Proof Assistant Projects
Collection
Digesting proof assistant libraries for AI ingestion. • 103 items • Updated • 3
statement stringlengths 9 9.22k | proof stringlengths 0 27.3k | type stringclasses 4
values | symbolic_name stringlengths 1 111 | library stringclasses 39
values | filename stringlengths 24 124 | imports listlengths 0 227 | deps listlengths 0 61 | docstring stringclasses 1
value | source_url stringclasses 1
value | commit stringclasses 1
value |
|---|---|---|---|---|---|---|---|---|---|---|
is-complete-prop-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → Prop (l1 ⊔ l2) | is-complete-prop-Metric-Ab G =
is-complete-prop-Metric-Space (metric-space-Metric-Ab G) | function | is-complete-prop-Metric-Ab | analysis | src/analysis/complete-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"foundation.dependent-pair-types",
"foundation.propositions",
"foundation.subtypes",
"foundation.universe-levels",
"metric-spaces.complete-metric-spaces",
"metric-spaces.metric-spaces"
] | [
"Metric-Ab",
"Prop"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-complete-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → UU (l1 ⊔ l2) | is-complete-Metric-Ab G = is-complete-Metric-Space (metric-space-Metric-Ab G) | function | is-complete-Metric-Ab | analysis | src/analysis/complete-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"foundation.dependent-pair-types",
"foundation.propositions",
"foundation.subtypes",
"foundation.universe-levels",
"metric-spaces.complete-metric-spaces",
"metric-spaces.metric-spaces"
] | [
"Metric-Ab"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Complete-Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | Complete-Metric-Ab l1 l2 = type-subtype (is-complete-prop-Metric-Ab {l1} {l2}) | function | Complete-Metric-Ab | analysis | src/analysis/complete-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"foundation.dependent-pair-types",
"foundation.propositions",
"foundation.subtypes",
"foundation.universe-levels",
"metric-spaces.complete-metric-spaces",
"metric-spaces.metric-spaces"
] | [
"is-complete-prop-Metric-Ab"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
convergent-series-Complete-Metric-Ab :
{l1 l2 : Level} (G : Complete-Metric-Ab l1 l2) → UU (l1 ⊔ l2) | convergent-series-Complete-Metric-Ab G =
type-subtype (is-convergent-prop-series-Complete-Metric-Ab G) | function | convergent-series-Complete-Metric-Ab | analysis | src/analysis/convergent-series-complete-metric-abelian-groups.lagda.md | [
"analysis.complete-metric-abelian-groups",
"analysis.convergent-series-metric-abelian-groups",
"analysis.series-complete-metric-abelian-groups",
"foundation.dependent-pair-types",
"foundation.inhabited-types",
"foundation.propositions",
"foundation.subtypes",
"foundation.universe-levels",
"metric-sp... | [
"Complete-Metric-Ab"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
convergent-series-Metric-Ab :
{l1 l2 : Level} (G : Metric-Ab l1 l2) → UU (l1 ⊔ l2) | convergent-series-Metric-Ab G =
type-subtype (is-convergent-prop-series-Metric-Ab {G = G}) | function | convergent-series-Metric-Ab | analysis | src/analysis/convergent-series-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"analysis.series-metric-abelian-groups",
"elementary-number-theory.addition-natural-numbers",
"elementary-number-theory.difference-natural-numbers",
"elementary-number-theory.inequality-natural-numbers",
"elementary-number-theory.natural-numbers",
"foundation.action-on-... | [
"Metric-Ab"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-metric-ab-prop-Ab-Pseudometric-Structure :
{l1 l2 : Level} (G : Ab l1) (M : Pseudometric-Structure l2 (type-Ab G)) →
Prop (l1 ⊔ l2) | is-metric-ab-prop-Ab-Pseudometric-Structure G M =
let
MS = (type-Ab G , M)
in
is-extensional-prop-Pseudometric-Space MS ∧
is-isometry-prop-Pseudometric-Space MS MS (neg-Ab G) ∧
Π-Prop
( type-Ab G)
( λ x → is-isometry-prop-Pseudometric-Space MS MS (add-Ab G x)) | function | is-metric-ab-prop-Ab-Pseudometric-Structure | analysis | src/analysis/metric-abelian-groups.lagda.md | [
"elementary-number-theory.positive-rational-numbers",
"foundation.action-on-identifications-binary-functions",
"foundation.binary-relations",
"foundation.cartesian-product-types",
"foundation.conjunction",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.fun... | [
"Ab",
"Prop",
"Pseudometric-Structure"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-metric-ab-Ab-Pseudometric-Structure :
{l1 l2 : Level} (G : Ab l1) (M : Pseudometric-Structure l2 (type-Ab G)) →
UU (l1 ⊔ l2) | is-metric-ab-Ab-Pseudometric-Structure G M =
type-Prop (is-metric-ab-prop-Ab-Pseudometric-Structure G M) | function | is-metric-ab-Ab-Pseudometric-Structure | analysis | src/analysis/metric-abelian-groups.lagda.md | [
"elementary-number-theory.positive-rational-numbers",
"foundation.action-on-identifications-binary-functions",
"foundation.binary-relations",
"foundation.cartesian-product-types",
"foundation.conjunction",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.fun... | [
"Ab",
"Pseudometric-Structure",
"is-metric-ab-prop-Ab-Pseudometric-Structure"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Metric-Ab : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | Metric-Ab l1 l2 =
Σ ( Ab l1)
( λ G →
Σ ( Pseudometric-Structure l2 (type-Ab G))
( is-metric-ab-Ab-Pseudometric-Structure G)) | function | Metric-Ab | analysis | src/analysis/metric-abelian-groups.lagda.md | [
"elementary-number-theory.positive-rational-numbers",
"foundation.action-on-identifications-binary-functions",
"foundation.binary-relations",
"foundation.cartesian-product-types",
"foundation.conjunction",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.fun... | [
"Ab",
"Pseudometric-Structure",
"is-metric-ab-Ab-Pseudometric-Structure"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
sequence-type-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → UU l1 | sequence-type-Metric-Ab G = sequence (type-Metric-Ab G) | function | sequence-type-Metric-Ab | analysis | src/analysis/sequences-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"foundation.universe-levels",
"lists.sequences"
] | [
"Metric-Ab",
"sequence"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
add-sequence-type-Metric-Ab :
{l1 l2 : Level} (G : Metric-Ab l1 l2) →
sequence-type-Metric-Ab G → sequence-type-Metric-Ab G →
sequence-type-Metric-Ab G | add-sequence-type-Metric-Ab G = binary-map-sequence (add-Metric-Ab G) | function | add-sequence-type-Metric-Ab | analysis | src/analysis/sequences-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"foundation.universe-levels",
"lists.sequences"
] | [
"Metric-Ab",
"binary-map-sequence",
"sequence-type-Metric-Ab"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
neg-sequence-type-Metric-Ab :
{l1 l2 : Level} (G : Metric-Ab l1 l2) →
sequence-type-Metric-Ab G → sequence-type-Metric-Ab G | neg-sequence-type-Metric-Ab G = map-sequence (neg-Metric-Ab G) | function | neg-sequence-type-Metric-Ab | analysis | src/analysis/sequences-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"foundation.universe-levels",
"lists.sequences"
] | [
"Metric-Ab",
"map-sequence",
"sequence-type-Metric-Ab"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
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 | series-Metric-Ab | analysis | src/analysis/series-metric-abelian-groups.lagda.md | [
"analysis.metric-abelian-groups",
"elementary-number-theory.addition-natural-numbers",
"elementary-number-theory.natural-numbers",
"foundation.action-on-identifications-functions",
"foundation.dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"foundation.function-... | [
"Metric-Ab",
"sequence"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
anafunctor-Category :
{l1 l2 l3 l4 : Level} (l : Level) →
Category l1 l2 → Category l3 l4 → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l) | anafunctor-Category l C D =
anafunctor-Precategory l (precategory-Category C) (precategory-Category D) | function | anafunctor-Category | category-theory | src/category-theory/anafunctors-categories.lagda.md | [
"category-theory.anafunctors-precategories",
"category-theory.categories",
"category-theory.functors-categories",
"foundation.dependent-pair-types",
"foundation.propositional-truncations",
"foundation.universe-levels"
] | [
"Category",
"anafunctor-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
image-object-anafunctor-Category :
{l1 l2 l3 l4 l5 : Level} (C : Category l1 l2) (D : Category l3 l4) →
anafunctor-Category l5 C D → obj-Category C → UU (l3 ⊔ l5) | image-object-anafunctor-Category C D F X =
Σ ( obj-Category D)
( λ U → type-trunc-Prop (object-anafunctor-Category C D F X U)) | function | image-object-anafunctor-Category | category-theory | src/category-theory/anafunctors-categories.lagda.md | [
"category-theory.anafunctors-precategories",
"category-theory.categories",
"category-theory.functors-categories",
"foundation.dependent-pair-types",
"foundation.propositional-truncations",
"foundation.universe-levels"
] | [
"Category",
"anafunctor-Category",
"type-trunc-Prop"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
anafunctor-Precategory :
{l1 l2 l3 l4 : Level} (l : Level) →
Precategory l1 l2 → Precategory l3 l4 → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4 ⊔ lsuc l) | anafunctor-Precategory l C D =
Σ ( obj-Precategory C → obj-Precategory D → UU l)
( λ F₀ →
Σ ( ( X Y : obj-Precategory C)
( U : obj-Precategory D) (u : F₀ X U) →
( V : obj-Precategory D) (v : F₀ Y V) →
( f : hom-Precategory C X Y) → hom-Precategory D U V)
( λ F₁ →
... | function | anafunctor-Precategory | category-theory | src/category-theory/anafunctors-precategories.lagda.md | [
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"foundation.action-on-identifications-functions",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.propositional-... | [
"Precategory",
"type-trunc-Prop"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
obj-augmented-simplex-Category : UU lzero | obj-augmented-simplex-Category = ℕ | function | obj-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-set-augmented-simplex-Category :
obj-augmented-simplex-Category → obj-augmented-simplex-Category → Set lzero | hom-set-augmented-simplex-Category n m =
hom-set-Poset (Fin-Poset n) (Fin-Poset m) | function | hom-set-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Fin-Poset",
"Set",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-augmented-simplex-Category :
obj-augmented-simplex-Category → obj-augmented-simplex-Category → UU lzero | hom-augmented-simplex-Category n m =
type-Set (hom-set-augmented-simplex-Category n m) | function | hom-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"hom-set-augmented-simplex-Category",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
comp-hom-augmented-simplex-Category :
{n m r : obj-augmented-simplex-Category} →
hom-augmented-simplex-Category m r →
hom-augmented-simplex-Category n m →
hom-augmented-simplex-Category n r | comp-hom-augmented-simplex-Category {n} {m} {r} =
comp-hom-Poset (Fin-Poset n) (Fin-Poset m) (Fin-Poset r) | function | comp-hom-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Fin-Poset",
"hom-augmented-simplex-Category",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
associative-comp-hom-augmented-simplex-Category :
{n m r s : obj-augmented-simplex-Category}
(h : hom-augmented-simplex-Category r s)
(g : hom-augmented-simplex-Category m r)
(f : hom-augmented-simplex-Category n m) →
comp-hom-augmented-simplex-Category {n} {m} {s}
( comp-hom-augmented-simplex-Category {m... | associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} =
associative-comp-hom-Poset
( Fin-Poset n)
( Fin-Poset m)
( Fin-Poset r)
( Fin-Poset s) | function | associative-comp-hom-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Fin-Poset",
"comp-hom-augmented-simplex-Category",
"hom-augmented-simplex-Category",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
involutive-eq-associative-comp-hom-augmented-simplex-Category :
{n m r s : obj-augmented-simplex-Category}
(h : hom-augmented-simplex-Category r s)
(g : hom-augmented-simplex-Category m r)
(f : hom-augmented-simplex-Category n m) →
comp-hom-augmented-simplex-Category {n} {m} {s}
( comp-hom-augmented-simpl... | involutive-eq-associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} =
involutive-eq-associative-comp-hom-Poset
( Fin-Poset n)
( Fin-Poset m)
( Fin-Poset r)
( Fin-Poset s) | function | involutive-eq-associative-comp-hom-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Fin-Poset",
"comp-hom-augmented-simplex-Category",
"hom-augmented-simplex-Category",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
associative-composition-operation-augmented-simplex-Category :
associative-composition-operation-binary-family-Set
hom-set-augmented-simplex-Category | function | associative-composition-operation-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"hom-set-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
id-hom-augmented-simplex-Category :
(n : obj-augmented-simplex-Category) → hom-augmented-simplex-Category n n | id-hom-augmented-simplex-Category n = id-hom-Poset (Fin-Poset n) | function | id-hom-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Fin-Poset",
"hom-augmented-simplex-Category",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
left-unit-law-comp-hom-augmented-simplex-Category :
{n m : obj-augmented-simplex-Category}
(f : hom-augmented-simplex-Category n m) →
comp-hom-augmented-simplex-Category {n} {m} {m}
( id-hom-augmented-simplex-Category m)
( f) =
f | left-unit-law-comp-hom-augmented-simplex-Category {n} {m} =
left-unit-law-comp-hom-Poset (Fin-Poset n) (Fin-Poset m) | function | left-unit-law-comp-hom-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Fin-Poset",
"comp-hom-augmented-simplex-Category",
"hom-augmented-simplex-Category",
"id-hom-augmented-simplex-Category",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
right-unit-law-comp-hom-augmented-simplex-Category :
{n m : obj-augmented-simplex-Category}
(f : hom-augmented-simplex-Category n m) →
comp-hom-augmented-simplex-Category {n} {n} {m}
( f)
( id-hom-augmented-simplex-Category n) =
f | right-unit-law-comp-hom-augmented-simplex-Category {n} {m} =
right-unit-law-comp-hom-Poset (Fin-Poset n) (Fin-Poset m) | function | right-unit-law-comp-hom-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Fin-Poset",
"comp-hom-augmented-simplex-Category",
"hom-augmented-simplex-Category",
"id-hom-augmented-simplex-Category",
"obj-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-unital-composition-operation-augmented-simplex-Category :
is-unital-composition-operation-binary-family-Set
( hom-set-augmented-simplex-Category)
( λ {n} {m} {r} → comp-hom-augmented-simplex-Category {n} {m} {r}) | function | is-unital-composition-operation-augmented-simplex-Category | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"comp-hom-augmented-simplex-Category",
"hom-set-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
augmented-simplex-Precategory : Precategory lzero lzero | function | augmented-simplex-Precategory | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
pr1 associative-composition-operation-augmented-simplex-Category {n} {m} {r} =
comp-hom-augmented-simplex-Category {n} {m} {r} | pr1 is-unital-composition-operation-augmented-simplex-Category =
id-hom-augmented-simplex-Category
pr1 (pr2 is-unital-composition-operation-augmented-simplex-Category) {n} {m} =
left-unit-law-comp-hom-augmented-simplex-Category {n} {m}
pr1 augmented-simplex-Precategory = obj-augmented-simplex-Category
pr1 (pr2 augm... | function | pr1 | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"associative-composition-operation-augmented-simplex-Category",
"augmented-simplex-Precategory",
"comp-hom-augmented-simplex-Category",
"hom-set-augmented-simplex-Category",
"id-hom-augmented-simplex-Category",
"is-unital-composition-operation-augmented-simplex-Category",
"left-unit-law-comp-hom-augment... | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
pr2 associative-composition-operation-augmented-simplex-Category
{ n} {m} {r} {s} =
involutive-eq-associative-comp-hom-augmented-simplex-Category {n} {m} {r} {s} | pr2 (pr2 is-unital-composition-operation-augmented-simplex-Category) {n} {m} =
right-unit-law-comp-hom-augmented-simplex-Category {n} {m}
pr2 (pr2 (pr2 augmented-simplex-Precategory)) =
is-unital-composition-operation-augmented-simplex-Category | function | pr2 | category-theory | src/category-theory/augmented-simplex-category.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.precategories",
"elementary-number-theory.inequality-standard-finite-types",
"elementary-number-theory.natural-numbers",
"foundation.dependent-pair-types",
"foundation.identity-types",
"foundation.sets",
"foundation.... | [
"associative-composition-operation-augmented-simplex-Category",
"augmented-simplex-Precategory",
"involutive-eq-associative-comp-hom-augmented-simplex-Category",
"is-unital-composition-operation-augmented-simplex-Category",
"right-unit-law-comp-hom-augmented-simplex-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | Category l1 l2 = Σ (Precategory l1 l2) (is-category-Precategory) | function | Category | category-theory | src/category-theory/categories.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theory.precategories",
"category-theory.preunivalent-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.1-t... | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
total-hom-Category :
{l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2) | total-hom-Category C = total-hom-Precategory (precategory-Category C) | function | total-hom-Category | category-theory | src/category-theory/categories.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theory.precategories",
"category-theory.preunivalent-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.1-t... | [
"Category",
"total-hom-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
obj-total-hom-Category :
{l1 l2 : Level} (C : Category l1 l2) →
total-hom-Category C →
obj-Category C × obj-Category C | obj-total-hom-Category C = obj-total-hom-Precategory (precategory-Category C) | function | obj-total-hom-Category | category-theory | src/category-theory/categories.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theory.precategories",
"category-theory.preunivalent-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.1-t... | [
"Category",
"obj-total-hom-Precategory",
"total-hom-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
precomp-hom-Category :
{l1 l2 : Level} (C : Category l1 l2) {x y : obj-Category C}
(f : hom-Category C x y) (z : obj-Category C) →
hom-Category C y z → hom-Category C x z | precomp-hom-Category C = precomp-hom-Precategory (precategory-Category C) | function | precomp-hom-Category | category-theory | src/category-theory/categories.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theory.precategories",
"category-theory.preunivalent-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.1-t... | [
"Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
postcomp-hom-Category :
{l1 l2 : Level} (C : Category l1 l2) {x y : obj-Category C}
(f : hom-Category C x y) (z : obj-Category C) →
hom-Category C z x → hom-Category C z y | postcomp-hom-Category C = postcomp-hom-Precategory (precategory-Category C) | function | postcomp-hom-Category | category-theory | src/category-theory/categories.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theory.precategories",
"category-theory.preunivalent-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.1-t... | [
"Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
sSet-Large-Precategory :
Large-Precategory lsuc (_⊔_) | sSet-Large-Precategory =
presheaf-large-precategory-Precategory simplex-Precategory | function | sSet-Large-Precategory | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"Large-Precategory",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-large-category-sSet-Large-Category :
is-large-category-Large-Precategory sSet-Large-Precategory | is-large-category-sSet-Large-Category =
is-large-category-presheaf-large-category-Precategory simplex-Precategory | function | is-large-category-sSet-Large-Category | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"is-large-category-Large-Precategory",
"sSet-Large-Precategory",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
sSet-Large-Category :
Large-Category lsuc (_⊔_) | sSet-Large-Category =
presheaf-large-category-Precategory simplex-Precategory | function | sSet-Large-Category | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"Large-Category",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
sSet : (l : Level) → UU (lsuc l) | sSet = obj-Large-Category sSet-Large-Category | function | sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"sSet-Large-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-set-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → Set (l1 ⊔ l2) | hom-set-sSet = hom-set-Large-Category sSet-Large-Category | function | hom-set-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"Set",
"sSet",
"sSet-Large-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-sSet : {l1 l2 : Level} (X : sSet l1) (Y : sSet l2) → UU (l1 ⊔ l2) | hom-sSet = hom-Large-Category sSet-Large-Category | function | hom-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"sSet",
"sSet-Large-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
comp-hom-sSet :
{l1 l2 l3 : Level} (X : sSet l1) (Y : sSet l2) (Z : sSet l3) →
hom-sSet Y Z → hom-sSet X Y → hom-sSet X Z | comp-hom-sSet = comp-hom-presheaf-Precategory simplex-Precategory | function | comp-hom-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"hom-sSet",
"sSet",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
id-hom-sSet : {l1 : Level} (X : sSet l1) → hom-sSet X X | id-hom-sSet = id-hom-presheaf-Precategory simplex-Precategory | function | id-hom-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"hom-sSet",
"sSet",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
associative-comp-hom-sSet :
{l1 l2 l3 l4 : Level}
(X : sSet l1) (Y : sSet l2) (Z : sSet l3) (W : sSet l4)
(h : hom-sSet Z W) (g : hom-sSet Y Z) (f : hom-sSet X Y) →
comp-hom-sSet X Y W (comp-hom-sSet Y Z W h g) f =
comp-hom-sSet X Z W h (comp-hom-sSet X Y Z g f) | associative-comp-hom-sSet =
associative-comp-hom-presheaf-Precategory simplex-Precategory | function | associative-comp-hom-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"comp-hom-sSet",
"hom-sSet",
"sSet",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
involutive-eq-associative-comp-hom-sSet :
{l1 l2 l3 l4 : Level}
(X : sSet l1) (Y : sSet l2) (Z : sSet l3) (W : sSet l4)
(h : hom-sSet Z W) (g : hom-sSet Y Z) (f : hom-sSet X Y) →
comp-hom-sSet X Y W (comp-hom-sSet Y Z W h g) f =ⁱ
comp-hom-sSet X Z W h (comp-hom-sSet X Y Z g f) | involutive-eq-associative-comp-hom-sSet =
involutive-eq-associative-comp-hom-presheaf-Precategory simplex-Precategory | function | involutive-eq-associative-comp-hom-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"comp-hom-sSet",
"hom-sSet",
"sSet",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
left-unit-law-comp-hom-sSet :
{l1 l2 : Level} (X : sSet l1) (Y : sSet l2) (f : hom-sSet X Y) →
comp-hom-sSet X Y Y (id-hom-sSet Y) f = f | left-unit-law-comp-hom-sSet =
left-unit-law-comp-hom-presheaf-Precategory simplex-Precategory | function | left-unit-law-comp-hom-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"comp-hom-sSet",
"hom-sSet",
"id-hom-sSet",
"sSet",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
right-unit-law-comp-hom-sSet :
{l1 l2 : Level} (X : sSet l1) (Y : sSet l2) (f : hom-sSet X Y) →
comp-hom-sSet X X Y f (id-hom-sSet X) = f | right-unit-law-comp-hom-sSet =
right-unit-law-comp-hom-presheaf-Precategory simplex-Precategory | function | right-unit-law-comp-hom-sSet | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"comp-hom-sSet",
"hom-sSet",
"id-hom-sSet",
"sSet",
"simplex-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
sSet-Precategory : (l : Level) → Precategory (lsuc l) l | sSet-Precategory = precategory-Large-Category sSet-Large-Category | function | sSet-Precategory | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"Precategory",
"sSet-Large-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
sSet-Category : (l : Level) → Category (lsuc l) l | sSet-Category = category-Large-Category sSet-Large-Category | function | sSet-Category | category-theory | src/category-theory/category-of-simplicial-sets.lagda.md | [
"category-theory.categories",
"category-theory.large-categories",
"category-theory.large-precategories",
"category-theory.precategories",
"category-theory.presheaf-categories",
"category-theory.simplex-category",
"foundation.commuting-squares-of-maps",
"foundation.function-types",
"foundation.homoto... | [
"Category",
"sSet-Large-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
coherence-square-hom-Large-Precategory :
{l1 l2 l3 l4 : Level}
{α : Level → Level}
{β : Level → Level → Level}
(C : Large-Precategory α β)
{x : obj-Large-Precategory C l1}
{y : obj-Large-Precategory C l2}
{z : obj-Large-Precategory C l3}
{w : obj-Large-Precategory C l4}
(top : hom-Large-Precategory C ... | coherence-square-hom-Large-Precategory C top left right bottom =
( comp-hom-Large-Precategory C bottom left) =
( comp-hom-Large-Precategory C right top) | function | coherence-square-hom-Large-Precategory | category-theory | src/category-theory/commuting-squares-of-morphisms-in-large-precategories.lagda.md | [
"category-theory.large-precategories",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Large-Precategory",
"comp-hom-Large-Precategory",
"obj-Large-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
coherence-square-hom-Precategory :
{l1 l2 : Level} (C : Precategory l1 l2) {x y z w : obj-Precategory C}
(top : hom-Precategory C x y)
(left : hom-Precategory C x z)
(right : hom-Precategory C y w)
(bottom : hom-Precategory C z w) →
UU l2 | coherence-square-hom-Precategory C =
coherence-square-hom-Set-Magmoid (set-magmoid-Precategory C) | function | coherence-square-hom-Precategory | category-theory | src/category-theory/commuting-squares-of-morphisms-in-precategories.lagda.md | [
"category-theory.commuting-squares-of-morphisms-in-set-magmoids",
"category-theory.precategories",
"foundation.action-on-identifications-functions",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Precategory",
"coherence-square-hom-Set-Magmoid"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
pasting-horizontal-coherence-square-hom-Precategory :
{l1 l2 : Level} (C : Precategory l1 l2)
{x y z w a b : obj-Precategory C}
(topleft : hom-Precategory C x y)
(topright : hom-Precategory C y a)
(left : hom-Precategory C x z)
(middle : hom-Precategory C y w)
(right : hom-Precategory C a b)
(bottomleft... | pasting-horizontal-coherence-square-hom-Precategory C
topleft topright left middle right bottomleft bottomright commleft commright =
( associative-comp-hom-Precategory C _ _ _) ∙
( ap (postcomp-hom-Precategory C bottomright _) commleft) ∙
( inv (associative-comp-hom-Precategory C _ _ _)) ∙
( ap (precomp-hom-P... | function | pasting-horizontal-coherence-square-hom-Precategory | category-theory | src/category-theory/commuting-squares-of-morphisms-in-precategories.lagda.md | [
"category-theory.commuting-squares-of-morphisms-in-set-magmoids",
"category-theory.precategories",
"foundation.action-on-identifications-functions",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Precategory",
"ap",
"coherence-square-hom-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
coherence-square-hom-Set-Magmoid :
{l1 l2 : Level} (C : Set-Magmoid l1 l2)
{x y z w : obj-Set-Magmoid C}
(top : hom-Set-Magmoid C x y)
(left : hom-Set-Magmoid C x z)
(right : hom-Set-Magmoid C y w)
(bottom : hom-Set-Magmoid C z w) →
UU l2 | coherence-square-hom-Set-Magmoid C top left right bottom =
( comp-hom-Set-Magmoid C bottom left) =
( comp-hom-Set-Magmoid C right top) | function | coherence-square-hom-Set-Magmoid | category-theory | src/category-theory/commuting-squares-of-morphisms-in-set-magmoids.lagda.md | [
"category-theory.set-magmoids",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Set-Magmoid"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
coherence-triangle-hom-Precategory :
{l1 l2 : Level} (C : Precategory l1 l2)
{x y z : obj-Precategory C}
(top : hom-Precategory C x y)
(left : hom-Precategory C x z)
(right : hom-Precategory C y z) →
UU l2 | coherence-triangle-hom-Precategory C =
coherence-triangle-hom-Set-Magmoid (set-magmoid-Precategory C) | function | coherence-triangle-hom-Precategory | category-theory | src/category-theory/commuting-triangles-of-morphisms-in-precategories.lagda.md | [
"category-theory.commuting-triangles-of-morphisms-in-set-magmoids",
"category-theory.precategories",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Precategory",
"coherence-triangle-hom-Set-Magmoid"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
coherence-triangle-hom-Set-Magmoid :
{l1 l2 : Level} (C : Set-Magmoid l1 l2)
{x y z : obj-Set-Magmoid C}
(top : hom-Set-Magmoid C x y)
(left : hom-Set-Magmoid C x z)
(right : hom-Set-Magmoid C y z) →
UU l2 | coherence-triangle-hom-Set-Magmoid C top left right =
left = comp-hom-Set-Magmoid C right top | function | coherence-triangle-hom-Set-Magmoid | category-theory | src/category-theory/commuting-triangles-of-morphisms-in-set-magmoids.lagda.md | [
"category-theory.set-magmoids",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Set-Magmoid"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-complete-Precategory :
(l1 l2 : Level) {l3 l4 : Level}
(D : Precategory l3 l4) →
UU (lsuc l1 ⊔ lsuc l2 ⊔ l3 ⊔ l4) | is-complete-Precategory l1 l2 D =
(C : Precategory l1 l2) (F : functor-Precategory C D) →
limit-Precategory C D F | function | is-complete-Precategory | category-theory | src/category-theory/complete-precategories.lagda.md | [
"category-theory.cones-precategories",
"category-theory.functors-precategories",
"category-theory.limits-precategories",
"category-theory.precategories",
"category-theory.terminal-objects-precategories",
"foundation.universe-levels"
] | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
conservative-functor-Precategory :
{l1 l2 l3 l4 : Level} (C : Precategory l1 l2) (D : Precategory l3 l4) →
UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) | conservative-functor-Precategory C D =
Σ ( functor-Precategory C D)
( is-conservative-functor-Precategory C D) | function | conservative-functor-Precategory | category-theory | src/category-theory/conservative-functors-precategories.lagda.md | [
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.iterated-dependent-product-types",
"foundation.propositions",
"foundation.telescopes"... | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
core-precategory-Category :
{l1 l2 : Level} (C : Category l1 l2) → Precategory l1 l2 | core-precategory-Category C =
core-precategory-Precategory (precategory-Category C) | function | core-precategory-Category | category-theory | src/category-theory/cores-categories.lagda.md | [
"category-theory.categories",
"category-theory.cores-precategories",
"category-theory.groupoids",
"category-theory.isomorphisms-in-categories",
"category-theory.precategories",
"category-theory.pregroupoids",
"category-theory.subcategories",
"category-theory.wide-subcategories",
"foundation.dependen... | [
"Category",
"Precategory",
"core-precategory-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
core-category-Category :
{l1 l2 : Level} (C : Category l1 l2) → Category l1 l2 | function | core-category-Category | category-theory | src/category-theory/cores-categories.lagda.md | [
"category-theory.categories",
"category-theory.cores-precategories",
"category-theory.groupoids",
"category-theory.isomorphisms-in-categories",
"category-theory.precategories",
"category-theory.pregroupoids",
"category-theory.subcategories",
"category-theory.wide-subcategories",
"foundation.dependen... | [
"Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
pr1 (core-category-Category C) = core-precategory-Category C | function | pr1 | category-theory | src/category-theory/cores-categories.lagda.md | [
"category-theory.categories",
"category-theory.cores-precategories",
"category-theory.groupoids",
"category-theory.isomorphisms-in-categories",
"category-theory.precategories",
"category-theory.pregroupoids",
"category-theory.subcategories",
"category-theory.wide-subcategories",
"foundation.dependen... | [
"core-category-Category",
"core-precategory-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
pr2 (core-category-Category C) =
is-category-core-is-category-Precategory
( precategory-Category C)
( is-category-Category C) | function | pr2 | category-theory | src/category-theory/cores-categories.lagda.md | [
"category-theory.categories",
"category-theory.cores-precategories",
"category-theory.groupoids",
"category-theory.isomorphisms-in-categories",
"category-theory.precategories",
"category-theory.pregroupoids",
"category-theory.subcategories",
"category-theory.wide-subcategories",
"foundation.dependen... | [
"core-category-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
core-precategory-Precategory :
{l1 l2 : Level} (C : Precategory l1 l2) → Precategory l1 l2 | core-precategory-Precategory C =
precategory-Wide-Subprecategory C (core-wide-subprecategory-Precategory C) | function | core-precategory-Precategory | category-theory | src/category-theory/cores-precategories.lagda.md | [
"category-theory.categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"category-theory.pregroupoids",
"category-theory.replete-subprecategories",
"category-theory.subprecategories",
"category-theory.wide-subprecate... | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom D : hom C x y → obj D x → obj D y → Set | function | hom | category-theory | src/category-theory/dependent-composition-operations-over-precategories.lagda.md | [
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.nonunital-precategories",
"category-theory.precategories",
"category-theory.set-magmoids",
"foundation.cartesian-product-types",
"foundation.dependent-identifications",
"foundation.dependent-pair-types",
"foundation.... | [
"Set"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
essentially-injective-functor-Precategory :
{l1 l2 l3 l4 : Level} (C : Precategory l1 l2) (D : Precategory l3 l4) →
UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) | essentially-injective-functor-Precategory C D =
Σ ( functor-Precategory C D)
( is-essentially-injective-functor-Precategory C D) | function | essentially-injective-functor-Precategory | category-theory | src/category-theory/essentially-injective-functors-precategories.lagda.md | [
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"foundation.dependent-pair-types",
"foundation.universe-levels"
] | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
essentially-surjective-functor-Precategory :
{l1 l2 l3 l4 : Level} (C : Precategory l1 l2) (D : Precategory l3 l4) →
UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) | essentially-surjective-functor-Precategory C D =
Σ ( functor-Precategory C D)
( is-essentially-surjective-functor-Precategory C D) | function | essentially-surjective-functor-Precategory | category-theory | src/category-theory/essentially-surjective-functors-precategories.lagda.md | [
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.existential-quantification",
"foundation.propositions",
"foundation.universe-levels"
... | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
faithful-functor-Precategory :
{l1 l2 l3 l4 : Level}
(C : Precategory l1 l2)
(D : Precategory l3 l4) → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) | faithful-functor-Precategory C D =
Σ (functor-Precategory C D) (is-faithful-functor-Precategory C D) | function | faithful-functor-Precategory | category-theory | src/category-theory/faithful-functors-precategories.lagda.md | [
"category-theory.faithful-maps-precategories",
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.embeddings",
"foundation.equivalences"... | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Full-Subcategory :
{l1 l2 : Level} (l3 : Level) (C : Category l1 l2) → UU (l1 ⊔ lsuc l3) | Full-Subcategory l3 C = Full-Subprecategory l3 (precategory-Category C) | function | Full-Subcategory | category-theory | src/category-theory/full-subcategories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.embeddings-precategories",
"category-theory.full-subprecategories",
"category-theory.fully-faithful-functors-precategories",
"category-theory.functors-categories",
"category-theory.maps-ca... | [
"Category",
"Full-Subprecategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Full-Subprecategory :
{l1 l2 : Level} (l3 : Level) (C : Precategory l1 l2) → UU (l1 ⊔ lsuc l3) | Full-Subprecategory l3 C = subtype l3 (obj-Precategory C) | function | Full-Subprecategory | category-theory | src/category-theory/full-subprecategories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.embeddings-precategories",
"category-theory.fully-faithful-functors-precategories",
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-categories",
"category-theory... | [
"Precategory",
"subtype"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
id-functor-Category :
{l1 l2 : Level} (C : Category l1 l2) → functor-Category C C | id-functor-Category C = id-functor-Precategory (precategory-Category C) | function | id-functor-Category | category-theory | src/category-theory/functors-categories.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-categories",
"category-theory.maps-categories",
"foundation.dependent-products-propositions",
"foundation.equivalences",
"foundation.function-types",
"foundation.homotopies",
"foundation.identity... | [
"Category",
"id-functor-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
comp-functor-Category :
{l1 l2 l3 l4 l5 l6 : Level}
(C : Category l1 l2) (D : Category l3 l4) (E : Category l5 l6) →
functor-Category D E → functor-Category C D → functor-Category C E | comp-functor-Category C D E =
comp-functor-Precategory
( precategory-Category C)
( precategory-Category D)
( precategory-Category E) | function | comp-functor-Category | category-theory | src/category-theory/functors-categories.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.isomorphisms-in-categories",
"category-theory.maps-categories",
"foundation.dependent-products-propositions",
"foundation.equivalences",
"foundation.function-types",
"foundation.homotopies",
"foundation.identity... | [
"Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
id-functor-Large-Category :
{αC : Level → Level} {βC : Level → Level → Level} →
(C : Large-Category αC βC) →
functor-Large-Category (λ l → l) C C | id-functor-Large-Category C =
id-functor-Large-Precategory (large-precategory-Large-Category C) | function | id-functor-Large-Category | category-theory | src/category-theory/functors-large-categories.lagda.md | [
"category-theory.functors-large-precategories",
"category-theory.large-categories",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Large-Category",
"id-functor-Large-Precategory",
"large-precategory-Large-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
comp-functor-Large-Category :
{αC αD αE γG γF : Level → Level}
{βC βD βE : Level → Level → Level} →
(C : Large-Category αC βC)
(D : Large-Category αD βD)
(E : Large-Category αE βE) →
functor-Large-Category γG D E →
functor-Large-Category γF C D →
functor-Large-Category (λ l → γG (γF l)) C E | comp-functor-Large-Category C D E =
comp-functor-Large-Precategory
( large-precategory-Large-Category C)
( large-precategory-Large-Category D)
( large-precategory-Large-Category E) | function | comp-functor-Large-Category | category-theory | src/category-theory/functors-large-categories.lagda.md | [
"category-theory.functors-large-precategories",
"category-theory.large-categories",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Large-Category",
"large-precategory-Large-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
id-functor-Large-Precategory :
{αC : Level → Level} {βC : Level → Level → Level} →
(C : Large-Precategory αC βC) →
functor-Large-Precategory (λ l → l) C C | function | id-functor-Large-Precategory | category-theory | src/category-theory/functors-large-precategories.lagda.md | [
"category-theory.large-precategories",
"foundation.action-on-identifications-functions",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Large-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
obj-functor-Large-Precategory
( id-functor-Large-Precategory C) =
id | function | obj-functor-Large-Precategory | category-theory | src/category-theory/functors-large-precategories.lagda.md | [
"category-theory.large-precategories",
"foundation.action-on-identifications-functions",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"id",
"id-functor-Large-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
hom-functor-Large-Precategory
( id-functor-Large-Precategory C) =
id | function | hom-functor-Large-Precategory | category-theory | src/category-theory/functors-large-precategories.lagda.md | [
"category-theory.large-precategories",
"foundation.action-on-identifications-functions",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"id",
"id-functor-Large-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
preserves-comp-functor-Large-Precategory
( id-functor-Large-Precategory C) g f =
refl | function | preserves-comp-functor-Large-Precategory | category-theory | src/category-theory/functors-large-precategories.lagda.md | [
"category-theory.large-precategories",
"foundation.action-on-identifications-functions",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"id-functor-Large-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
preserves-id-functor-Large-Precategory
( id-functor-Large-Precategory C) =
refl | function | preserves-id-functor-Large-Precategory | category-theory | src/category-theory/functors-large-precategories.lagda.md | [
"category-theory.large-precategories",
"foundation.action-on-identifications-functions",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"id-functor-Large-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
id-functor-Nonunital-Precategory :
{l1 l2 : Level} (C : Nonunital-Precategory l1 l2) →
functor-Nonunital-Precategory C C | id-functor-Nonunital-Precategory C =
id-functor-Set-Magmoid (set-magmoid-Nonunital-Precategory C) | function | id-functor-Nonunital-Precategory | category-theory | src/category-theory/functors-nonunital-precategories.lagda.md | [
"category-theory.functors-set-magmoids",
"category-theory.maps-set-magmoids",
"category-theory.nonunital-precategories",
"foundation.dependent-pair-types",
"foundation.equivalences",
"foundation.function-types",
"foundation.homotopies",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"Nonunital-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
id-functor-Precategory :
{l1 l2 : Level} (C : Precategory l1 l2) → functor-Precategory C C | function | id-functor-Precategory | category-theory | src/category-theory/functors-precategories.lagda.md | [
"category-theory.functors-set-magmoids",
"category-theory.isomorphisms-in-precategories",
"category-theory.maps-precategories",
"category-theory.opposite-precategories",
"category-theory.precategories",
"foundation.action-on-identifications-functions",
"foundation.dependent-pair-types",
"foundation.de... | [
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
pr1 (id-functor-Precategory C) = id | pr1 (pr2 (id-functor-Precategory C)) = id
pr1 (pr2 (pr2 (id-functor-Precategory C))) g f = refl | function | pr1 | category-theory | src/category-theory/functors-precategories.lagda.md | [
"category-theory.functors-set-magmoids",
"category-theory.isomorphisms-in-precategories",
"category-theory.maps-precategories",
"category-theory.opposite-precategories",
"category-theory.precategories",
"foundation.action-on-identifications-functions",
"foundation.dependent-pair-types",
"foundation.de... | [
"id",
"id-functor-Precategory",
"pr2"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
pr2 (pr2 (pr2 (id-functor-Precategory C))) x = refl | function | pr2 | category-theory | src/category-theory/functors-precategories.lagda.md | [
"category-theory.functors-set-magmoids",
"category-theory.isomorphisms-in-precategories",
"category-theory.maps-precategories",
"category-theory.opposite-precategories",
"category-theory.precategories",
"foundation.action-on-identifications-functions",
"foundation.dependent-pair-types",
"foundation.de... | [
"id-functor-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
Gaunt-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | Gaunt-Category l1 l2 = Σ (Category l1 l2) (is-gaunt-Category) | function | Gaunt-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
nonunital-precategory-Gaunt-Category :
{l1 l2 : Level} → Gaunt-Category l1 l2 → Nonunital-Precategory l1 l2 | nonunital-precategory-Gaunt-Category C =
nonunital-precategory-Category (category-Gaunt-Category C) | function | nonunital-precategory-Gaunt-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Gaunt-Category",
"Nonunital-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
precategory-Gaunt-Category :
{l1 l2 : Level} → Gaunt-Category l1 l2 → Precategory l1 l2 | precategory-Gaunt-Category C = precategory-Category (category-Gaunt-Category C) | function | precategory-Gaunt-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Gaunt-Category",
"Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
strongly-preunivalent-category-Gaunt-Category :
{l1 l2 : Level} → Gaunt-Category l1 l2 → Strongly-Preunivalent-Category l1 l2 | strongly-preunivalent-category-Gaunt-Category C =
strongly-preunivalent-category-Category (category-Gaunt-Category C) | function | strongly-preunivalent-category-Gaunt-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Gaunt-Category",
"Strongly-Preunivalent-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
total-hom-Gaunt-Category :
{l1 l2 : Level} (C : Gaunt-Category l1 l2) → UU (l1 ⊔ l2) | total-hom-Gaunt-Category C =
total-hom-Category (category-Gaunt-Category C) | function | total-hom-Gaunt-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Gaunt-Category",
"total-hom-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
obj-total-hom-Gaunt-Category :
{l1 l2 : Level} (C : Gaunt-Category l1 l2) →
total-hom-Gaunt-Category C →
obj-Gaunt-Category C × obj-Gaunt-Category C | obj-total-hom-Gaunt-Category C =
obj-total-hom-Category (category-Gaunt-Category C) | function | obj-total-hom-Gaunt-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Gaunt-Category",
"obj-total-hom-Category",
"total-hom-Gaunt-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-strict-category-is-prop-iso-Strongly-Preunivalent-Category :
{l1 l2 : Level} (C : Strongly-Preunivalent-Category l1 l2) →
is-prop-iso-Precategory (precategory-Strongly-Preunivalent-Category C) →
is-strict-category-Strongly-Preunivalent-Category C | is-strict-category-is-prop-iso-Strongly-Preunivalent-Category
C is-prop-iso-C x y =
is-prop-emb (emb-iso-eq-Strongly-Preunivalent-Category C) (is-prop-iso-C x y) | function | is-strict-category-is-prop-iso-Strongly-Preunivalent-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Strongly-Preunivalent-Category",
"is-prop-emb"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-strict-category-is-gaunt-Category :
{l1 l2 : Level} (C : Category l1 l2) →
is-gaunt-Category C → is-strict-category-Category C | is-strict-category-is-gaunt-Category C =
is-strict-category-is-prop-iso-Strongly-Preunivalent-Category
( strongly-preunivalent-category-Category C) | function | is-strict-category-is-gaunt-Category | category-theory | src/category-theory/gaunt-categories.lagda.md | [
"category-theory.categories",
"category-theory.composition-operations-on-binary-families-of-sets",
"category-theory.isomorphism-induction-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.nonunital-precategories",
"category-theor... | [
"Category",
"is-strict-category-is-prop-iso-Strongly-Preunivalent-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-groupoid-prop-Category :
{l1 l2 : Level} (C : Category l1 l2) → Prop (l1 ⊔ l2) | is-groupoid-prop-Category C =
is-pregroupoid-prop-Precategory (precategory-Category C) | function | is-groupoid-prop-Category | category-theory | src/category-theory/groupoids.lagda.md | [
"category-theory.categories",
"category-theory.functors-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"category-theory.pregroupoids",
"foundation.1-types",
"foundation.contractible-types",
"foundation.depe... | [
"Category",
"Prop"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-groupoid-Category :
{l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2) | is-groupoid-Category C =
is-pregroupoid-Precategory (precategory-Category C) | function | is-groupoid-Category | category-theory | src/category-theory/groupoids.lagda.md | [
"category-theory.categories",
"category-theory.functors-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"category-theory.pregroupoids",
"foundation.1-types",
"foundation.contractible-types",
"foundation.depe... | [
"Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Groupoid : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | Groupoid l1 l2 = Σ (Category l1 l2) is-groupoid-Category | function | Groupoid | category-theory | src/category-theory/groupoids.lagda.md | [
"category-theory.categories",
"category-theory.functors-categories",
"category-theory.isomorphisms-in-categories",
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"category-theory.pregroupoids",
"foundation.1-types",
"foundation.contractible-types",
"foundation.depe... | [
"Category",
"is-groupoid-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-section-indiscrete-Precategory :
{l : Level} → obj-Precategory ∘ indiscrete-Precategory {l} ~ id | is-section-indiscrete-Precategory X = refl | function | is-section-indiscrete-Precategory | category-theory | src/category-theory/indiscrete-precategories.lagda.md | [
"category-theory.isomorphisms-in-precategories",
"category-theory.precategories",
"category-theory.pregroupoids",
"category-theory.preunivalent-categories",
"category-theory.strict-categories",
"category-theory.subterminal-precategories",
"foundation.contractible-types",
"foundation.dependent-pair-typ... | [
"id"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
obj-initial-Category : UU lzero | obj-initial-Category = empty | function | obj-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"empty"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-set-initial-Category :
obj-initial-Category → obj-initial-Category → Set lzero | hom-set-initial-Category _ _ = unit-Set | function | hom-set-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"Set",
"obj-initial-Category",
"unit-Set"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-initial-Category :
obj-initial-Category → obj-initial-Category → UU lzero | hom-initial-Category x y = type-Set (hom-set-initial-Category x y) | function | hom-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"hom-set-initial-Category",
"obj-initial-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
comp-hom-initial-Category :
{x y z : obj-initial-Category} →
hom-initial-Category y z → hom-initial-Category x y → hom-initial-Category x z | comp-hom-initial-Category {x} {y} {z} =
comp-hom-indiscrete-Precategory empty {x} {y} {z} | function | comp-hom-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"empty",
"hom-initial-Category",
"obj-initial-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
associative-comp-hom-initial-Category :
{x y z w : obj-initial-Category}
(h : hom-initial-Category z w)
(g : hom-initial-Category y z)
(f : hom-initial-Category x y) →
comp-hom-initial-Category {x} {y} {w}
( comp-hom-initial-Category {y} {z} {w} h g)
( f) =
comp-hom-initial-Category {x} {z} {w}
... | associative-comp-hom-initial-Category {x} {y} {z} {w} =
associative-comp-hom-indiscrete-Precategory empty {x} {y} {z} {w} | function | associative-comp-hom-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"comp-hom-initial-Category",
"empty",
"hom-initial-Category",
"obj-initial-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
id-hom-initial-Category : {x : obj-initial-Category} → hom-initial-Category x x | id-hom-initial-Category {x} = id-hom-indiscrete-Precategory empty {x} | function | id-hom-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"empty",
"hom-initial-Category",
"obj-initial-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
left-unit-law-comp-hom-initial-Category :
{x y : obj-initial-Category}
(f : hom-initial-Category x y) →
comp-hom-initial-Category {x} {y} {y} (id-hom-initial-Category {y}) f = f | left-unit-law-comp-hom-initial-Category {x} {y} =
left-unit-law-comp-hom-indiscrete-Precategory empty {x} {y} | function | left-unit-law-comp-hom-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"comp-hom-initial-Category",
"empty",
"hom-initial-Category",
"id-hom-initial-Category",
"obj-initial-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
right-unit-law-comp-hom-initial-Category :
{x y : obj-initial-Category}
(f : hom-initial-Category x y) →
comp-hom-initial-Category {x} {x} {y} f (id-hom-initial-Category {x}) = f | right-unit-law-comp-hom-initial-Category {x} {y} =
right-unit-law-comp-hom-indiscrete-Precategory empty {x} {y} | function | right-unit-law-comp-hom-initial-Category | category-theory | src/category-theory/initial-category.lagda.md | [
"category-theory.categories",
"category-theory.functors-precategories",
"category-theory.gaunt-categories",
"category-theory.indiscrete-precategories",
"category-theory.precategories",
"category-theory.strict-categories",
"category-theory.strongly-preunivalent-categories",
"foundation.contractible-typ... | [
"comp-hom-initial-Category",
"empty",
"hom-initial-Category",
"id-hom-initial-Category",
"obj-initial-Category"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 |
Structured dataset from agda-unimath — Univalent mathematics.
c85d7fb834778f96a66576318cdc4ef3d4b80a26| Column | Type | Description |
|---|---|---|
| statement | string | Declaration signature/claim with the leading keyword removed (verbatim slice); the full declaration minus its proof |
| proof | string | Verbatim proof/body, empty if the declaration has none |
| type | string | Declaration keyword |
| symbolic_name | string | Declaration identifier |
| library | string | Sub-library |
| filename | string | Repository-relative source path |
| imports | list[string] | File-level Require/Import modules |
| deps | list[string] | Intra-corpus identifiers referenced |
| docstring | string | Preceding documentation comment, empty if absent |
| source_url | string | Upstream repository |
| commit | string | Upstream commit extracted |
| Type | Count |
|---|---|
| function | 11,307 |
| record | 133 |
| postulate | 80 |
| data | 62 |
is-complete-prop-Metric-Ab : {l1 l2 : Level} → Metric-Ab l1 l2 → Prop (l1 ⊔ l2)
is-complete-prop-Metric-Ab G =
is-complete-prop-Metric-Space (metric-space-Metric-Ab G)
is-complete-prop-Metric-Ab | src/analysis/complete-metric-abelian-groups.lagda.mdEach declaration is split into a statement (signature/claim) and a proof (body) that are disjoint
and together form the complete declaration, for proof modeling, autoformalization, retrieval, and
dependency analysis via deps.
@misc{agda_unimath_dataset,
title = {Agda-UniMath},
author = {Norton, Charles},
year = {2026},
note = {Extracted from https://github.com/UniMath/agda-unimath, commit c85d7fb83477},
url = {https://huggingface.co/datasets/phanerozoic/Agda-UniMath}
}