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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
End of preview. Expand in Data Studio

Agda-UniMath

Structured dataset from agda-unimath — Univalent mathematics.

Source

Schema

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

Statistics

  • Entries: 11,582
  • With proof: 9,595 (82.8%)
  • With docstring: 0 (0.0%)
  • Libraries: 39

By type

Type Count
function 11,307
record 133
postulate 80
data 62

Example

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)
  • type: function | symbolic_name: is-complete-prop-Metric-Ab | src/analysis/complete-metric-abelian-groups.lagda.md

Use

Each 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.

Citation

@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}
}
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