in \Ob(\mathcal{C})$, $x\in \Ob(\mathcal{X}_U)$, $y\in \Ob(\mathcal{Y}_U)$, and $f : F(x) \to G(y)$ is an isomorphism in $\mathcal{S}_U$, \item a morphism $(U, x, y, f) \to (U', x', y', f')$ is given by a pair $(a, b)$, where $a : x \to x'$ is a morphism in $\mathcal{X}$, and $b : y \to y'$ is a morphism in $\mathcal{Y}$ such that \begin{enumerate} \item $a$ and $b$ induce the same morphism $U \to U'$, and \item the diagram $$ \xymatrix{ F(x) \ar[r]^f \ar[d]^{F(a)} & G(y) \ar[d]^{G(b)} \\ F(x') \ar[r]^{f'} & G(y') } $$ is commutative. \end{enumerate} \end{enumerate} The functors $p : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{X}$ and $q : \mathcal{X} \times_\mathcal{S}\mathcal{Y} \to \mathcal{Y}$ are the forgetful functors in this case. The transformation $\psi : F \circ p \to G \circ q$ is given on the object $\xi = (U, x, y, f)$ by $\psi_\xi = f : F(p(\xi)) = F(x) \to G(y) = G(q(\xi))$.


categories-lemma-fibre-2-fibre-product-categories-over-C

Let $\mathcal{C}$ be a category. Let $f : \mathcal{X} \to \mathcal{S}$ and $g : \mathcal{Y} \to \mathcal{S}$ be morphisms of categories over $\mathcal{C}$. For any object $U$ of $\mathcal{C}$ we have the following identity of fibre categories $$ \left(\mathcal{X} \times_\mathcal{S}\mathcal{Y}\right)_U = \mathcal{X}_U \times_{\mathcal{S}_U} \mathcal{Y}_U $$


categories-lemma-composition-cartesian

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. \begin{enumerate} \item The composition of two strongly cartesian morphisms is strongly cartesian. \item Any isomorphism of $\mathcal{S}$ is strongly cartesian. \item Any strongly cartesian morphism $\varphi$ such that $p(\varphi)$ is an isomorphism, is an isomorphism. \end{enumerate}


categories-lemma-cartesian-over-cartesian

Let $F : \mathcal{A} \to \mathcal{B}$ and $G : \mathcal{B} \to \mathcal{C}$ be composable functors between categories. Let $x \to y$ be a morphism of $\mathcal{A}$. If $x \to y$ is strongly $\mathcal{B}$-cartesian and $F(x) \to F(y)$ is strongly $\mathcal{C}$-cartesian, then $x \to y$ is strongly $\mathcal{C}$-cartesian.


categories-lemma-strongly-cartesian-fibre-product

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category over $\mathcal{C}$. Let $x \to y$ and $z \to y$ be morphisms of $\mathcal{S}$. Assume \begin{enumerate} \item $x \to y$ is strongly cartesian, \item $p(x) \times_{p(y)} p(z)$ exists, and \item there exists a strongly cartesian morphism $a : w \to z$ in $\mathcal{S}$ with $p(w) = p(x) \times_{p(y)} p(z)$ and $p(a) = \text{pr}_2 : p(x) \times_{p(y)} p(z) \to p(z)$. \end{enumerate} Then the fibre product $x \times_y z$ exists and is isomorphic to $w$.


categories-lemma-fibred

Assume $p : \mathcal{S} \to \mathcal{C}$ is a fibred category. Assume given a choice of pullbacks for $p : \mathcal{S} \to \mathcal{C}$. \begin{enumerate} \item For any pair of composable morphisms $f : V \to U$, $g : W \to V$ there is a unique isomorphism $$ \alpha_{g, f} : (f \circ g)^\ast \longrightarrow g^\ast \circ f^\ast $$ as functors $\mathcal{S}_U \to \mathcal{S}_W$ such that for every $y\in \Ob(\mathcal{S}_U)$ the following diagram commutes $$ \xymatrix{ g^\ast f^\ast y \ar[r] & f^\ast y \ar[d] \\ (f \circ g)^\ast y \ar[r] \ar[u]^{(\alpha_{g, f})_y} & y } $$ \item If $f = \text{id}_U$, then there is a canonical isomorphism $\alpha_U : \text{id} \to (\text{id}_U)^*$ as functors $\mathcal{S}_U \to \mathcal{S}_U$. \item The quadruple $(U \mapsto \mathcal{S}_U, f \mapsto f^*, \alpha_{g, f}, \alpha_U)$ defines a pseudo functor from $\mathcal{C}^{opp}$ to the $(2, 1)$-category of categories, see Definition \ref{definition-functor-into-2-category}. \end{enumerate}


categories-lemma-fibred-equivalent

Let $\mathcal{C}$ be a category. Let $\mathcal{S}_1$, $\mathcal{S}_2$ be categories over $\mathcal{C}$. Suppose that $\mathcal{S}_1$ and $\mathcal{S}_2$ are equivalent as categories over $\mathcal{C}$. Then $\mathcal{S}_1$ is fibred over $\mathcal{C}$ if and only if $\mathcal{S}_2$ is fibred over $\mathcal{C}$.


categories-lemma-2-product-fibred-categories-over-C

Let $\mathcal{C}$ be a category. The $(2, 1)$-category of fibred categories over $\mathcal{C}$ has 2-fibre products, and they are described as in Lemma \ref{lemma-2-product-categories-over-C}.


categories-lemma-cute

Let $\mathcal{C}$ be a category. Let $U \in \Ob(\mathcal{C})$. If $p : \mathcal{S} \to \mathcal{C}$ is a fibred category and $p$ factors through $p' : \mathcal{S} \to \mathcal{C}/U$ then $p' : \mathcal{S} \to \mathcal{C}/U$ is a fibred category.


categories-lemma-fibred-over-fibred

Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be functors between categories. If $\mathcal{A}$ is fibred over $\mathcal{B}$ and $\mathcal{B}$ is fibred over $\mathcal{C}$, then $\mathcal{A}$ is fibred over $\mathcal{C}$.


categories-lemma-fibred-category-representable-goes-up

Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $x \to y$ and $z \to y$ be morphisms of $\mathcal{S}$ with $x \to y$ strongly cartesian. If $p(x) \times_{p(y)} p(z)$ exists, then $x \times_y z$ exists, $p(x \times_y z) = p(x) \times_{p(y)} p(z)$, and $x \times_y z \to z$ is strongly cartesian.


categories-lemma-ameliorate-morphism-fibred-categories

Let $\mathcal{C}$ be a category. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of fibred categories over $\mathcal{C}$. There exist $1$-morphisms of fibred categories over $\mathcal{C}$ $$ \xymatrix{ \mathcal{X} \ar@<1ex>[r]^u & \mathcal{X}' \ar[r]^v \ar@<1ex>[l]^w & \mathcal{Y} } $$ such that $F = v \circ u$ and such that \begin{enumerate} \item $u : \mathcal{X} \to \mathcal{X}'$ is fully faithful, \item $w$ is left adjoint to $u$, and \item $v : \mathcal{X}' \to \mathcal{Y}$ is a fibred category. \end{enumerate}


categories-lemma-inertia-fibred-category

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ and $p' : \mathcal{S}' \to \mathcal{C}$ be fibred categories. Let $F : \mathcal{S} \to \mathcal{S}'$ be a $1$-morphism of fibred categories over $\mathcal{C}$. Consider the category $\mathcal{I}_{\mathcal{S}/\mathcal{S}'}$ over $\mathcal{C}$ whose \begin{enumerate} \item objects are pairs $(x, \alpha)$ where $x \in \Ob(\mathcal{S})$ and $\alpha : x \to x$ is an automorphism with $F(\alpha) = \text{id}$, \item morphisms $(x, \alpha) \to (y, \beta)$ are given by morphisms $\phi : x \to y$ such that $$ \xymatrix{ x\ar[r]_\phi\ar[d]_\alpha & y\ar[d]^{\beta} \\ x\ar[r]^\phi & y \\ } $$ commutes, and \item the functor $\mathcal{I}_{\mathcal{S}/\mathcal{S}'} \to \mathcal{C}$ is given by $(x, \alpha) \mapsto p(x)$. \end{enumerate} Then \begin{enumerate} \item there is an equivalence $$ \mathcal{I}_{\mathcal{S}/\mathcal{S}'} \longrightarrow \mathcal{S} \times_{\Delta, (\mathcal{S} \times_{\mathcal{S}'} \mathcal{S}), \Delta} \mathcal{S} $$ in the $(2, 1)$-category of categories over $\mathcal{C}$, and \item $\mathcal{I}_{\mathcal{S}/\mathcal{S}'}$ is a fibred category over $\mathcal{C}$. \end{enumerate}


categories-lemma-relative-inertia-as-fibre-product

Let $F : \mathcal{S} \to \mathcal{S}'$ be a $1$-morphism of categories fibred over a category $\mathcal{C}$. Then the diagram $$ \xymatrix{ \mathcal{I}_{\mathcal{S}/\mathcal{S}'} \ar[d]_{F \circ (\ref{equation-inertia-structure-map})} \ar[rr]_{(\ref{equation-comparison})} & & \mathcal{I}_\mathcal{S} \ar[d]^{(\ref{equation-functorial})} \\ \mathcal{S}' \ar[rr]^e & & \mathcal{I}_{\mathcal{S}'} } $$ is a $2$-fibre product.


categories-lemma-fibred-groupoids

Let $p : \mathcal{S} \to \mathcal{C}$ be a functor. The following are equivalent \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in groupoids, and \item all fibre categories are groupoids and $\mathcal{S}$ is a fibred category over $\mathcal{C}$. \end{enumerate} Moreover, in this case every morphism of $\mathcal{S}$ is strongly cartesian. In addition, given $f^\ast x \to x$ lying over $f$ for all $f: V \to U = p(x)$ the data $(U \mapsto \mathcal{S}_U, f \mapsto f^*, \alpha_{f, g}, \alpha_U)$ constructed in Lemma \ref{lemma-fibred} defines a pseudo functor from $\mathcal{C}^{opp}$ in to the $(2, 1)$-category of groupoids.


categories-lemma-fibred-gives-fibred-groupoids

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a fibred category. Let $\mathcal{S}'$ be the subcategory of $\mathcal{S}$ defined as follows \begin{enumerate} \item $\Ob(\mathcal{S}') = \Ob(\mathcal{S})$, and \item for $x, y \in \Ob(\mathcal{S}')$ the set of morphisms between $x$ and $y$ in $\mathcal{S}'$ is the set of strongly cartesian morphisms between $x$ and $y$ in $\mathcal{S}$. \end{enumerate} Let $p' : \mathcal{S}' \to \mathcal{C}$ be the restriction of $p$ to $\mathcal{S}'$. Then $p' : \mathcal{S}' \to \mathcal{C}$ is fibred in groupoids.


categories-lemma-2-product-fibred-categories

Let $\mathcal{C}$ be a category. The $2$-category of categories fibred in groupoids over $\mathcal{C}$ has 2-fibre products, and they are described as in Lemma \ref{lemma-2-product-categories-over-C}.


categories-lemma-equivalence-fibred-categories

Let $p : \mathcal{S}\to \mathcal{C}$ and $p' : \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids, and suppose that $G : \mathcal{S}\to \mathcal {S}'$ is a functor over $\mathcal{C}$. \begin{enumerate} \item Then $G$ is faithful (resp.\ fully faithful, resp.\ an equivalence) if and only if for each $U\in\Ob(\mathcal{C})$ the induced functor $G_U : \mathcal{S}_U\to \mathcal{S}'_U$ is faithful (resp.\ fully faithful, resp.\ an equivalence). \item If $G$ is an equivalence, then $G$ is an equivalence in the $2$-category of categories fibred in groupoids over $\mathcal{C}$. \end{enumerate}


categories-lemma-fully-faithful-diagonal-equivalence

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S}\to \mathcal{C}$ and $p' : \mathcal{S'}\to \mathcal{C}$ be categories fibred in groupoids. Let $G : \mathcal{S}\to \mathcal {S}'$ be a functor over $\mathcal{C}$. Then $G$ is fully faithful if and only if the diagonal $$ \Delta_G : \mathcal{S} \longrightarrow \mathcal{S} \times_{G, \mathcal{S}', G} \mathcal{S} $$ is an equivalence.


categories-lemma-morphisms-equivalent-fibred-groupoids

Let $\mathcal{C}$ be a category. Let $\mathcal{S}_i$, $i = 1, 2, 3, 4$ be categories fibred in groupoids over $\mathcal{C}$. Suppose that $\varphi : \mathcal{S}_1 \to \mathcal{S}_2$ and $\psi : \mathcal{S}_3 \to \mathcal{S}_4$ are equivalences over $\mathcal{C}$. Then $$ \Mor_{\textit{Cat}/\mathcal{C}}(\mathcal{S}_2, \mathcal{S}_3) \longrightarrow \Mor_{\textit{Cat}/\mathcal{C}}(\mathcal{S}_1, \mathcal{S}_4), \quad \alpha \longmapsto \psi \circ \alpha \circ \varphi $$ is an equivalence of categories.


categories-lemma-cute-groupoids

Let $\mathcal{C}$ be a category. Let $U \in \Ob(\mathcal{C})$. If $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in groupoids and $p$ factors through $p' : \mathcal{S} \to \mathcal{C}/U$ then $p' : \mathcal{S} \to \mathcal{C}/U$ is fibred in groupoids.


categories-lemma-fibred-in-groupoids-over-fibred-in-groupoids

Let $\mathcal{A} \to \mathcal{B} \to \mathcal{C}$ be functors between categories. If $\mathcal{A}$ is fibred in groupoids over $\mathcal{B}$ and $\mathcal{B}$ is fibred in groupoids over $\mathcal{C}$, then $\mathcal{A}$ is fibred in groupoids over $\mathcal{C}$.


categories-lemma-fibred-groupoids-fibre-product-goes-up

Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. Let $x \to y$ and $z \to y$ be morphisms of $\mathcal{S}$. If $p(x) \times_{p(y)} p(z)$ exists, then $x \times_y z$ exists and $p(x \times_y z) = p(x) \times_{p(y)} p(z)$.


categories-lemma-ameliorate-morphism-categories-fibred-groupoids

Let $\mathcal{C}$ be a category. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. There exists a factorization $\mathcal{X} \to \mathcal{X}' \to \mathcal{Y}$ by $1$-morphisms of categories fibred in groupoids over $\mathcal{C}$ such that $\mathcal{X} \to \mathcal{X}'$ is an equivalence over $\mathcal{C}$ and such that $\mathcal{X}'$ is a category fibred in groupoids over $\mathcal{Y}$.


categories-lemma-amelioration-unique

Let $\mathcal{C}$ be a category. Let $F : \mathcal{X} \to \mathcal{Y}$ be a $1$-morphism of categories fibred in groupoids over $\mathcal{C}$. Assume we have a $2$-commutative diagram $$ \xymatrix{ \mathcal{X}' \ar[rd]_f & \mathcal{X} \ar[l]^a \ar[d]^F \ar[r]_b & \mathcal{X}'' \ar[ld]^g \\ & \mathcal{Y} } $$ where $a$ and $b$ are equivalences of categories over $\mathcal{C}$ and $f$ and $g$ are categories fibred in groupoids. Then there exists an equivalence $h : \mathcal{X}'' \to \mathcal{X}'$ of categories over $\mathcal{Y}$ such that $h \circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\mathcal{C}$. If the diagram above actually commutes, then we can arrange it so that $h \circ b$ is $2$-isomorphic to $a$ as $1$-morphisms of categories over $\mathcal{Y}$.


categories-lemma-when-split

Let $\mathcal{C}$ be a category. Let $\mathcal{S}$ be a fibred category over $\mathcal{C}$. Then $\mathcal{S}$ is split if and only if for some choice of pullbacks (see Definition \ref{definition-pullback-functor-fibred-category}) the pullback functors $(f \circ g)^*$ and $g^* \circ f^*$ are equal.


categories-lemma-fibred-strict

Let $ p : \mathcal{S} \to \mathcal{C}$ be a fibred category. There exists a contravariant functor $F : \mathcal{C} \to \textit{Cat}$ such that $\mathcal{S}$ is equivalent to $\mathcal{S}_F$ in the $2$-category of fibred categories over $\mathcal{C}$. In other words, every fibred category is equivalent to a split one.


categories-lemma-fibred-groupoids-strict

Let $ p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. There exists a contravariant functor $F : \mathcal{C} \to \textit{Groupoids}$ such that $\mathcal{S}$ is equivalent to $\mathcal{S}_F$ over $\mathcal{C}$. In other words, every category fibred in groupoids is equivalent to a split one.


categories-lemma-2-product-categories-fibred-sets

Let $\mathcal{C}$ be a category. The 2-category of categories fibred in sets over $\mathcal{C}$ has 2-fibre products. More precisely, the 2-fibre product described in Lemma \ref{lemma-2-product-categories-over-C} returns a category fibred in sets if one starts out with such.


categories-lemma-2-category-fibred-sets

\begin{slogan} Categories fibred in sets are precisely presheaves. \end{slogan} Let $\mathcal{C}$ be a category. The only $2$-morphisms between categories fibred in sets are identities. In other words, the $2$-category of categories fibred in sets is a category. Moreover, there is an equivalence of categories $$ \left\{ \begin{matrix} \text{the category of presheaves}\\ \text{of sets over }\mathcal{C} \end{matrix} \right\} \leftrightarrow \left\{ \begin{matrix} \text{the category of categories}\\ \text{fibred in sets over }\mathcal{C} \end{matrix} \right\} $$ The functor from left to right is the construction $F \to \mathcal{S}_F$ discussed in Example \ref{example-presheaf}. The functor from right to left assigns to $p : \mathcal{S} \to \mathcal{C}$ the presheaf of objects $U \mapsto \Ob(\mathcal{S}_U)$.


categories-lemma-2-product-categories-fibred-setoids

Let $\mathcal{C}$ be a category. The 2-category of categories fibred in setoids over $\mathcal{C}$ has 2-fibre products. More precisely, the 2-fibre product described in Lemma \ref{lemma-2-product-categories-over-C} returns a category fibred in setoids if one starts out with such.


categories-lemma-setoid-fibres

Let $\mathcal{C}$ be a category. Let $\mathcal{S}$ be a category over $\mathcal{C}$. \begin{enumerate} \item If $\mathcal{S} \to \mathcal{S}'$ is an equivalence over $\mathcal{C}$ with $\mathcal{S}'$ fibred in sets over $\mathcal{C}$, then \begin{enumerate} \item $\mathcal{S}$ is fibred in setoids over $\mathcal{C}$, and \item for each $U \in \Ob(\mathcal{C})$ the map $\Ob(\mathcal{S}_U) \to \Ob(\mathcal{S}'_U)$ identifies the target as the set of isomorphism classes of the source. \end{enumerate} \item If $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in setoids, then there exists a category fibred in sets $p' : \mathcal{S}' \to \mathcal{C}$ and an equivalence $\text{can} : \mathcal{S} \to \mathcal{S}'$ over $\mathcal{C}$. \end{enumerate}


categories-lemma-2-category-fibred-setoids

Let $\mathcal{C}$ be a category. The construction of Lemma \ref{lemma-setoid-fibres} part (2) gives a functor $$ F : \left\{ \begin{matrix} \text{the 2-category of categories}\\ \text{fibred in setoids over }\mathcal{C} \end{matrix} \right\} \longrightarrow \left\{ \begin{matrix} \text{the category of categories}\\ \text{fibred in sets over }\mathcal{C} \end{matrix} \right\} $$ (see Definition \ref{definition-functor-into-2-category}). This functor is an equivalence in the following sense: \begin{enumerate} \item for any two 1-morphisms $f, g : \mathcal{S}_1 \to \mathcal{S}_2$ with $F(f) = F(g)$ there exists a unique 2-isomorphism $f \to g$, \item for any morphism $h : F(\mathcal{S}_1) \to F(\mathcal{S}_2)$ there exists a 1-morphism $f : \mathcal{S}_1 \to \mathcal{S}_2$ with $F(f) = h$, and \item any category fibred in sets $\mathcal{S}$ is equal to $F(\mathcal{S})$. \end{enumerate} In particular, defining $F_i \in \textit{PSh}(\mathcal{C})$ by the rule $F_i(U) = \Ob(\mathcal{S}_{i, U})/\cong$, we have $$ \Mor_{\textit{Cat}/\mathcal{C}}(\mathcal{S}_1, \mathcal{S}_2) \Big/ 2\text{-isomorphism} = \Mor_{\textit{PSh}(\mathcal{C})}(F_1, F_2) $$ More precisely, given any map $\phi : F_1 \to F_2$ there exists a $1$-morphism $f : \mathcal{S}_1 \to \mathcal{S}_2$ which induces $\phi$ on isomorphism classes of objects and which is unique up to unique $2$-isomorphism.


categories-lemma-characterize-fibred-setoids-inertia

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. The following are equivalent: \begin{enumerate} \item $p : \mathcal{S} \to \mathcal{C}$ is a category fibred in setoids, and \item the canonical $1$-morphism $\mathcal{I}_\mathcal{S} \to \mathcal{S}$, see (\ref{equation-inertia-structure-map}), is an equivalence (of categories over $\mathcal{C}$). \end{enumerate}


categories-lemma-category-fibred-setoids-presheaves-products

Let $\mathcal{C}$ be a category. The construction of Lemma \ref{lemma-2-category-fibred-setoids} which associates to a category fibred in setoids a presheaf is compatible with products, in the sense that the presheaf associated to a $2$-fibre product $\mathcal{X} \times_\mathcal{Y} \mathcal{Z}$ is the fibre product of the presheaves associated to $\mathcal{X}, \mathcal{Y}, \mathcal{Z}$.


categories-lemma-characterize-representable-fibred-category

Let $\mathcal{C}$ be a category. Let $p : \mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. \begin{enumerate} \item $\mathcal{S}$ is representable if and only if the following conditions are satisfied: \begin{enumerate} \item $\mathcal{S}$ is fibred in setoids, and \item the presheaf $U \mapsto \Ob(\mathcal{S}_U)/\cong$ is representable. \end{enumerate} \item If $\mathcal{S}$ is representable the pair $(X, j)$, where $j$ is the equivalence $j : \mathcal{S} \to \mathcal{C}/X$, is uniquely determined up to isomorphism. \end{enumerate}


categories-lemma-yoneda-2category

Let $\mathcal{S}\to \mathcal{C}$ be fibred in groupoids. Let $U \in \Ob(\mathcal{C})$. The functor $$ \Mor_{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \longrightarrow \mathcal{S}_U $$ given by $G \mapsto G(\text{id}_U)$ is an equivalence.


categories-lemma-identify-fibre-product

In the situation above the fibre category of $(\mathcal{C}/U) \times_\mathcal{Y} \mathcal{X}$ over an object $f : V \to U$ of $\mathcal{C}/U$ is the category described as follows: \begin{enumerate} \item objects are pairs $(x, \phi)$, where $x \in \Ob(\mathcal{X}_V)$, and $\phi : f^*y \to F(x)$ is a morphism in $\mathcal{Y}_V$, \item the set of morphisms between $(x, \phi)$ and $(x', \phi')$ is the set of morphisms $\psi : x \to x'$ in $\mathcal{X}_V$ such that $F(\psi) = \phi' \circ \phi^{-1}$. \end{enumerate}


categories-lemma-representable-diagonal-groupoids

Let $\mathcal{C}$ be a category. Let $\mathcal{S} \to \mathcal{C}$ be a category fibred in groupoids. Assume $\mathcal{C}$ has products of pairs of objects and fibre products. The following are equivalent: \begin{enumerate} \item The diagonal $\mathcal{S} \to \mathcal{S} \times \mathcal{S}$ is representable. \item For every $U$ in $\mathcal{C}$, any $G : \mathcal{C}/U \to \mathcal{S}$ is representable. \end{enumerate}


categories-lemma-invertible

Let $\mathcal{C}$ be a monoidal category. Let $X$ be an object of $\mathcal{C}$. The following are equivalent \begin{enumerate} \item the functor $L : Y \mapsto X \otimes Y$ is an equivalence, \item the functor $R : Y \mapsto Y \otimes X$ is an equivalence, \item there exists an object $X'$ such that $X \otimes X' \cong X' \otimes X \cong \mathbf{1}$. \end{enumerate}


categories-lemma-left-dual

Let $\mathcal{C}$ be a monoidal category. If $Y$ is a left dual to $X$, then $$ \Mor(Z' \otimes X, Z) = \Mor(Z', Z \otimes Y) \quad\text{and}\quad \Mor(Y \otimes Z', Z) = \Mor(Z', X \otimes Z) $$ functorially in $Z$ and $Z'$.


categories-lemma-tensor-dual

Let $\mathcal{C}$ be a monoidal category. If $Y_i$, $i = 1, 2$ are left duals of $X_i$, $i = 1, 2$, then $Y_2 \otimes Y_1$ is a left dual of $X_1 \otimes X_2$.


categories-lemma-dual-symmetric

Let $(\mathcal{C}, \otimes, \phi, \psi)$ be a symmetric monoidal category. Let $X$ be an object of $\mathcal{C}$ and let $Y$, $\eta : \mathbf{1} \to X \otimes Y$, and $\epsilon : Y \otimes X \to \mathbf{1}$ be a left dual of $X$ as in Definition \ref{definition-dual}. Then $\eta' = \psi \circ \eta : \mathbf{1} \to Y \otimes X$ and $\epsilon' = \epsilon \circ \psi : X \otimes Y \to \mathbf{1}$ makes $X$ into a left dual of $Y$.


topologies-lemma-zariski

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is a Zariski covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is a Zariski covering and for each $i$ we have a Zariski covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a Zariski covering. \item If $\{T_i \to T\}_{i\in I}$ is a Zariski covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is a Zariski covering. \end{enumerate}


topologies-lemma-zariski-affine

Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be a Zariski covering of $T$. Then there exists a Zariski covering $\{U_j \to T\}_{j = 1, \ldots, m}$ which is a refinement of $\{T_i \to T\}_{i \in I}$ such that each $U_j$ is a standard open of $T$, see Schemes, Definition \ref{schemes-definition-standard-covering}. Moreover, we may choose each $U_j$ to be an open of one of the $T_i$.


topologies-lemma-zariski-induced

Let $\Sch_{Zar}$ be a big Zariski site as in Definition \ref{definition-big-zariski-site}. Let $T \in \Ob(\Sch_{Zar})$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary Zariski covering of $T$. There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{Zar}$ which is tautologically equivalent (see Sites, Definition \ref{sites-definition-combinatorial-tautological}) to $\{T_i \to T\}_{i \in I}$.


topologies-lemma-fibre-products-Zariski

Let $S$ be a scheme. Let $\Sch_{Zar}$ be a big Zariski site containing $S$. The underlying categories of the sites $\Sch_{Zar}$, $(\Sch/S)_{Zar}$, $S_{Zar}$, and $(\textit{Aff}/S)_{Zar}$ have fibre products. In each case the obvious functor into the category $\Sch$ of all schemes commutes with taking fibre products. The categories $(\Sch/S)_{Zar}$, and $S_{Zar}$ both have a final object, namely $S/S$.


topologies-lemma-affine-big-site-Zariski

Let $S$ be a scheme. Let $\Sch_{Zar}$ be a big Zariski site containing $S$. The functor $(\textit{Aff}/S)_{Zar} \to (\Sch/S)_{Zar}$ is a special cocontinuous functor. Hence it induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_{Zar})$ to $\Sh((\Sch/S)_{Zar})$.


topologies-lemma-Zariski-usual

The category of sheaves on $S_{Zar}$ is equivalent to the category of sheaves on the underlying topological space of $S$.


topologies-lemma-put-in-T

Let $\Sch_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\Sch_{Zar}$. The functor $T_{Zar} \to (\Sch/S)_{Zar}$ is cocontinuous and induces a morphism of topoi $$ i_f : \Sh(T_{Zar}) \longrightarrow \Sh((\Sch/S)_{Zar}) $$ For a sheaf $\mathcal{G}$ on $(\Sch/S)_{Zar}$ we have the formula $(i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.


topologies-lemma-at-the-bottom

Let $S$ be a scheme. Let $\Sch_{Zar}$ be a big Zariski site containing $S$. The inclusion functor $S_{Zar} \to (\Sch/S)_{Zar}$ satisfies the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \pi_S : (\Sch/S)_{Zar} \longrightarrow S_{Zar} $$ and a morphism of topoi $$ i_S : \Sh(S_{Zar}) \longrightarrow \Sh((\Sch/S)_{Zar}) $$ such that $\pi_S \circ i_S = \text{id}$. Moreover, $i_S = i_{\text{id}_S}$ with $i_{\text{id}_S}$ as in Lemma \ref{lemma-put-in-T}. In particular the functor $i_S^{-1} = \pi_{S, *}$ is described by the rule $i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.


topologies-lemma-morphism-big

Let $\Sch_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\Sch_{Zar}$. The functor $$ u : (\Sch/T)_{Zar} \longrightarrow (\Sch/S)_{Zar}, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_{Zar} \longrightarrow (\Sch/T)_{Zar}, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_{Zar}) \longrightarrow \Sh((\Sch/S)_{Zar}) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


topologies-lemma-morphism-big-small

Let $\Sch_{Zar}$ be a big Zariski site. Let $f : T \to S$ be a morphism in $\Sch_{Zar}$. \begin{enumerate} \item We have $i_f = f_{big} \circ i_T$ with $i_f$ as in Lemma \ref{lemma-put-in-T} and $i_T$ as in Lemma \ref{lemma-at-the-bottom}. \item The functor $S_{Zar} \to T_{Zar}$, $(U \to S) \mapsto (U \times_S T \to T)$ is continuous and induces a morphism of topoi $$ f_{small} : \Sh(T_{Zar}) \longrightarrow \Sh(S_{Zar}). $$ The functors $f_{small}^{-1}$ and $f_{small, *}$ agree with the usual notions $f^{-1}$ and $f_*$ is we identify sheaves on $T_{Zar}$, resp.\ $S_{Zar}$ with sheaves on $T$, resp.\ $S$ via Lemma \ref{lemma-Zariski-usual}. \item We have a commutative diagram of morphisms of sites $$ \xymatrix{ T_{Zar} \ar[d]_{f_{small}} & (\Sch/T)_{Zar} \ar[d]^{f_{big}} \ar[l]^{\pi_T} \\ S_{Zar} & (\Sch/S)_{Zar} \ar[l]_{\pi_S} } $$ so that $f_{small} \circ \pi_T = \pi_S \circ f_{big}$ as morphisms of topoi. \item We have $f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f$. \end{enumerate}


topologies-lemma-zariski-etale

Any Zariski covering is an \'etale covering.


topologies-lemma-etale

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is an \'etale covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is an \'etale covering and for each $i$ we have an \'etale covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is an \'etale covering. \item If $\{T_i \to T\}_{i\in I}$ is an \'etale covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is an \'etale covering. \end{enumerate}


topologies-lemma-etale-affine

Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be an \'etale covering of $T$. Then there exists an \'etale covering $\{U_j \to T\}_{j = 1, \ldots, m}$ which is a refinement of $\{T_i \to T\}_{i \in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$.


topologies-lemma-etale-induced

Let $\Sch_\etale$ be a big \'etale site as in Definition \ref{definition-big-etale-site}. Let $T \in \Ob(\Sch_\etale)$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary \'etale covering of $T$. \begin{enumerate} \item There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_\etale$ which refines $\{T_i \to T\}_{i \in I}$. \item If $\{T_i \to T\}_{i \in I}$ is a standard \'etale covering, then it is tautologically equivalent to a covering in $\Sch_\etale$. \item If $\{T_i \to T\}_{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering in $\Sch_\etale$. \end{enumerate}


topologies-lemma-verify-site-etale

Let $S$ be a scheme. Let $\Sch_\etale$ be a big \'etale site containing $S$. Both $S_\etale$ and $(\textit{Aff}/S)_\etale$ are sites.


topologies-lemma-fibre-products-etale

Let $S$ be a scheme. Let $\Sch_\etale$ be a big \'etale site containing $S$. The underlying categories of the sites $\Sch_\etale$, $(\Sch/S)_\etale$, $S_\etale$, and $(\textit{Aff}/S)_\etale$ have fibre products. In each case the obvious functor into the category $\Sch$ of all schemes commutes with taking fibre products. The categories $(\Sch/S)_\etale$, and $S_\etale$ both have a final object, namely $S/S$.


topologies-lemma-affine-big-site-etale

Let $S$ be a scheme. Let $\Sch_\etale$ be a big \'etale site containing $S$. The functor $(\textit{Aff}/S)_\etale \to (\Sch/S)_\etale$ is special cocontinuous and induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_\etale)$ to $\Sh((\Sch/S)_\etale)$.


topologies-lemma-put-in-T-etale

Let $\Sch_\etale$ be a big \'etale site. Let $f : T \to S$ be a morphism in $\Sch_\etale$. The functor $T_\etale \to (\Sch/S)_\etale$ is cocontinuous and induces a morphism of topoi $$ i_f : \Sh(T_\etale) \longrightarrow \Sh((\Sch/S)_\etale) $$ For a sheaf $\mathcal{G}$ on $(\Sch/S)_\etale$ we have the formula $(i_f^{-1}\mathcal{G})(U/T) = \mathcal{G}(U/S)$. The functor $i_f^{-1}$ also has a left adjoint $i_{f, !}$ which commutes with fibre products and equalizers.


topologies-lemma-at-the-bottom-etale

Let $S$ be a scheme. Let $\Sch_\etale$ be a big \'etale site containing $S$. The inclusion functor $S_\etale \to (\Sch/S)_\etale$ satisfies the hypotheses of Sites, Lemma \ref{sites-lemma-bigger-site} and hence induces a morphism of sites $$ \pi_S : (\Sch/S)_\etale \longrightarrow S_\etale $$ and a morphism of topoi $$ i_S : \Sh(S_\etale) \longrightarrow \Sh((\Sch/S)_\etale) $$ such that $\pi_S \circ i_S = \text{id}$. Moreover, $i_S = i_{\text{id}_S}$ with $i_{\text{id}_S}$ as in Lemma \ref{lemma-put-in-T-etale}. In particular the functor $i_S^{-1} = \pi_{S, *}$ is described by the rule $i_S^{-1}(\mathcal{G})(U/S) = \mathcal{G}(U/S)$.


topologies-lemma-morphism-big-etale

Let $\Sch_\etale$ be a big \'etale site. Let $f : T \to S$ be a morphism in $\Sch_\etale$. The functor $$ u : (\Sch/T)_\etale \longrightarrow (\Sch/S)_\etale, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_\etale \longrightarrow (\Sch/T)_\etale, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_\etale) \longrightarrow \Sh((\Sch/S)_\etale) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


topologies-lemma-morphism-big-small-etale

Let $\Sch_\etale$ be a big \'etale site. Let $f : T \to S$ be a morphism in $\Sch_\etale$. \begin{enumerate} \item We have $i_f = f_{big} \circ i_T$ with $i_f$ as in Lemma \ref{lemma-put-in-T-etale} and $i_T$ as in Lemma \ref{lemma-at-the-bottom-etale}. \item The functor $S_\etale \to T_\etale$, $(U \to S) \mapsto (U \times_S T \to T)$ is continuous and induces a morphism of sites $$ f_{small} : T_\etale \longrightarrow S_\etale $$ We have $f_{small, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. \item We have a commutative diagram of morphisms of sites $$ \xymatrix{ T_\etale \ar[d]_{f_{small}} & (\Sch/T)_\etale \ar[d]^{f_{big}} \ar[l]^{\pi_T}\\ S_\etale & (\Sch/S)_\etale \ar[l]_{\pi_S} } $$ so that $f_{small} \circ \pi_T = \pi_S \circ f_{big}$ as morphisms of topoi. \item We have $f_{small} = \pi_S \circ f_{big} \circ i_T = \pi_S \circ i_f$. \end{enumerate}


topologies-lemma-morphism-big-small-cartesian-diagram-etale

Let $\Sch_\etale$ be a big \'etale site. Consider a cartesian diagram $$ \xymatrix{ T' \ar[r]_{g'} \ar[d]_{f'} & T \ar[d]^f \\ S' \ar[r]^g & S } $$ in $\Sch_\etale$. Then $i_g^{-1} \circ f_{big, *} = f'_{small, *} \circ (i_{g'})^{-1}$ and $g_{big}^{-1} \circ f_{big, *} = f'_{big, *} \circ (g'_{big})^{-1}$.


topologies-lemma-characterize-sheaf-big-etale

Let $S$ be a scheme contained in a big \'etale site $\Sch_\etale$. A sheaf $\mathcal{F}$ on the big \'etale site $(\Sch/S)_\etale$ is given by the following data: \begin{enumerate} \item for every $T/S \in \Ob((\Sch/S)_\etale)$ a sheaf $\mathcal{F}_T$ on $T_\etale$, \item for every $f : T' \to T$ in $(\Sch/S)_\etale$ a map $c_f : f_{small}^{-1}\mathcal{F}_T \to \mathcal{F}_{T'}$. \end{enumerate} These data are subject to the following conditions: \begin{enumerate} \item[(a)] given any $f : T' \to T$ and $g : T'' \to T'$ in $(\Sch/S)_\etale$ the composition $c_g \circ g_{small}^{-1}c_f$ is equal to $c_{f \circ g}$, and \item[(b)] if $f : T' \to T$ in $(\Sch/S)_\etale$ is \'etale then $c_f$ is an isomorphism. \end{enumerate}


topologies-lemma-zariski-etale-smooth

Any \'etale covering is a smooth covering, and a fortiori, any Zariski covering is a smooth covering.


topologies-lemma-smooth

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is a smooth covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is a smooth covering and for each $i$ we have a smooth covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a smooth covering. \item If $\{T_i \to T\}_{i\in I}$ is a smooth covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is a smooth covering. \end{enumerate}


topologies-lemma-smooth-affine

Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be a smooth covering of $T$. Then there exists a smooth covering $\{U_j \to T\}_{j = 1, \ldots, m}$ which is a refinement of $\{T_i \to T\}_{i \in I}$ such that each $U_j$ is an affine scheme, and such that each morphism $U_j \to T$ is standard smooth, see Morphisms, Definition \ref{morphisms-definition-smooth}. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$.


topologies-lemma-smooth-induced

Let $\Sch_{smooth}$ be a big smooth site as in Definition \ref{definition-big-smooth-site}. Let $T \in \Ob(\Sch_{smooth})$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary smooth covering of $T$. \begin{enumerate} \item There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{smooth}$ which refines $\{T_i \to T\}_{i \in I}$. \item If $\{T_i \to T\}_{i \in I}$ is a standard smooth covering, then it is tautologically equivalent to a covering of $\Sch_{smooth}$. \item If $\{T_i \to T\}_{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\Sch_{smooth}$. \end{enumerate}


topologies-lemma-affine-big-site-smooth

Let $S$ be a scheme. Let $\Sch_\etale$ be a big smooth site containing $S$. The functor $(\textit{Aff}/S)_{smooth} \to (\Sch/S)_{smooth}$ is special cocontinuous and induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_{smooth})$ to $\Sh((\Sch/S)_{smooth})$.


topologies-lemma-morphism-big-smooth

Let $\Sch_{smooth}$ be a big smooth site. Let $f : T \to S$ be a morphism in $\Sch_{smooth}$. The functor $$ u : (\Sch/T)_{smooth} \longrightarrow (\Sch/S)_{smooth}, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_{smooth} \longrightarrow (\Sch/T)_{smooth}, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_{smooth}) \longrightarrow \Sh((\Sch/S)_{smooth}) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


topologies-lemma-zariski-etale-smooth-syntomic

Any smooth covering is a syntomic covering, and a fortiori, any \'etale or Zariski covering is a syntomic covering.


topologies-lemma-syntomic

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is a syntomic covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is a syntomic covering and for each $i$ we have a syntomic covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a syntomic covering. \item If $\{T_i \to T\}_{i\in I}$ is a syntomic covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is a syntomic covering. \end{enumerate}


topologies-lemma-syntomic-affine

Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be a syntomic covering of $T$. Then there exists a syntomic covering $\{U_j \to T\}_{j = 1, \ldots, m}$ which is a refinement of $\{T_i \to T\}_{i \in I}$ such that each $U_j$ is an affine scheme, and such that each morphism $U_j \to T$ is standard syntomic, see Morphisms, Definition \ref{morphisms-definition-syntomic}. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$.


topologies-lemma-syntomic-induced

Let $\Sch_{syntomic}$ be a big syntomic site as in Definition \ref{definition-big-syntomic-site}. Let $T \in \Ob(\Sch_{syntomic})$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary syntomic covering of $T$. \begin{enumerate} \item There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{syntomic}$ which refines $\{T_i \to T\}_{i \in I}$. \item If $\{T_i \to T\}_{i \in I}$ is a standard syntomic covering, then it is tautologically equivalent to a covering in $\Sch_{syntomic}$. \item If $\{T_i \to T\}_{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering in $\Sch_{syntomic}$. \end{enumerate}


topologies-lemma-affine-big-site-syntomic

Let $S$ be a scheme. Let $\Sch_{syntomic}$ be a big syntomic site containing $S$. The functor $(\textit{Aff}/S)_{syntomic} \to (\Sch/S)_{syntomic}$ is special cocontinuous and induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_{syntomic})$ to $\Sh((\Sch/S)_{syntomic})$.


topologies-lemma-morphism-big-syntomic

Let $\Sch_{syntomic}$ be a big syntomic site. Let $f : T \to S$ be a morphism in $\Sch_{syntomic}$. The functor $$ u : (\Sch/T)_{syntomic} \longrightarrow (\Sch/S)_{syntomic}, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_{syntomic} \longrightarrow (\Sch/T)_{syntomic}, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_{syntomic}) \longrightarrow \Sh((\Sch/S)_{syntomic}) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


topologies-lemma-zariski-etale-smooth-syntomic-fppf

Any syntomic covering is an fppf covering, and a fortiori, any smooth, \'etale, or Zariski covering is an fppf covering.


topologies-lemma-fppf

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is an fppf covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is an fppf covering and for each $i$ we have an fppf covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is an fppf covering. \item If $\{T_i \to T\}_{i\in I}$ is an fppf covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is an fppf covering. \end{enumerate}


topologies-lemma-fppf-affine

Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be an fppf covering of $T$. Then there exists an fppf covering $\{U_j \to T\}_{j = 1, \ldots, m}$ which is a refinement of $\{T_i \to T\}_{i \in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$.


topologies-lemma-fppf-induced

Let $\Sch_{fppf}$ be a big fppf site as in Definition \ref{definition-big-fppf-site}. Let $T \in \Ob(\Sch_{fppf})$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary fppf covering of $T$. \begin{enumerate} \item There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{fppf}$ which refines $\{T_i \to T\}_{i \in I}$. \item If $\{T_i \to T\}_{i \in I}$ is a standard fppf covering, then it is tautologically equivalent to a covering of $\Sch_{fppf}$. \item If $\{T_i \to T\}_{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\Sch_{fppf}$. \end{enumerate}


topologies-lemma-fibre-products-fppf

Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$. The underlying categories of the sites $\Sch_{fppf}$, $(\Sch/S)_{fppf}$, and $(\textit{Aff}/S)_{fppf}$ have fibre products. In each case the obvious functor into the category $\Sch$ of all schemes commutes with taking fibre products. The category $(\Sch/S)_{fppf}$ has a final object, namely $S/S$.


topologies-lemma-affine-big-site-fppf

Let $S$ be a scheme. Let $\Sch_{fppf}$ be a big fppf site containing $S$. The functor $(\textit{Aff}/S)_{fppf} \to (\Sch/S)_{fppf}$ is cocontinuous and induces an equivalence of topoi from $\Sh((\textit{Aff}/S)_{fppf})$ to $\Sh((\Sch/S)_{fppf})$.


topologies-lemma-morphism-big-fppf

Let $\Sch_{fppf}$ be a big fppf site. Let $f : T \to S$ be a morphism in $\Sch_{fppf}$. The functor $$ u : (\Sch/T)_{fppf} \longrightarrow (\Sch/S)_{fppf}, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_{fppf} \longrightarrow (\Sch/T)_{fppf}, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_{fppf}) \longrightarrow \Sh((\Sch/S)_{fppf}) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


topologies-lemma-base-change-standard-ph

Let $\{f_j : U_j \to T\}_{j = 1, \ldots, m}$ be a standard ph covering. Let $T' \to T$ be a morphism of affine schemes. Then $\{U_j \times_T T' \to T'\}_{j = 1, \ldots, m}$ is a standard ph covering.


topologies-lemma-refine-by-standard-ph

Let $T$ be an affine scheme. Each of the following types of families of maps with target $T$ has a refinement by a standard ph covering: \begin{enumerate} \item any Zariski open covering of $T$, \item $\{W_{ji} \to T\}_{j = 1, \ldots, m, i = 1, \ldots n_j}$ where $\{W_{ji} \to U_j\}_{i = 1, \ldots, n_j}$ and $\{U_j \to T\}_{j = 1, \ldots, m}$ are standard ph coverings. \end{enumerate}


topologies-lemma-zariski-ph

A Zariski covering is a ph covering\footnote{We will see in More on Morphisms, Lemma \ref{more-morphisms-lemma-fppf-ph} that fppf coverings (and hence syntomic, smooth, or \'etale coverings) are ph coverings as well.}.


topologies-lemma-surjective-proper-ph

Let $f : Y \to X$ be a surjective proper morphism of schemes. Then $\{Y \to X\}$ is a ph covering.


topologies-lemma-refine-by-ph

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms such that $f_i$ is locally of finite type for all $i$. The following are equivalent \begin{enumerate} \item $\{T_i \to T\}_{i \in I}$ is a ph covering, \item there is a ph covering which refines $\{T_i \to T\}_{i \in I}$, and \item $\{\coprod_{i \in I} T_i \to T\}$ is a ph covering. \end{enumerate}


topologies-lemma-ph

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is a ph covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is a ph covering and for each $i$ we have a ph covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a ph covering. \item If $\{T_i \to T\}_{i\in I}$ is a ph covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is a ph covering. \end{enumerate}


topologies-lemma-ph-induced

Let $\Sch_{ph}$ be a big ph site as in Definition \ref{definition-big-ph-site}. Let $T \in \Ob(\Sch_{ph})$. Let $\{T_i \to T\}_{i \in I}$ be an arbitrary ph covering of $T$. \begin{enumerate} \item There exists a covering $\{U_j \to T\}_{j \in J}$ of $T$ in the site $\Sch_{ph}$ which refines $\{T_i \to T\}_{i \in I}$. \item If $\{T_i \to T\}_{i \in I}$ is a standard ph covering, then it is tautologically equivalent to a covering of $\Sch_{ph}$. \item If $\{T_i \to T\}_{i \in I}$ is a Zariski covering, then it is tautologically equivalent to a covering of $\Sch_{ph}$. \end{enumerate}


topologies-lemma-characterize-sheaf

Let $\mathcal{F}$ be a presheaf on $(\Sch/S)_{ph}$. Then $\mathcal{F}$ is a sheaf if and only if \begin{enumerate} \item $\mathcal{F}$ satisfies the sheaf condition for Zariski coverings, and \item if $f : V \to U$ is proper surjective, then $\mathcal{F}(U)$ maps bijectively to the equalizer of the two maps $\mathcal{F}(V) \to \mathcal{F}(V \times_U V)$. \end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \begin{enumerate} \item[(2')] the sheaf property for $\{V \to U\}$ as in (2) with $U$ affine. \end{enumerate}


topologies-lemma-morphism-big-ph

Let $\Sch_{ph}$ be a big ph site. Let $f : T \to S$ be a morphism in $\Sch_{ph}$. The functor $$ u : (\Sch/T)_{ph} \longrightarrow (\Sch/S)_{ph}, \quad V/T \longmapsto V/S $$ is cocontinuous, and has a continuous right adjoint $$ v : (\Sch/S)_{ph} \longrightarrow (\Sch/T)_{ph}, \quad (U \to S) \longmapsto (U \times_S T \to T). $$ They induce the same morphism of topoi $$ f_{big} : \Sh((\Sch/T)_{ph}) \longrightarrow \Sh((\Sch/S)_{ph}) $$ We have $f_{big}^{-1}(\mathcal{G})(U/T) = \mathcal{G}(U/S)$. We have $f_{big, *}(\mathcal{F})(U/S) = \mathcal{F}(U \times_S T/T)$. Also, $f_{big}^{-1}$ has a left adjoint $f_{big!}$ which commutes with fibre products and equalizers.


topologies-lemma-recognize-fpqc-covering

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent \begin{enumerate} \item $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering, \item each $f_i$ is flat and for every affine open $U \subset T$ there exist quasi-compact opens $U_i \subset T_i$ which are almost all empty, such that $U = \bigcup f_i(U_i)$, \item each $f_i$ is flat and there exists an affine open covering $T = \bigcup_{\alpha \in A} U_\alpha$ and for each $\alpha \in A$ there exist $i_{\alpha, 1}, \ldots, i_{\alpha, n(\alpha)} \in I$ and quasi-compact opens $U_{\alpha, j} \subset T_{i_{\alpha, j}}$ such that $U_\alpha = \bigcup_{j = 1, \ldots, n(\alpha)} f_{i_{\alpha, j}}(U_{\alpha, j})$. \end{enumerate} If $T$ is quasi-separated, these are also equivalent to \begin{enumerate} \item[(4)] each $f_i$ is flat, and for every $t \in T$ there exist $i_1, \ldots, i_n \in I$ and quasi-compact opens $U_j \subset T_{i_j}$ such that $\bigcup_{j = 1, \ldots, n} f_{i_j}(U_j)$ is a (not necessarily open) neighbourhood of $t$ in $T$. \end{enumerate}


topologies-lemma-disjoint-union-is-fpqc-covering

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with target $T$. The following are equivalent \begin{enumerate} \item $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering, and \item setting $T' = \coprod_{i \in I} T_i$, and $f = \coprod_{i \in I} f_i$ the family $\{f : T' \to T\}$ is an fpqc covering. \end{enumerate}


topologies-lemma-family-flat-dominated-covering

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with target $T$. Assume that \begin{enumerate} \item each $f_i$ is flat, and \item the family $\{f_i : T_i \to T\}_{i \in I}$ can be refined by an fpqc covering of $T$. \end{enumerate} Then $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering of $T$.


topologies-lemma-family-flat-fpqc-local-covering

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms of schemes with target $T$. Assume that \begin{enumerate} \item each $f_i$ is flat, and \item there exists an fpqc covering $\{g_j : S_j \to T\}_{j \in J}$ such that each $\{S_j \times_T T_i \to S_j\}_{i \in I}$ is an fpqc covering. \end{enumerate} Then $\{f_i : T_i \to T\}_{i \in I}$ is an fpqc covering of $T$.


topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc

Any fppf covering is an fpqc covering, and a fortiori, any syntomic, smooth, \'etale or Zariski covering is an fpqc covering.


topologies-lemma-fpqc

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is an fpqc covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is an fpqc covering and for each $i$ we have an fpqc covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is an fpqc covering. \item If $\{T_i \to T\}_{i\in I}$ is an fpqc covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is an fpqc covering. \end{enumerate}


topologies-lemma-fpqc-affine

Let $T$ be an affine scheme. Let $\{T_i \to T\}_{i \in I}$ be an fpqc covering of $T$. Then there exists an fpqc covering $\{U_j \to T\}_{j = 1, \ldots, n}$ which is a refinement of $\{T_i \to T\}_{i \in I}$ such that each $U_j$ is an affine scheme. Moreover, we may choose each $U_j$ to be open affine in one of the $T_i$.


topologies-lemma-fpqc-affine-axioms

Let $T$ be an affine scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is a standard fpqc covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is a standard fpqc covering and for each $i$ we have a standard fpqc covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a standard fpqc covering. \item If $\{T_i \to T\}_{i\in I}$ is a standard fpqc covering and $T' \to T$ is a morphism of affine schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is a standard fpqc covering. \end{enumerate}


topologies-lemma-sheaf-property-fpqc

Let $F$ be a contravariant functor on the category of schemes with values in sets. Then $F$ satisfies the sheaf property for the fpqc topology if and only if it satisfies \begin{enumerate} \item the sheaf property for every Zariski covering, and \item the sheaf property for any standard fpqc covering. \end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \begin{enumerate} \item[(2')] the sheaf property for $\{V \to U\}$ with $V$, $U$ affine and $V \to U$ faithfully flat. \end{enumerate}


topologies-lemma-no-set-of-fpqc-covers-is-initial

Let $R$ be a nonzero ring. There does not exist a set $A$ of fpqc-coverings of $\Spec(R)$ such that every fpqc-covering can be refined by an element of $A$.


topologies-lemma-standard-fpqc-standard-V

A standard fpqc covering is a standard V covering.


topologies-lemma-standard-ph-standard-V

A standard ph covering is a standard V covering.


topologies-lemma-base-change-standard-V

Let $\{T_j \to T\}_{j = 1, \ldots, m}$ be a standard V covering. Let $T' \to T$ be a morphism of affine schemes. Then $\{T_j \times_T T' \to T'\}_{j = 1, \ldots, m}$ is a standard V covering.


topologies-lemma-composition-standard-V

Let $T$ be an affine scheme. Let $\{T_j \to T\}_{j = 1, \ldots, m}$ be a standard V covering. Let $\{T_{ji} \to T_j\}_{i = 1, \ldots n_j}$ be a standard V covering. Then $\{T_{ji} \to T\}_{i, j}$ is a standard V covering.


topologies-lemma-refine-standard-V

Let $T$ be an affine scheme. Let $\{T_j \to T\}_{j = 1, \ldots, m}$ be a family of morphisms with $T_j$ affine for all $j$. The following are equivalent \begin{enumerate} \item $\{T_j \to T\}_{j = 1, \ldots, m}$ is a standard V covering, \item there is a standard V covering which refines $\{T_j \to T\}_{j = 1, \ldots, m}$, and \item $\{\coprod_{j = 1, \ldots, m} T_j \to T\}$ is a standard V covering. \end{enumerate}


topologies-lemma-refine-by-V

Let $T$ be a scheme. Let $\{f_i : T_i \to T\}_{i \in I}$ be a family of morphisms. The following are equivalent \begin{enumerate} \item $\{T_i \to T\}_{i \in I}$ is a V covering, \item there is a V covering which refines $\{T_i \to T\}_{i \in I}$, and \item $\{\coprod_{i \in I} T_i \to T\}$ is a V covering. \end{enumerate}


topologies-lemma-V

Let $T$ be a scheme. \begin{enumerate} \item If $T' \to T$ is an isomorphism then $\{T' \to T\}$ is a V covering of $T$. \item If $\{T_i \to T\}_{i\in I}$ is a V covering and for each $i$ we have a V covering $\{T_{ij} \to T_i\}_{j\in J_i}$, then $\{T_{ij} \to T\}_{i \in I, j\in J_i}$ is a V covering. \item If $\{T_i \to T\}_{i\in I}$ is a V covering and $T' \to T$ is a morphism of schemes then $\{T' \times_T T_i \to T'\}_{i\in I}$ is a V covering. \end{enumerate}


topologies-lemma-zariski-etale-smooth-syntomic-fppf-fpqc-ph-V

Any fpqc covering is a V covering. A fortiori, any fppf, syntomic, smooth, \'etale or Zariski covering is a V covering. Also, a ph covering is a V covering.


topologies-lemma-sheaf-property-V

Let $F$ be a contravariant functor on the category of schemes with values in sets. Then $F$ satisfies the sheaf property for the V topology if and only if it satisfies \begin{enumerate} \item the sheaf property for every Zariski covering, and \item the sheaf property for any standard V covering. \end{enumerate} Moreover, in the presence of (1) property (2) is equivalent to property \begin{enumerate} \item[(2')] the sheaf property for a standard V covering of the form $\{V \to U\}$, i.e., consisting of a single arrow. \end{enumerate}


topologies-lemma-refine-qcqs-V

Let $X \to Y$ be a quasi-compact morphism of schemes. The following are equivalent \begin{enumerate} \item $\{X \to Y\}$ is a V covering, \item for any valuation ring $V$ and morphism $g : \Spec(V) \to Y$ there exists an extension of valuation rings $V \subset W$ and a commutative diagram $$ \xymatrix{ \Spec(W) \ar[r] \ar[d] & X \ar[d] \\ \Spec(V) \ar[r] & Y } $$ \item for any morphism $Z \to Y$ and specialization $z' \leadsto z$ of points in $Z$, there is a specialization $w' \leadsto w$ of points in $Z \times_Y X$ mapping to $z' \leadsto z$. \end{enumerate}


topologies-lemma-V-covering-universally-submersive

Let $\{f_i : X_i \to X\}_{i \in I}$ be a V covering. Then $$ \coprod\nolimits_{i \in I} f_i : \coprod\nolimits_{i \in I} X_i \longrightarrow X $$ is a universally submersive morphism of schemes (Morphisms, Definition \ref{morphisms-definition-submersive}).


topologies-lemma-contained-in

Any set of big Zariski sites is contained in a common big Zariski site. The same is true, mutatis mutandis, for big fppf and big \'etale sites.