Datasets:

phanerozoic
/

Agda-UniMath

fact stringlengths 9 34.3k	type stringclasses 3 values	library stringclasses 2 values	imports listlengths 0 227	filename stringlengths 22 99	symbolic_name stringlengths 1 57
concept : ...	function	docs	[ "open import foundation-core.template public", "open import ..." ]	docs/TEMPLATE.lagda.md	concept
satisfies-property-concept : ...	function	docs	[ "open import foundation-core.template public", "open import ..." ]	docs/TEMPLATE.lagda.md	satisfies-property-concept
concept-subconcept : ...	function	docs	[ "open import foundation-core.template public", "open import ..." ]	docs/TEMPLATE.lagda.md	concept-subconcept
is-complete-prop-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → Prop (l1 ⊔ l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]	src/analysis/complete-metric-abelian-groups.lagda.md	is-complete-prop-Metric-Ab
is-complete-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → UU (l1 ⊔ l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]	src/analysis/complete-metric-abelian-groups.lagda.md	is-complete-Metric-Ab
Complete-Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "open import foundation.universe-levels", "open import metric-spaces.complete-metric-spaces", "open import metric-spaces.metric-spaces" ]	src/analysis/complete-metric-abelian-groups.lagda.md	Complete-Metric-Ab
convergent-series-Complete-Metric-Ab : {l1 l2 : Level} (G : Complete-Metric-Ab l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import analysis.complete-metric-abelian-groups", "open import analysis.convergent-series-metric-abelian-groups", "open import analysis.series-complete-metric-abelian-groups", "open import foundation.dependent-pair-types", "open import foundation.inhabited-types", "open import foundation.propositions...	src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md	convergent-series-Complete-Metric-Ab
convergent-series-Metric-Ab : {l1 l2 : Level} (G : Metric-Ab l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import analysis.metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.inequality-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import founda...	src/analysis/convergent-series-metric-abelian-groups.lagda.md	convergent-series-Metric-Ab
convergent-series-ℝ : (l : Level) → UU (lsuc l)	function	src	[ "open import analysis.convergent-series-complete-metric-abelian-groups", "open import analysis.convergent-series-metric-abelian-groups", "open import analysis.series-real-numbers", "open import foundation.dependent-pair-types", "open import foundation.propositions", "open import foundation.subtypes", "o...	src/analysis/convergent-series-real-numbers.lagda.md	convergent-series-ℝ
Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import elementary-number-theory.positive-rational-numbers", "open import foundation.action-on-identifications-binary-functions", "open import foundation.binary-relations", "open import foundation.cartesian-product-types", "open import foundation.conjunction", "open import foundation.dependent-pair-t...	src/analysis/metric-abelian-groups.lagda.md	Metric-Ab
is-nonnegative-prop-series-ℝ : {l : Level} → series-ℝ l → Prop l	function	src	[ "open import analysis.series-real-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import order-theory.increasin...	src/analysis/nonnegative-series-real-numbers.lagda.md	is-nonnegative-prop-series-ℝ
is-nonnegative-series-ℝ : {l : Level} → series-ℝ l → UU l	function	src	[ "open import analysis.series-real-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import order-theory.increasin...	src/analysis/nonnegative-series-real-numbers.lagda.md	is-nonnegative-series-ℝ
series-Metric-Ab {l1 l2 : Level} (G : Metric-Ab l1 l2) : UU l1 where constructor series-terms-Metric-Ab field term-series-Metric-Ab : sequence (type-Metric-Ab G)	record	src	[ "open import analysis.metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.action-on-identifications-functions", "open import foundation.dependent-pair-types", "open import foundation.equiva...	src/analysis/series-metric-abelian-groups.lagda.md	series-Metric-Ab
series-ℝ : (l : Level) → UU (lsuc l)	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation...	src/analysis/series-real-numbers.lagda.md	series-ℝ
series-terms-ℝ : {l : Level} → sequence (ℝ l) → series-ℝ l	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation...	src/analysis/series-real-numbers.lagda.md	series-terms-ℝ
term-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation...	src/analysis/series-real-numbers.lagda.md	term-series-ℝ
partial-sum-series-ℝ : {l : Level} → series-ℝ l → sequence (ℝ l)	function	src	[ "open import analysis.series-complete-metric-abelian-groups", "open import analysis.series-metric-abelian-groups", "open import elementary-number-theory.addition-natural-numbers", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "open import foundation...	src/analysis/series-real-numbers.lagda.md	partial-sum-series-ℝ
obj-augmented-simplex-Category : UU lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "ope...	src/category-theory/augmented-simplex-category.lagda.md	obj-augmented-simplex-Category
hom-set-augmented-simplex-Category : obj-augmented-simplex-Category → obj-augmented-simplex-Category → Set lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "ope...	src/category-theory/augmented-simplex-category.lagda.md	hom-set-augmented-simplex-Category
hom-augmented-simplex-Category : obj-augmented-simplex-Category → obj-augmented-simplex-Category → UU lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "ope...	src/category-theory/augmented-simplex-category.lagda.md	hom-augmented-simplex-Category
id-hom-augmented-simplex-Category : (n : obj-augmented-simplex-Category) → hom-augmented-simplex-Category n n	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "ope...	src/category-theory/augmented-simplex-category.lagda.md	id-hom-augmented-simplex-Category
augmented-simplex-Precategory : Precategory lzero lzero	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.precategories", "open import elementary-number-theory.inequality-standard-finite-types", "open import elementary-number-theory.natural-numbers", "open import foundation.dependent-pair-types", "ope...	src/category-theory/augmented-simplex-category.lagda.md	augmented-simplex-Precategory
Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open imp...	src/category-theory/categories.lagda.md	Category
total-hom-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.nonunital-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open imp...	src/category-theory/categories.lagda.md	total-hom-Category
sSet-Large-Precategory : Large-Precategory lsuc (_⊔_)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Large-Precategory
is-large-category-sSet-Large-Category : is-large-category-Large-Precategory sSet-Large-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	is-large-category-sSet-Large-Category
sSet-Large-Category : Large-Category lsuc (_⊔_)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Large-Category
sSet : (l : Level) → UU (lsuc l)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	sSet
hom-set-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → Set (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	hom-set-sSet
hom-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	hom-sSet
id-hom-sSet : {l1 : Level} (X : sSet l1) → hom-sSet X X	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	id-hom-sSet
sSet-Precategory : (l : Level) → Precategory (lsuc l) l	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Precategory
sSet-Category : (l : Level) → Category (lsuc l) l	function	src	[ "open import category-theory.categories", "open import category-theory.large-categories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import category-theory.presheaf-categories", "open import category-theory.simplex-category", "open import founda...	src/category-theory/category-of-simplicial-sets.lagda.md	sSet-Category
extensions : the identity map on `D` trivially gives a right extension of `D`	function	src	[ "open import category-theory.algebras-monads-on-precategories", "open import category-theory.functors-precategories", "open import category-theory.monads-on-precategories", "open import category-theory.natural-transformations-functors-precategories", "open import category-theory.precategories", "open impo...	src/category-theory/codensity-monads-on-precategories.lagda.md	extensions
core-precategory-Category : {l1 l2 : Level} (C : Category l1 l2) → Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.cores-precategories", "open import category-theory.groupoids", "open import category-theory.isomorphisms-in-categories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-t...	src/category-theory/cores-categories.lagda.md	core-precategory-Category
core-category-Category : {l1 l2 : Level} (C : Category l1 l2) → Category l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.cores-precategories", "open import category-theory.groupoids", "open import category-theory.isomorphisms-in-categories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-t...	src/category-theory/cores-categories.lagda.md	core-category-Category
core-precategory-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.replete-subprecategories", ...	src/category-theory/cores-precategories.lagda.md	core-precategory-Precategory
Full-Subcategory : {l1 l2 : Level} (l3 : Level) (C : Category l1 l2) → UU (l1 ⊔ lsuc l3)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.embeddings-precategories", "open import category-theory.full-subprecategories", "open import category-theory.fully-faithful-functors-precategories", "open...	src/category-theory/full-subcategories.lagda.md	Full-Subcategory
Full-Subprecategory : {l1 l2 : Level} (l3 : Level) (C : Precategory l1 l2) → UU (l1 ⊔ lsuc l3)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.embeddings-precategories", "open import category-theory.fully-faithful-functors-precategories", "open import category-theory.functors-precategories", "ope...	src/category-theory/full-subprecategories.lagda.md	Full-Subprecategory
id-functor-Category : {l1 l2 : Level} (C : Category l1 l2) → functor-Category C C	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.maps-categories", "open import foundation.equivalences", "open import foundation.function-types", "open import foundation...	src/category-theory/functors-categories.lagda.md	id-functor-Category
of : - a map `F₀ : C → D` on objects at some chosen universe level `γ`,	function	src	[ "open import category-theory.functors-precategories", "open import category-theory.large-precategories", "open import category-theory.maps-from-small-to-large-precategories", "open import category-theory.precategories", "open import foundation.cartesian-product-types", "open import foundation.dependent-pa...	src/category-theory/functors-from-small-to-large-precategories.lagda.md	of
id-functor-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → functor-Precategory C C	function	src	[ "open import category-theory.functors-set-magmoids", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.maps-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.action-on-identific...	src/category-theory/functors-precategories.lagda.md	id-functor-Precategory
Gaunt-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", ...	src/category-theory/gaunt-categories.lagda.md	Gaunt-Category
nonunital-precategory-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Nonunital-Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", ...	src/category-theory/gaunt-categories.lagda.md	nonunital-precategory-Gaunt-Category
precategory-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Precategory l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", ...	src/category-theory/gaunt-categories.lagda.md	precategory-Gaunt-Category
strongly-preunivalent-category-Gaunt-Category : {l1 l2 : Level} → Gaunt-Category l1 l2 → Strongly-Preunivalent-Category l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", ...	src/category-theory/gaunt-categories.lagda.md	strongly-preunivalent-category-Gaunt-Category
total-hom-Gaunt-Category : {l1 l2 : Level} (C : Gaunt-Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphism-induction-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", ...	src/category-theory/gaunt-categories.lagda.md	total-hom-Gaunt-Category
is-groupoid-prop-Category : {l1 l2 : Level} (C : Category l1 l2) → Prop (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "op...	src/category-theory/groupoids.lagda.md	is-groupoid-prop-Category
is-groupoid-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "op...	src/category-theory/groupoids.lagda.md	is-groupoid-Category
Groupoid : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.categories", "open import category-theory.functors-categories", "open import category-theory.isomorphisms-in-categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "op...	src/category-theory/groupoids.lagda.md	Groupoid
is-section-indiscrete-Precategory : {l : Level} → obj-Precategory ∘ indiscrete-Precategory {l} ~ id	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import category-theory.pregroupoids", "open import category-theory.preunivalent-categories", "open import category-theory.strict-categories", "open import category-theory.subterminal-precategori...	src/category-theory/indiscrete-precategories.lagda.md	is-section-indiscrete-Precategory
obj-initial-Category : UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	obj-initial-Category
hom-set-initial-Category : obj-initial-Category → obj-initial-Category → Set lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	hom-set-initial-Category
hom-initial-Category : obj-initial-Category → obj-initial-Category → UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	hom-initial-Category
id-hom-initial-Category : {x : obj-initial-Category} → hom-initial-Category x x	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	id-hom-initial-Category
initial-Precategory : Precategory lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	initial-Precategory
is-category-initial-Category : is-category-Precategory initial-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	is-category-initial-Category
initial-Category : Category lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	initial-Category
is-strongly-preunivalent-initial-Category : is-strongly-preunivalent-Precategory initial-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	is-strongly-preunivalent-initial-Category
is-strict-category-initial-Category : is-strict-category-Precategory initial-Precategory	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	is-strict-category-initial-Category
initial-Strict-Category : Strict-Category lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	initial-Strict-Category
is-gaunt-initial-Category : is-gaunt-Category initial-Category	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	is-gaunt-initial-Category
initial-Gaunt-Category : Gaunt-Category lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.functors-precategories", "open import category-theory.gaunt-categories", "open import category-theory.indiscrete-precategories", "open import category-theory.precategories", "open import category-theory.strict-categories", "open impo...	src/category-theory/initial-category.lagda.md	initial-Gaunt-Category
initial-obj-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.propositions", "open import foundation.universe-levels", "open import foundation-core.identity-types" ...	src/category-theory/initial-objects-precategories.lagda.md	initial-obj-Precategory
Large-Category (α : Level → Level) (β : Level → Level → Level) : UUω where constructor make-Large-Category field large-precategory-Large-Category : Large-Precategory α β is-large-category-Large-Category : is-large-category-Large-Precategory large-precategory-Large-Category	record	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-large-precategories", "open import category-theory.large-precategories", "open import category-theory.precategories", "open import foundation.action-on-identifications-binary-functions", "open import foundation.dependen...	src/category-theory/large-categories.lagda.md	Large-Category
Large-Precategory (α : Level → Level) (β : Level → Level → Level) : UUω where field obj-Large-Precategory : (l : Level) → UU (α l) hom-set-Large-Precategory : {l1 l2 : Level} → obj-Large-Precategory l1 → obj-Large-Precategory l2 → Set (β l1 l2) hom-Large-Precategory : {l1 l2 : Level} → obj-Large-Precategory l1 → obj-La...	record	src	[ "open import category-theory.precategories", "open import foundation.action-on-identifications-binary-functions", "open import foundation.dependent-pair-types", "open import foundation.function-types", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation....	src/category-theory/large-precategories.lagda.md	Large-Precategory
Large-Subprecategory {α : Level → Level} {β : Level → Level → Level} (γ : Level → Level) (δ : Level → Level → Level) (C : Large-Precategory α β) : UUω where field subtype-obj-Large-Subprecategory : (l : Level) → subtype (γ l) (obj-Large-Precategory C l) is-in-obj-Large-Subprecategory : {l : Level} → obj-Large-Precatego...	record	src	[ "open import category-theory.large-precategories", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.sets", "open import foundation.strictly-involutive-identity-types", "open import foundation.subtypes", "open import foundation.universe-level...	src/category-theory/large-subprecategories.lagda.md	Large-Subprecategory
extensions : the identity map gives a left extension (with the identity natural	function	src	[ "open import category-theory.functors-precategories", "open import category-theory.natural-transformations-functors-precategories", "open import category-theory.precategories", "open import foundation.action-on-equivalences-functions", "open import foundation.action-on-identifications-functions", "open im...	src/category-theory/left-extensions-precategories.lagda.md	extensions
Nonunital-Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.set-magmoids", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "o...	src/category-theory/nonunital-precategories.lagda.md	Nonunital-Precategory
total-hom-Nonunital-Precategory : {l1 l2 : Level} (C : Nonunital-Precategory l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.set-magmoids", "open import foundation.cartesian-product-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "o...	src/category-theory/nonunital-precategories.lagda.md	total-hom-Nonunital-Precategory
is-one-object-prop-Precategory : {l1 l2 : Level} → Precategory l1 l2 → Prop l1	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.s...	src/category-theory/one-object-precategories.lagda.md	is-one-object-prop-Precategory
is-one-object-Precategory : {l1 l2 : Level} → Precategory l1 l2 → UU l1	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.s...	src/category-theory/one-object-precategories.lagda.md	is-one-object-Precategory
One-Object-Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.s...	src/category-theory/one-object-precategories.lagda.md	One-Object-Precategory
monoid-One-Object-Precategory : {l1 l2 : Level} → One-Object-Precategory l1 l2 → Monoid l2	function	src	[ "open import category-theory.endomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.contractible-types", "open import foundation.dependent-pair-types", "open import foundation.identity-types", "open import foundation.propositions", "open import foundation.s...	src/category-theory/one-object-precategories.lagda.md	monoid-One-Object-Precategory
is-involution-opposite-Category : {l1 l2 : Level} → is-involution (opposite-Category {l1} {l2})	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import fou...	src/category-theory/opposite-categories.lagda.md	is-involution-opposite-Category
involution-opposite-Category : (l1 l2 : Level) → involution (Category l1 l2)	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import fou...	src/category-theory/opposite-categories.lagda.md	involution-opposite-Category
is-equiv-opposite-Category : {l1 l2 : Level} → is-equiv (opposite-Category {l1} {l2})	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import fou...	src/category-theory/opposite-categories.lagda.md	is-equiv-opposite-Category
equiv-opposite-Category : (l1 l2 : Level) → Category l1 l2 ≃ Category l1 l2	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import fou...	src/category-theory/opposite-categories.lagda.md	equiv-opposite-Category
is-involution-opposite-Precategory : {l1 l2 : Level} → is-involution (opposite-Precategory {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involution...	src/category-theory/opposite-precategories.lagda.md	is-involution-opposite-Precategory
involution-opposite-Precategory : (l1 l2 : Level) → involution (Precategory l1 l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involution...	src/category-theory/opposite-precategories.lagda.md	involution-opposite-Precategory
is-equiv-opposite-Precategory : {l1 l2 : Level} → is-equiv (opposite-Precategory {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involution...	src/category-theory/opposite-precategories.lagda.md	is-equiv-opposite-Precategory
equiv-opposite-Precategory : (l1 l2 : Level) → Precategory l1 l2 ≃ Precategory l1 l2	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.homotopies", "open import foundation.identity-types", "open import foundation.involution...	src/category-theory/opposite-precategories.lagda.md	equiv-opposite-Precategory
is-involution-opposite-Preunivalent-Category : {l1 l2 : Level} → is-involution (opposite-Preunivalent-Category {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open...	src/category-theory/opposite-preunivalent-categories.lagda.md	is-involution-opposite-Preunivalent-Category
involution-opposite-Preunivalent-Category : (l1 l2 : Level) → involution (Preunivalent-Category l1 l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open...	src/category-theory/opposite-preunivalent-categories.lagda.md	involution-opposite-Preunivalent-Category
is-equiv-opposite-Preunivalent-Category : {l1 l2 : Level} → is-equiv (opposite-Preunivalent-Category {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open...	src/category-theory/opposite-preunivalent-categories.lagda.md	is-equiv-opposite-Preunivalent-Category
equiv-opposite-Preunivalent-Category : (l1 l2 : Level) → Preunivalent-Category l1 l2 ≃ Preunivalent-Category l1 l2	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings", "open...	src/category-theory/opposite-preunivalent-categories.lagda.md	equiv-opposite-Preunivalent-Category
involution-opposite-Strongly-Preunivalent-Category : (l1 l2 : Level) → involution (Strongly-Preunivalent-Category l1 l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings"...	src/category-theory/opposite-strongly-preunivalent-categories.lagda.md	involution-opposite-Strongly-Preunivalent-Category
is-equiv-opposite-Strongly-Preunivalent-Category : {l1 l2 : Level} → is-equiv (opposite-Strongly-Preunivalent-Category {l1} {l2})	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.opposite-precategories", "open import category-theory.precategories", "open import category-theory.strongly-preunivalent-categories", "open import foundation.dependent-pair-types", "open import foundation.embeddings"...	src/category-theory/opposite-strongly-preunivalent-categories.lagda.md	is-equiv-opposite-Strongly-Preunivalent-Category
Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.nonunital-precategories", "open import category-theory.set-magmoids", "open import foundation.action-on-identifications-functions", "open import foundation.cartesian-product-types", "open import f...	src/category-theory/precategories.lagda.md	Precategory
total-hom-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.nonunital-precategories", "open import category-theory.set-magmoids", "open import foundation.action-on-identifications-functions", "open import foundation.cartesian-product-types", "open import f...	src/category-theory/precategories.lagda.md	total-hom-Precategory
Pregroupoid : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.dependent-pair-types", "open import foundation.equivalences", "open import foundation.identity-types", "open import foundation.iterated-dependent-product-types", "open import...	src/category-theory/pregroupoids.lagda.md	Pregroupoid
Preunivalent-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-p...	src/category-theory/preunivalent-categories.lagda.md	Preunivalent-Category
total-hom-Preunivalent-Category : {l1 l2 : Level} (𝒞 : Preunivalent-Category l1 l2) → UU (l1 ⊔ l2)	function	src	[ "open import category-theory.composition-operations-on-binary-families-of-sets", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.1-types", "open import foundation.cartesian-product-types", "open import foundation.dependent-p...	src/category-theory/preunivalent-categories.lagda.md	total-hom-Preunivalent-Category
that : - sends an object `x` of `C` to the [set](foundation-core.sets.md) `hom c x` and	function	src	[ "open import category-theory.functors-large-precategories", "open import category-theory.large-precategories", "open import category-theory.natural-transformations-functors-large-precategories", "open import foundation.category-of-sets", "open import foundation.function-extensionality", "open import found...	src/category-theory/representable-functors-large-precategories.lagda.md	that
that : - sends an object `x` of `C` to the [set](foundation-core.sets.md) `hom c x` and	function	src	[ "open import category-theory.copresheaf-categories", "open import category-theory.functors-precategories", "open import category-theory.maps-precategories", "open import category-theory.natural-transformations-functors-precategories", "open import category-theory.opposite-precategories", "open import cate...	src/category-theory/representable-functors-precategories.lagda.md	that
obj-representing-arrow-Category : UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.e...	src/category-theory/representing-arrow-category.lagda.md	obj-representing-arrow-Category
hom-set-representing-arrow-Category : obj-representing-arrow-Category → obj-representing-arrow-Category → Set lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.e...	src/category-theory/representing-arrow-category.lagda.md	hom-set-representing-arrow-Category
hom-representing-arrow-Category : obj-representing-arrow-Category → obj-representing-arrow-Category → UU lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.e...	src/category-theory/representing-arrow-category.lagda.md	hom-representing-arrow-Category
id-hom-representing-arrow-Category : {x : obj-representing-arrow-Category} → hom-representing-arrow-Category x x	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.e...	src/category-theory/representing-arrow-category.lagda.md	id-hom-representing-arrow-Category
representing-arrow-Precategory : Precategory lzero lzero	function	src	[ "open import category-theory.categories", "open import category-theory.isomorphisms-in-precategories", "open import category-theory.precategories", "open import foundation.booleans", "open import foundation.decidable-propositions", "open import foundation.dependent-pair-types", "open import foundation.e...	src/category-theory/representing-arrow-category.lagda.md	representing-arrow-Precategory

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Agda-UniMath

Structured dataset from agda-unimath — Univalent mathematics.

4,490 declarations extracted from Agda source files.

Applications

Training language models on formal proofs
Fine-tuning theorem provers
Retrieval-augmented generation for proof assistants
Learning proof embeddings and representations

Source

Repository: https://github.com/UniMath/agda-unimath

Schema

Column	Type	Description
fact	string	Declaration body
type	string	data, record, function
library	string	Source module
imports	list	Required imports
filename	string	Source file path
symbolic_name	string	Identifier

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Digesting proof assistant libraries for AI ingestion. • 84 items • Updated Jan 15 • 3