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
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|---|---|---|---|---|---|---|---|---|---|---|
is-untruncated-π-finite-retract :
{l1 l2 : Level} (k : ℕ) {A : UU l1} {B : UU l2} →
A retract-of B → is-untruncated-π-finite k B → is-untruncated-π-finite k A | is-untruncated-π-finite-retract zero-ℕ =
has-finitely-many-connected-components-retract | function | is-untruncated-π-finite-retract | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-equiv :
{l1 l2 : Level} (k : ℕ) {A : UU l1} {B : UU l2} →
A ≃ B → is-untruncated-π-finite k B → is-untruncated-π-finite k A | is-untruncated-π-finite-equiv k e =
is-untruncated-π-finite-retract k (retract-equiv e) | function | is-untruncated-π-finite-equiv | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"is-untruncated-π-finite",
"is-untruncated-π-finite-retract"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-equiv' :
{l1 l2 : Level} (k : ℕ) {A : UU l1} {B : UU l2} →
A ≃ B → is-untruncated-π-finite k A → is-untruncated-π-finite k B | is-untruncated-π-finite-equiv' k e =
is-untruncated-π-finite-retract k (retract-inv-equiv e) | function | is-untruncated-π-finite-equiv' | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"is-untruncated-π-finite",
"is-untruncated-π-finite-retract"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-empty : (k : ℕ) → is-untruncated-π-finite k empty | is-untruncated-π-finite-empty zero-ℕ =
has-finitely-many-connected-components-empty
is-untruncated-π-finite-empty (succ-ℕ k) =
( is-untruncated-π-finite-empty zero-ℕ , ind-empty) | function | is-untruncated-π-finite-empty | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"empty",
"has-finitely-many-connected-components-empty",
"ind-empty",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
empty-Untruncated-π-Finite-Type : (k : ℕ) → Untruncated-π-Finite-Type lzero k | empty-Untruncated-π-Finite-Type k = (empty , is-untruncated-π-finite-empty k) | function | empty-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Untruncated-π-Finite-Type",
"empty",
"is-untruncated-π-finite-empty"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-is-empty :
{l : Level} (k : ℕ) {A : UU l} → is-empty A → is-untruncated-π-finite k A | is-untruncated-π-finite-is-empty zero-ℕ =
has-finitely-many-connected-components-is-empty
is-untruncated-π-finite-is-empty (succ-ℕ k) f =
( is-untruncated-π-finite-is-empty zero-ℕ f , ex-falso ∘ f) | function | is-untruncated-π-finite-is-empty | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"ex-falso",
"has-finitely-many-connected-components-is-empty",
"is-empty",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-is-contr :
{l : Level} (k : ℕ) {A : UU l} → is-contr A → is-untruncated-π-finite k A | is-untruncated-π-finite-is-contr zero-ℕ =
has-finitely-many-connected-components-is-contr | function | is-untruncated-π-finite-is-contr | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"has-finitely-many-connected-components-is-contr",
"is-contr",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-unit : (k : ℕ) → is-untruncated-π-finite k unit | is-untruncated-π-finite-unit k =
is-untruncated-π-finite-is-contr k is-contr-unit | function | is-untruncated-π-finite-unit | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"is-contr-unit",
"is-untruncated-π-finite",
"is-untruncated-π-finite-is-contr",
"unit"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
unit-Untruncated-π-Finite-Type : (k : ℕ) → Untruncated-π-Finite-Type lzero k | unit-Untruncated-π-Finite-Type k =
( unit , is-untruncated-π-finite-unit k) | function | unit-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Untruncated-π-Finite-Type",
"is-untruncated-π-finite-unit",
"unit"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-coproduct :
{l1 l2 : Level} (k : ℕ) {A : UU l1} {B : UU l2} →
is-untruncated-π-finite k A → is-untruncated-π-finite k B →
is-untruncated-π-finite k (A + B) | is-untruncated-π-finite-coproduct zero-ℕ =
has-finitely-many-connected-components-coproduct | function | is-untruncated-π-finite-coproduct | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"has-finitely-many-connected-components-coproduct",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
coproduct-Untruncated-π-Finite-Type :
{l1 l2 : Level} (k : ℕ) →
Untruncated-π-Finite-Type l1 k →
Untruncated-π-Finite-Type l2 k →
Untruncated-π-Finite-Type (l1 ⊔ l2) k | function | coproduct-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Untruncated-π-Finite-Type"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
is-untruncated-π-finite-Maybe :
{l : Level} (k : ℕ) {A : UU l} →
is-untruncated-π-finite k A → is-untruncated-π-finite k (Maybe A) | is-untruncated-π-finite-Maybe k H =
is-untruncated-π-finite-coproduct k H (is-untruncated-π-finite-unit k) | function | is-untruncated-π-finite-Maybe | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Maybe",
"is-untruncated-π-finite",
"is-untruncated-π-finite-coproduct",
"is-untruncated-π-finite-unit"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Maybe-Untruncated-π-Finite-Type :
{l : Level} (k : ℕ) →
Untruncated-π-Finite-Type l k →
Untruncated-π-Finite-Type l k | Maybe-Untruncated-π-Finite-Type k A =
coproduct-Untruncated-π-Finite-Type k A (unit-Untruncated-π-Finite-Type k) | function | Maybe-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Untruncated-π-Finite-Type",
"coproduct-Untruncated-π-Finite-Type",
"unit-Untruncated-π-Finite-Type"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-Fin :
(k n : ℕ) → is-untruncated-π-finite k (Fin n) | is-untruncated-π-finite-Fin k zero-ℕ =
is-untruncated-π-finite-empty k
is-untruncated-π-finite-Fin k (succ-ℕ n) =
is-untruncated-π-finite-Maybe k (is-untruncated-π-finite-Fin k n) | function | is-untruncated-π-finite-Fin | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Fin",
"is-untruncated-π-finite",
"is-untruncated-π-finite-Maybe",
"is-untruncated-π-finite-empty"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Fin-Untruncated-π-Finite-Type :
(k : ℕ) (n : ℕ) → Untruncated-π-Finite-Type lzero k | Fin-Untruncated-π-Finite-Type k n = (Fin n , is-untruncated-π-finite-Fin k n) | function | Fin-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Fin",
"Untruncated-π-Finite-Type",
"is-untruncated-π-finite-Fin"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-count :
{l : Level} (k : ℕ) {A : UU l} → count A → is-untruncated-π-finite k A | is-untruncated-π-finite-count k (n , e) =
is-untruncated-π-finite-equiv' k e (is-untruncated-π-finite-Fin k n) | function | is-untruncated-π-finite-count | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"count",
"is-untruncated-π-finite",
"is-untruncated-π-finite-Fin",
"is-untruncated-π-finite-equiv'"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-is-finite :
{l : Level} (k : ℕ) {A : UU l} → is-finite A → is-untruncated-π-finite k A | is-untruncated-π-finite-is-finite k {A} H =
apply-universal-property-trunc-Prop H
( is-untruncated-π-finite-Prop k A)
( is-untruncated-π-finite-count k) | function | is-untruncated-π-finite-is-finite | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"apply-universal-property-trunc-Prop",
"is-finite",
"is-untruncated-π-finite",
"is-untruncated-π-finite-Prop",
"is-untruncated-π-finite-count"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
untruncated-π-finite-type-Finite-Type :
{l : Level} (k : ℕ) → Finite-Type l → Untruncated-π-Finite-Type l k | function | untruncated-π-finite-type-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Finite-Type",
"Untruncated-π-Finite-Type"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
is-untruncated-π-finite-Type-With-Cardinality-ℕ :
{l : Level} (k n : ℕ) →
is-untruncated-π-finite k (Type-With-Cardinality-ℕ l n) | is-untruncated-π-finite-Type-With-Cardinality-ℕ zero-ℕ n =
has-finitely-many-connected-components-Type-With-Cardinality-ℕ n | function | is-untruncated-π-finite-Type-With-Cardinality-ℕ | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Type-With-Cardinality-ℕ",
"has-finitely-many-connected-components-Type-With-Cardinality-ℕ",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Type-With-Cardinality-ℕ-Untruncated-π-Finite-Type :
(l : Level) (k n : ℕ) → Untruncated-π-Finite-Type (lsuc l) k | Type-With-Cardinality-ℕ-Untruncated-π-Finite-Type l k n =
( Type-With-Cardinality-ℕ l n ,
is-untruncated-π-finite-Type-With-Cardinality-ℕ k n) | function | Type-With-Cardinality-ℕ-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Type-With-Cardinality-ℕ",
"Untruncated-π-Finite-Type",
"is-untruncated-π-finite-Type-With-Cardinality-ℕ"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-is-untruncated-π-finite-succ-ℕ :
{l : Level} (k : ℕ) {A : UU l} →
is-untruncated-π-finite (succ-ℕ k) A → is-untruncated-π-finite k A | is-untruncated-π-finite-is-untruncated-π-finite-succ-ℕ zero-ℕ H =
has-finitely-many-connected-components-is-untruncated-π-finite 1 H | function | is-untruncated-π-finite-is-untruncated-π-finite-succ-ℕ | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"has-finitely-many-connected-components-is-untruncated-π-finite",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
untruncated-π-finite-type-succ-Untruncated-π-Finite-Type :
{l : Level} (k : ℕ) →
Untruncated-π-Finite-Type l (succ-ℕ k) → Untruncated-π-Finite-Type l k | untruncated-π-finite-type-succ-Untruncated-π-Finite-Type k =
tot (λ A → is-untruncated-π-finite-is-untruncated-π-finite-succ-ℕ k) | function | untruncated-π-finite-type-succ-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Untruncated-π-Finite-Type",
"is-untruncated-π-finite-is-untruncated-π-finite-succ-ℕ"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-one-is-untruncated-π-finite-succ-ℕ :
{l : Level} (k : ℕ) {A : UU l} →
is-untruncated-π-finite (succ-ℕ k) A → is-untruncated-π-finite 1 A | is-untruncated-π-finite-one-is-untruncated-π-finite-succ-ℕ zero-ℕ H = H
is-untruncated-π-finite-one-is-untruncated-π-finite-succ-ℕ (succ-ℕ k) H =
is-untruncated-π-finite-one-is-untruncated-π-finite-succ-ℕ k
( is-untruncated-π-finite-is-untruncated-π-finite-succ-ℕ (succ-ℕ k) H) | function | is-untruncated-π-finite-one-is-untruncated-π-finite-succ-ℕ | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"is-untruncated-π-finite",
"is-untruncated-π-finite-is-untruncated-π-finite-succ-ℕ"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-finite-is-untruncated-π-finite :
{l : Level} (k : ℕ) {A : UU l} → is-set A →
is-untruncated-π-finite k A → is-finite A | is-finite-is-untruncated-π-finite k H K =
is-finite-equiv'
( equiv-unit-trunc-Set (_ , H))
( has-finitely-many-connected-components-is-untruncated-π-finite k K) | function | is-finite-is-untruncated-π-finite | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"equiv-unit-trunc-Set",
"has-finitely-many-connected-components-is-untruncated-π-finite",
"is-finite",
"is-finite-equiv'",
"is-set",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-Π :
{l1 l2 : Level} (k : ℕ) {A : UU l1} {B : A → UU l2} →
is-finite A → ((a : A) → is-untruncated-π-finite k (B a)) →
is-untruncated-π-finite k ((a : A) → B a) | is-untruncated-π-finite-Π zero-ℕ =
has-finitely-many-connected-components-finite-Π | function | is-untruncated-π-finite-Π | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"has-finitely-many-connected-components-finite-Π",
"is-finite",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
finite-Π-Untruncated-π-Finite-Type :
{l1 l2 : Level} (k : ℕ) (A : Finite-Type l1)
(B : type-Finite-Type A → Untruncated-π-Finite-Type l2 k) →
Untruncated-π-Finite-Type (l1 ⊔ l2) k | function | finite-Π-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Finite-Type",
"Untruncated-π-Finite-Type",
"type-Finite-Type"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
has-finitely-many-connected-components-Σ-is-0-connected :
{l1 l2 : Level} {A : UU l1} {B : A → UU l2} →
is-0-connected A →
((a : A) → has-finitely-many-connected-components (a ＝ a)) →
((x : A) → has-finitely-many-connected-components (B x)) →
has-finitely-many-connected-components (Σ A B) | has-finitely-many-connected-components-Σ-is-0-connected {A = A} {B} C H K =
apply-universal-property-trunc-Prop
( is-inhabited-is-0-connected C)
( has-finitely-many-connected-components-Prop (Σ A B))
( α)
where
α : A → has-finitely-many-connected-components (Σ A B)
α a =
is-finite-codomain
... | function | has-finitely-many-connected-components-Σ-is-0-connected | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Prop",
"Prop-Set",
"ap",
"apply-universal-property-trunc-Prop",
"center",
"dependent-identification",
"has-decidable-equality-is-finite",
"has-finitely-many-connected-components",
"has-finitely-many-connected-components-Prop",
"hom-Set",
"hom-set-Set",
"is-0-connected",
"is-contr",
"is-de... | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
has-finitely-many-connected-components-Σ' :
{l1 l2 : Level} (k : ℕ) {A : UU l1} {B : A → UU l2} →
(Fin k ≃ type-trunc-Set A) →
((x y : A) → has-finitely-many-connected-components (x ＝ y)) →
((x : A) → has-finitely-many-connected-components (B x)) →
has-finitely-many-connected-components (Σ A B) | has-finitely-many-connected-components-Σ' zero-ℕ e H K =
has-finitely-many-connected-components-is-empty
( is-empty-is-empty-trunc-Set (map-inv-equiv e) ∘ pr1)
has-finitely-many-connected-components-Σ' (succ-ℕ k) {A} {B} e H K =
apply-universal-property-trunc-Prop
( has-presentation-of-cardinality-h... | function | has-finitely-many-connected-components-Σ' | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Fin",
"apply-universal-property-trunc-Prop",
"equiv-trunc-Set",
"has-finitely-many-connected-components",
"has-finitely-many-connected-components-Prop",
"has-finitely-many-connected-components-is-empty",
"has-finitely-many-connected-components-Σ-is-0-connected",
"has-presentation-of-cardinality-has-c... | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
has-finitely-many-connected-components-Σ :
{l1 l2 : Level} {A : UU l1} {B : A → UU l2} →
is-untruncated-π-finite 1 A →
((x : A) → has-finitely-many-connected-components (B x)) →
has-finitely-many-connected-components (Σ A B) | has-finitely-many-connected-components-Σ {A = A} {B} H K =
apply-universal-property-trunc-Prop
( pr1 H)
( has-finitely-many-connected-components-Prop (Σ A B))
( λ (k , e) →
has-finitely-many-connected-components-Σ' k e
( λ x y → is-untruncated-π-finite-Id 0 H x y)
( K)) | function | has-finitely-many-connected-components-Σ | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"apply-universal-property-trunc-Prop",
"has-finitely-many-connected-components",
"has-finitely-many-connected-components-Prop",
"has-finitely-many-connected-components-Σ'",
"is-untruncated-π-finite",
"is-untruncated-π-finite-Id",
"pr1"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-untruncated-π-finite-Σ :
{l1 l2 : Level} (k : ℕ) {A : UU l1} {B : A → UU l2} →
is-untruncated-π-finite (succ-ℕ k) A →
((x : A) → is-untruncated-π-finite k (B x)) →
is-untruncated-π-finite k (Σ A B) | is-untruncated-π-finite-Σ zero-ℕ =
has-finitely-many-connected-components-Σ | function | is-untruncated-π-finite-Σ | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"has-finitely-many-connected-components-Σ",
"is-untruncated-π-finite"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Σ-Untruncated-π-Finite-Type :
{l1 l2 : Level} (k : ℕ) (A : Untruncated-π-Finite-Type l1 (succ-ℕ k))
(B :
(x : type-Untruncated-π-Finite-Type (succ-ℕ k) A) →
Untruncated-π-Finite-Type l2 k) →
Untruncated-π-Finite-Type (l1 ⊔ l2) k | function | Σ-Untruncated-π-Finite-Type | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"Untruncated-π-Finite-Type",
"type-Untruncated-π-Finite-Type"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
pr1 (is-untruncated-π-finite-retract (succ-ℕ k) r H) =
is-untruncated-π-finite-retract zero-ℕ r
( has-finitely-many-connected-components-is-untruncated-π-finite
( succ-ℕ k)
( H)) | pr1 (is-untruncated-π-finite-is-contr (succ-ℕ k) H) =
is-untruncated-π-finite-is-contr zero-ℕ H
pr1 (is-untruncated-π-finite-coproduct (succ-ℕ k) H K) =
is-untruncated-π-finite-coproduct zero-ℕ (pr1 H) (pr1 K)
pr1 (coproduct-Untruncated-π-Finite-Type k A B) =
(type-Untruncated-π-Finite-Type k A + type-Untruncated... | function | pr1 | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"coproduct-Untruncated-π-Finite-Type",
"finite-Π-Untruncated-π-Finite-Type",
"has-finitely-many-connected-components-is-untruncated-π-finite",
"has-finitely-many-connected-components-Σ",
"is-untruncated-π-finite-Type-With-Cardinality-ℕ",
"is-untruncated-π-finite-coproduct",
"is-untruncated-π-finite-is-c... | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
pr2 (is-untruncated-π-finite-retract (succ-ℕ k) r H) x y =
is-untruncated-π-finite-retract k
( retract-eq r x y)
( is-untruncated-π-finite-Id k H
( inclusion-retract r x)
( inclusion-retract r y)) | pr2 (is-untruncated-π-finite-is-contr (succ-ℕ k) H) x y =
is-untruncated-π-finite-is-contr k (is-prop-is-contr H x y)
pr2 (is-untruncated-π-finite-coproduct (succ-ℕ k) H K) (inl x) (inl y) =
is-untruncated-π-finite-equiv k
( compute-eq-coproduct-inl-inl x y)
( is-untruncated-π-finite-Id k H x y)
pr2 (is-unt... | function | pr2 | univalent-combinatorics | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.0-connected-types",
"foundation.action-on-identifications-functions",
"foundation.contractible-types",
"foundation.coproduct-types",
"foundation.decidable-propositions",
"foundation.decidable-types",
"foundation.dependent-identifications",
"fou... | [
"coproduct-Untruncated-π-Finite-Type",
"equiv-equiv-eq-Type-With-Cardinality-ℕ",
"finite-Π-Untruncated-π-Finite-Type",
"is-finite-type-Finite-Type",
"is-finite-type-equiv",
"is-prop-is-contr",
"is-untruncated-π-finite-Id",
"is-untruncated-π-finite-Type-With-Cardinality-ℕ",
"is-untruncated-π-finite-c... | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Algebraic-Theory : {l1 : Level} (l2 : Level) → signature l1 → UU (l1 ⊔ lsuc l2) | Algebraic-Theory l2 σ = Σ (UU l2) (λ B → (B → abstract-equation σ)) | function | Algebraic-Theory | universal-algebra | src/universal-algebra/algebraic-theories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.universe-levels",
"universal-algebra.abstract-equations-over-signatures",
"universal-algebra.signatures"
] | [
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
group-ops : UU lzero where
unit-group-op mul-group-op inv-group-op : group-ops | data | group-ops | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
group-laws : UU lzero where
associative-group-laws : group-laws
invl-group-laws : group-laws
invr-group-laws : group-laws
idl-l-group-laws : group-laws
idr-group-laws : group-laws | data | group-laws | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
group-signature : signature lzero | function | group-signature | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
algebraic-theory-Group : Algebraic-Theory lzero group-signature | function | algebraic-theory-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"Algebraic-Theory",
"group-signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
algebra-Group : (l : Level) → UU (lsuc l) | algebra-Group l = Algebra l group-signature algebraic-theory-Group | function | algebra-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"Algebra",
"algebraic-theory-Group",
"group-signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
group-algebra-Group :
{l : Level} → algebra-Group l → Group l | group-algebra-Group ((A-Set , models-A) , satisfies-A) =
let
mul-A x y = models-A mul-group-op (x ∷ y ∷ empty-tuple)
inv-A x = models-A inv-group-op (x ∷ empty-tuple)
unit-A = models-A unit-group-op empty-tuple
associative-mul-A x y z =
satisfies-A
( associative-group-laws)
( fin... | function | group-algebra-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"Group",
"algebra-Group"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
algebra-group-Group :
{l : Level} → Group l → algebra-Group l | algebra-group-Group G =
let
fin : (i : ℕ) (k : ℕ) → {le-ℕ i k} → Fin k
fin i k {i<k} = standard-classical-Fin k (i , i<k)
in
( ( set-Group G ,
λ where
mul-group-op (x ∷ y ∷ empty-tuple) → mul-Group G x y
inv-group-op (x ∷ empty-tuple) → inv-Group G x
unit-group-op _... | function | algebra-group-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"Fin",
"Group",
"algebra-Group",
"le-ℕ",
"standard-classical-Fin",
"xs"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
equiv-group-algebra-Group :
{l : Level} → algebra-Group l ≃ Group l | function | equiv-group-algebra-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"Group",
"algebra-Group"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
hom-algebra-Group :
{l1 l2 : Level} → algebra-Group l1 → algebra-Group l2 → UU (l1 ⊔ l2) | hom-algebra-Group =
hom-Algebra group-signature algebraic-theory-Group | function | hom-algebra-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"algebra-Group",
"algebraic-theory-Group",
"group-signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-group-hom-algebra-Group :
{l1 l2 : Level} (G : algebra-Group l1) (H : algebra-Group l2) →
hom-algebra-Group G H →
hom-Group (group-algebra-Group G) (group-algebra-Group H) | hom-group-hom-algebra-Group G H (f , K) =
( f , λ {x} {y} → K mul-group-op (x ∷ y ∷ empty-tuple)) | function | hom-group-hom-algebra-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"algebra-Group",
"group-algebra-Group",
"hom-algebra-Group"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
hom-algebra-group-hom-Group :
{l1 l2 : Level} (G : Group l1) (H : Group l2) →
hom-Group G H →
hom-algebra-Group (algebra-group-Group G) (algebra-group-Group H) | hom-algebra-group-hom-Group G H (f , K) =
( f ,
λ where
unit-group-op empty-tuple → preserves-unit-hom-Group G H (f , K)
mul-group-op (x ∷ y ∷ empty-tuple) → K {x} {y}
inv-group-op (x ∷ empty-tuple) → preserves-inv-hom-Group G H (f , K)) | function | hom-algebra-group-hom-Group | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"Group",
"algebra-group-Group",
"hom-algebra-Group"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
pr1 group-signature = group-ops | pr1 algebraic-theory-Group = group-laws
pr1 equiv-group-algebra-Group = group-algebra-Group
pr1 (pr1 (pr2 equiv-group-algebra-Group)) = algebra-group-Group
pr1 (pr2 (pr2 equiv-group-algebra-Group)) = algebra-group-Group | function | pr1 | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"algebra-group-Group",
"algebraic-theory-Group",
"equiv-group-algebra-Group",
"group-algebra-Group",
"group-laws",
"group-ops",
"group-signature",
"pr2"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
pr2 group-signature unit-group-op = 0 | pr2 group-signature mul-group-op = 2
pr2 group-signature inv-group-op = 1
pr2 algebraic-theory-Group =
let
_*-term_ :
{k : ℕ} →
term group-signature k → term group-signature k → term group-signature k
_*-term_ x y =
op-term
( mul-group-op)
( x ∷ y ∷ empty-tuple)
inv-term ... | function | pr2 | universal-algebra | src/universal-algebra/algebraic-theory-of-groups.lagda.md | [
"elementary-number-theory.natural-numbers",
"elementary-number-theory.strict-inequality-natural-numbers",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equality-dependent-pair-types",
"foundation.equivalences",
"foundation.function-extensionality",
"found... | [
"algebraic-theory-Group",
"equiv-group-algebra-Group",
"group-signature",
"is-group-prop-Semigroup",
"le-ℕ",
"pr1",
"standard-classical-Fin"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Algebra :
{l1 l2 : Level} (l3 : Level)
(σ : signature l1) →
Algebraic-Theory l2 σ →
UU (l1 ⊔ l2 ⊔ lsuc l3) | Algebra l3 σ T =
type-subtype (is-algebra-prop-Model-Of-Signature σ T {l3}) | function | Algebra | universal-algebra | src/universal-algebra/algebras.lagda.md | [
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.fundamental-theorem-of-identity-types",
"foundation.identity-types",
"foundation.propositions",
"foundation.sets",
"foundation.subtype-identity-principle",
"foundation.subtypes",
"foundation.torsorial-type-f... | [
"Algebraic-Theory",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
congruence-Algebra :
{l1 l2 l3 : Level} (l4 : Level)
(σ : signature l1) (T : Algebraic-Theory l2 σ) (A : Algebra l3 σ T) →
UU (l1 ⊔ l3 ⊔ lsuc l4) | congruence-Algebra l4 σ T A =
Σ ( equivalence-relation l4 (type-Algebra σ T A))
( preserves-operations-equivalence-relation-Algebra σ T A) | function | congruence-Algebra | universal-algebra | src/universal-algebra/congruences.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equivalence-relations",
"foundation.propositions",
"foundation.raising-universe-levels-unit-type",
"foundation.unit-type",
"f... | [
"Algebra",
"Algebraic-Theory",
"equivalence-relation",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
equiv-Model-Of-Signature :
{l1 l2 l3 : Level} (σ : signature l1)
(X : Model-Of-Signature l2 σ)
(Y : Model-Of-Signature l3 σ) →
UU (l1 ⊔ l2 ⊔ l3) | equiv-Model-Of-Signature σ X Y =
Σ ( type-Model-Of-Signature σ X ≃ type-Model-Of-Signature σ Y)
( λ e → preserves-operations-Model-Of-Signature σ X Y (map-equiv e)) | function | equiv-Model-Of-Signature | universal-algebra | src/universal-algebra/equivalences-models-of-signatures.lagda.md | [
"foundation.action-on-identifications-functions",
"foundation.binary-homotopies",
"foundation.dependent-pair-types",
"foundation.dependent-products-propositions",
"foundation.equivalences-contractible-types",
"foundation.function-extensionality",
"foundation.functoriality-dependent-pair-types",
"found... | [
"Model-Of-Signature",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-extension-of-signature :
{l1 l2 : Level} →
signature l1 → signature l2 → UU (l1 ⊔ l2) | is-extension-of-signature σ τ =
Σ ( operation-signature σ ↪ operation-signature τ)
( λ f →
( (op : operation-signature σ) →
arity-operation-signature σ op ＝
arity-operation-signature τ (map-emb f op))) | function | is-extension-of-signature | universal-algebra | src/universal-algebra/extensions-signatures.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.embeddings",
"foundation.identity-types",
"foundation.universe-levels",
"universal-algebra.signatures"
] | [
"arity-operation-signature",
"operation-signature",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-model-of-signature-type :
{l1 l2 : Level} → signature l1 → UU l2 → UU (l1 ⊔ l2) | is-model-of-signature-type σ X =
(f : operation-signature σ) →
tuple X (arity-operation-signature σ f) → X | function | is-model-of-signature-type | universal-algebra | src/universal-algebra/models-of-signatures.lagda.md | [
"foundation.action-on-identifications-functions",
"foundation.dependent-pair-types",
"foundation.function-extensionality",
"foundation.sets",
"foundation.structure-identity-principle",
"foundation.universe-levels",
"foundation-core.cartesian-product-types",
"foundation-core.dependent-identifications",... | [
"arity-operation-signature",
"operation-signature",
"signature",
"tuple"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-model-of-signature : {l1 l2 : Level} → signature l1 → Set l2 → UU (l1 ⊔ l2) | is-model-of-signature σ X = is-model-of-signature-type σ (type-Set X) | function | is-model-of-signature | universal-algebra | src/universal-algebra/models-of-signatures.lagda.md | [
"foundation.action-on-identifications-functions",
"foundation.dependent-pair-types",
"foundation.function-extensionality",
"foundation.sets",
"foundation.structure-identity-principle",
"foundation.universe-levels",
"foundation-core.cartesian-product-types",
"foundation-core.dependent-identifications",... | [
"Set",
"is-model-of-signature-type",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Model-Of-Signature :
{l1 : Level} (l2 : Level) → signature l1 → UU (l1 ⊔ lsuc l2) | Model-Of-Signature l2 σ = Σ (Set l2) (is-model-of-signature σ) | function | Model-Of-Signature | universal-algebra | src/universal-algebra/models-of-signatures.lagda.md | [
"foundation.action-on-identifications-functions",
"foundation.dependent-pair-types",
"foundation.function-extensionality",
"foundation.sets",
"foundation.structure-identity-principle",
"foundation.universe-levels",
"foundation-core.cartesian-product-types",
"foundation-core.dependent-identifications",... | [
"Set",
"is-model-of-signature",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
signature : (l : Level) → UU (lsuc l) | signature l = Σ (UU l) (λ operations → (operations → ℕ)) | function | signature | universal-algebra | src/universal-algebra/signatures.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.embeddings",
"foundation.identity-types",
"foundation.universe-levels"
] | [] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
operation-signature : {l : Level} → signature l → UU l | operation-signature = pr1 | function | operation-signature | universal-algebra | src/universal-algebra/signatures.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.embeddings",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"pr1",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
arity-operation-signature :
{l : Level} (σ : signature l) → operation-signature σ → ℕ | arity-operation-signature = pr2 | function | arity-operation-signature | universal-algebra | src/universal-algebra/signatures.lagda.md | [
"elementary-number-theory.natural-numbers",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.embeddings",
"foundation.identity-types",
"foundation.universe-levels"
] | [
"operation-signature",
"pr2",
"signature"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-coinductive-iso-Noncoherent-Large-ω-Precategory
{α : Level → Level} {β : Level → Level → Level}
(𝒞 : Noncoherent-Large-ω-Precategory α β)
{l1 : Level} {x : obj-Noncoherent-Large-ω-Precategory 𝒞 l1}
{l2 : Level} {y : obj-Noncoherent-Large-ω-Precategory 𝒞 l2}
(f : hom-Noncoherent-Large-ω-Precategory 𝒞 x ... | record | is-coinductive-iso-Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/coinductive-isomorphisms-in-noncoherent-large-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.universe-levels",
"wild-category-theory.coinductive-isomorphisms-in-noncoherent-omega-precategories",
"wild-category-theory.noncoherent-large-omega-precategories"
] | [
"Noncoherent-Large-ω-Precategory",
"is-coinductive-iso-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
coinductive-iso-Noncoherent-Large-ω-Precategory :
{α : Level → Level} {β : Level → Level → Level}
(𝒞 : Noncoherent-Large-ω-Precategory α β)
{l1 : Level} (x : obj-Noncoherent-Large-ω-Precategory 𝒞 l1)
{l2 : Level} (y : obj-Noncoherent-Large-ω-Precategory 𝒞 l2) →
UU (β l1 l1 ⊔ β l1 l2 ⊔ β l2 l1 ⊔ β l2 l2) | coinductive-iso-Noncoherent-Large-ω-Precategory 𝒞 x y =
Σ ( hom-Noncoherent-Large-ω-Precategory 𝒞 x y)
( is-coinductive-iso-Noncoherent-Large-ω-Precategory 𝒞) | function | coinductive-iso-Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/coinductive-isomorphisms-in-noncoherent-large-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.universe-levels",
"wild-category-theory.coinductive-isomorphisms-in-noncoherent-omega-precategories",
"wild-category-theory.noncoherent-large-omega-precategories"
] | [
"Noncoherent-Large-ω-Precategory",
"is-coinductive-iso-Noncoherent-Large-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-coinductive-iso-Noncoherent-ω-Precategory
{l1 l2 : Level} (𝒞 : Noncoherent-ω-Precategory l1 l2)
{x y : obj-Noncoherent-ω-Precategory 𝒞}
(f : hom-Noncoherent-ω-Precategory 𝒞 x y) : UU l2
where
coinductive
field
hom-section-is-coinductive-iso-Noncoherent-ω-Precategory :
hom-Noncoherent-ω-Preca... | record | is-coinductive-iso-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/coinductive-isomorphisms-in-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.universe-levels",
"wild-category-theory.noncoherent-omega-precategories"
] | [
"Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
coinductive-iso-Noncoherent-ω-Precategory :
{l1 l2 : Level} (𝒞 : Noncoherent-ω-Precategory l1 l2)
(x y : obj-Noncoherent-ω-Precategory 𝒞) →
UU l2 | coinductive-iso-Noncoherent-ω-Precategory 𝒞 x y =
Σ ( hom-Noncoherent-ω-Precategory 𝒞 x y)
( is-coinductive-iso-Noncoherent-ω-Precategory 𝒞) | function | coinductive-iso-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/coinductive-isomorphisms-in-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.universe-levels",
"wild-category-theory.noncoherent-omega-precategories"
] | [
"Noncoherent-ω-Precategory",
"is-coinductive-iso-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-colax-functor-Noncoherent-Large-ω-Precategory
{α1 α2 : Level → Level}
{β1 β2 : Level → Level → Level}
{γ : Level → Level}
(𝒜 : Noncoherent-Large-ω-Precategory α1 β1)
(ℬ : Noncoherent-Large-ω-Precategory α2 β2)
(F : map-Noncoherent-Large-ω-Precategory γ 𝒜 ℬ) : UUω
where
constructor
make-is-cola... | record | is-colax-functor-Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-large-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types.large-colax-reflexive-globular-maps",
"globular-types.large-colax-transitive-globular-maps",
"gl... | [
"Noncoherent-Large-ω-Precategory",
"is-colax-functor-Noncoherent-ω-Precategory",
"is-colax-reflexive-1-cell-globular-map-is-colax-reflexive-large-globular-map",
"is-colax-transitive-1-cell-globular-map-is-colax-transitive-large-globular-map",
"map-Noncoherent-Large-ω-Precategory",
"preserves-comp-1-cell-i... | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
colax-functor-Noncoherent-Large-ω-Precategory
{α1 α2 : Level → Level}
{β1 β2 : Level → Level → Level}
(δ : Level → Level)
(𝒜 : Noncoherent-Large-ω-Precategory α1 β1)
(ℬ : Noncoherent-Large-ω-Precategory α2 β2) : UUω
where
constructor
make-colax-functor-Noncoherent-Large-ω-Precategory | record | colax-functor-Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-large-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types.large-colax-reflexive-globular-maps",
"globular-types.large-colax-transitive-globular-maps",
"gl... | [
"Noncoherent-Large-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
preserves-id-structure-map-Noncoherent-Large-ω-Precategory :
{α1 α2 γ : Level → Level}
{β1 β2 : Level → Level → Level}
(𝒜 : Noncoherent-Large-ω-Precategory α1 β1)
(ℬ : Noncoherent-Large-ω-Precategory α2 β2)
(F : map-Noncoherent-Large-ω-Precategory γ 𝒜 ℬ) → UUω | preserves-id-structure-map-Noncoherent-Large-ω-Precategory 𝒜 ℬ F =
is-colax-reflexive-large-globular-map
( large-reflexive-globular-type-Noncoherent-Large-ω-Precategory
𝒜)
( large-reflexive-globular-type-Noncoherent-Large-ω-Precategory
ℬ)
( F) | function | preserves-id-structure-map-Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-large-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types.large-colax-reflexive-globular-maps",
"globular-types.large-colax-transitive-globular-maps",
"gl... | [
"Noncoherent-Large-ω-Precategory",
"is-colax-reflexive-large-globular-map",
"map-Noncoherent-Large-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
preserves-comp-structure-map-Noncoherent-Large-ω-Precategory :
{α1 α2 γ : Level → Level}
{β1 β2 : Level → Level → Level}
(𝒜 : Noncoherent-Large-ω-Precategory α1 β1)
(ℬ : Noncoherent-Large-ω-Precategory α2 β2)
(F : map-Noncoherent-Large-ω-Precategory γ 𝒜 ℬ) → UUω | preserves-comp-structure-map-Noncoherent-Large-ω-Precategory 𝒜 ℬ F =
is-colax-transitive-large-globular-map
( large-transitive-globular-type-Noncoherent-Large-ω-Precategory
𝒜)
( large-transitive-globular-type-Noncoherent-Large-ω-Precategory
ℬ)
( F) | function | preserves-comp-structure-map-Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-large-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types.large-colax-reflexive-globular-maps",
"globular-types.large-colax-transitive-globular-maps",
"gl... | [
"Noncoherent-Large-ω-Precategory",
"is-colax-transitive-large-globular-map",
"map-Noncoherent-Large-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
is-colax-functor-Noncoherent-ω-Precategory
{l1 l2 l3 l4 : Level}
(𝒜 : Noncoherent-ω-Precategory l1 l2)
(ℬ : Noncoherent-ω-Precategory l3 l4)
(F : map-Noncoherent-ω-Precategory 𝒜 ℬ) : UU (l1 ⊔ l2 ⊔ l4)
where
constructor make-is-colax-functor-Noncoherent-ω-Precategory
coinductive
field
is... | record | is-colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"0-cell-globular-map",
"1-cell-globular-map-globular-map",
"Noncoherent-ω-Precategory",
"is-colax-reflexive-1-cell-globular-map-is-colax-reflexive-globular-map",
"is-colax-reflexive-globular-map",
"is-colax-transitive-1-cell-globular-map-is-colax-transitive-globular-map",
"is-colax-transitive-globular-m... | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
colax-functor-Noncoherent-ω-Precategory :
{l1 l2 l3 l4 : Level}
(𝒜 : Noncoherent-ω-Precategory l1 l2)
(ℬ : Noncoherent-ω-Precategory l3 l4) →
UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) | colax-functor-Noncoherent-ω-Precategory 𝒜 ℬ =
Σ ( map-Noncoherent-ω-Precategory 𝒜 ℬ)
( is-colax-functor-Noncoherent-ω-Precategory 𝒜 ℬ) | function | colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"Noncoherent-ω-Precategory",
"is-colax-functor-Noncoherent-ω-Precategory",
"map-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
map-id-colax-functor-Noncoherent-ω-Precategory :
{l1 l2 : Level} (𝒜 : Noncoherent-ω-Precategory l1 l2) →
map-Noncoherent-ω-Precategory 𝒜 𝒜 | map-id-colax-functor-Noncoherent-ω-Precategory 𝒜 =
id-map-Noncoherent-ω-Precategory 𝒜 | function | map-id-colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"Noncoherent-ω-Precategory",
"map-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
preserves-id-structure-id-colax-functor-Noncoherent-ω-Precategory :
{l1 l2 : Level} (𝒜 : Noncoherent-ω-Precategory l1 l2) →
preserves-id-structure-map-Noncoherent-ω-Precategory 𝒜 𝒜
( map-id-colax-functor-Noncoherent-ω-Precategory 𝒜) | preserves-id-structure-id-colax-functor-Noncoherent-ω-Precategory 𝒜 =
is-colax-reflexive-id-colax-reflexive-globular-map
( reflexive-globular-type-Noncoherent-ω-Precategory 𝒜) | function | preserves-id-structure-id-colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"Noncoherent-ω-Precategory",
"is-colax-reflexive-id-colax-reflexive-globular-map",
"map-id-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory :
{l1 l2 : Level} (𝒜 : Noncoherent-ω-Precategory l1 l2) →
preserves-comp-structure-map-Noncoherent-ω-Precategory 𝒜 𝒜
( map-id-colax-functor-Noncoherent-ω-Precategory 𝒜) | function | preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"Noncoherent-ω-Precategory",
"map-id-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
is-colax-functor-id-colax-functor-Noncoherent-ω-Precategory :
{l1 l2 : Level} (𝒜 : Noncoherent-ω-Precategory l1 l2) →
is-colax-functor-Noncoherent-ω-Precategory 𝒜 𝒜
( map-id-colax-functor-Noncoherent-ω-Precategory 𝒜) | is-colax-functor-id-colax-functor-Noncoherent-ω-Precategory 𝒜 =
make-is-colax-functor-Noncoherent-ω-Precategory
( preserves-id-structure-id-colax-functor-Noncoherent-ω-Precategory
𝒜)
( preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory
𝒜) | function | is-colax-functor-id-colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"Noncoherent-ω-Precategory",
"is-colax-functor-Noncoherent-ω-Precategory",
"map-id-colax-functor-Noncoherent-ω-Precategory",
"preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory",
"preserves-id-structure-id-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
id-colax-functor-Noncoherent-ω-Precategory :
{l1 l2 : Level} (𝒜 : Noncoherent-ω-Precategory l1 l2) →
colax-functor-Noncoherent-ω-Precategory 𝒜 𝒜 | id-colax-functor-Noncoherent-ω-Precategory 𝒜 =
( id-map-Noncoherent-ω-Precategory 𝒜 ,
is-colax-functor-id-colax-functor-Noncoherent-ω-Precategory 𝒜) | function | id-colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"Noncoherent-ω-Precategory",
"colax-functor-Noncoherent-ω-Precategory",
"is-colax-functor-id-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
preserves-id-structure-comp-colax-functor-Noncoherent-ω-Precategory :
{l1 l2 l3 l4 l5 l6 : Level}
(𝒜 : Noncoherent-ω-Precategory l1 l2)
(ℬ : Noncoherent-ω-Precategory l3 l4)
(𝒞 : Noncoherent-ω-Precategory l5 l6)
(G : colax-functor-Noncoherent-ω-Precategory ℬ 𝒞)
(F : colax-functor-Noncoherent-ω-Precategor... | function | preserves-id-structure-comp-colax-functor-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"Noncoherent-ω-Precategory",
"colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
preserves-comp-1-cell-is-colax-transitive-globular-map
( preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory
𝒜)
_ _ =
id-2-hom-Noncoherent-ω-Precategory 𝒜 | function | preserves-comp-1-cell-is-colax-transitive-globular-map | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
is-colax-transitive-1-cell-globular-map-is-colax-transitive-globular-map
( preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory
𝒜) =
preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory
( hom-noncoherent-ω-precategory-Noncoherent-ω-Precategory
( 𝒜)
( _)
( _)... | function | is-colax-transitive-1-cell-globular-map-is-colax-transitive-globular-map | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"preserves-comp-structure-id-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
preserves-refl-1-cell-is-colax-reflexive-globular-map
( preserves-id-structure-comp-colax-functor-Noncoherent-ω-Precategory
𝒜 ℬ 𝒞 G F)
x =
comp-2-hom-Noncoherent-ω-Precategory 𝒞
( preserves-id-hom-colax-functor-Noncoherent-ω-Precategory G _)
( 2-hom-colax-functor-Noncoherent-ω-Precategory G
... | function | preserves-refl-1-cell-is-colax-reflexive-globular-map | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"preserves-id-structure-comp-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
is-colax-reflexive-1-cell-globular-map-is-colax-reflexive-globular-map
( preserves-id-structure-comp-colax-functor-Noncoherent-ω-Precategory
𝒜 ℬ 𝒞 G F) =
preserves-id-structure-comp-colax-functor-Noncoherent-ω-Precategory
( hom-noncoherent-ω-precategory-Noncoherent-ω-Precategory
𝒜 _ _)
( hom-... | function | is-colax-reflexive-1-cell-globular-map-is-colax-reflexive-globular-map | wild-category-theory | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.colax-reflexive-globular-maps",
"globular-types.colax-transitive-globular-maps",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types... | [
"preserves-id-structure-comp-colax-functor-Noncoherent-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
map-Noncoherent-Large-ω-Precategory :
{α1 α2 : Level → Level} {β1 β2 : Level → Level → Level} (δ : Level → Level)
(𝒜 : Noncoherent-Large-ω-Precategory α1 β1)
(ℬ : Noncoherent-Large-ω-Precategory α2 β2) → UUω | map-Noncoherent-Large-ω-Precategory δ 𝒜 ℬ =
large-globular-map δ
( large-globular-type-Noncoherent-Large-ω-Precategory 𝒜)
( large-globular-type-Noncoherent-Large-ω-Precategory ℬ) | function | map-Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/maps-noncoherent-large-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.globular-maps",
"globular-types.globular-types",
"globular-types.large-globular-maps",
"globular-types.large-globular-types",
"wild-category-theory.maps-noncoher... | [
"Noncoherent-Large-ω-Precategory",
"large-globular-map",
"large-globular-type-Noncoherent-Large-ω-Precategory"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
map-Noncoherent-ω-Precategory :
{l1 l2 l3 l4 : Level}
(𝒜 : Noncoherent-ω-Precategory l1 l2)
(ℬ : Noncoherent-ω-Precategory l3 l4) → UU (l1 ⊔ l2 ⊔ l3 ⊔ l4) | map-Noncoherent-ω-Precategory 𝒜 ℬ =
globular-map
( globular-type-Noncoherent-ω-Precategory 𝒜)
( globular-type-Noncoherent-ω-Precategory ℬ) | function | map-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/maps-noncoherent-omega-precategories.lagda.md | [
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.identity-types",
"foundation.universe-levels",
"globular-types.globular-maps",
"globular-types.globular-types",
"wild-category-theory.noncoherent-omega-precategories"
] | [
"Noncoherent-ω-Precategory",
"globular-map"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
Noncoherent-Large-ω-Precategory
(α : Level → Level) (β : Level → Level → Level) : UUω
where | record | Noncoherent-Large-ω-Precategory | wild-category-theory | src/wild-category-theory/noncoherent-large-omega-precategories.lagda.md | [
"category-theory.precategories",
"foundation.action-on-identifications-binary-functions",
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.homotopies",
"foundation.identity-types",
"foundation.sets",
"foundation.strictly-involutive-identity-types",
"foundation.universe-lev... | [] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | ||
Noncoherent-ω-Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | Noncoherent-ω-Precategory l1 l2 =
Σ ( Globular-Type l1 l2)
( λ X → is-reflexive-Globular-Type X × is-transitive-Globular-Type X) | function | Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/noncoherent-omega-precategories.lagda.md | [
"category-theory.precategories",
"foundation.action-on-identifications-binary-functions",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.homotopies",
"foundation.identity-types",
"foundation.sets",
"foundation.strictly-involutive-ide... | [
"Globular-Type",
"is-reflexive-Globular-Type",
"is-transitive-Globular-Type"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 | |
make-Noncoherent-ω-Precategory :
{l1 l2 : Level} {X : Globular-Type l1 l2} → is-reflexive-Globular-Type X →
is-transitive-Globular-Type X → Noncoherent-ω-Precategory l1 l2 | make-Noncoherent-ω-Precategory id comp =
( _ , id , comp)
{-# INLINE make-Noncoherent-ω-Precategory #-} | function | make-Noncoherent-ω-Precategory | wild-category-theory | src/wild-category-theory/noncoherent-omega-precategories.lagda.md | [
"category-theory.precategories",
"foundation.action-on-identifications-binary-functions",
"foundation.cartesian-product-types",
"foundation.dependent-pair-types",
"foundation.function-types",
"foundation.homotopies",
"foundation.identity-types",
"foundation.sets",
"foundation.strictly-involutive-ide... | [
"Globular-Type",
"Noncoherent-ω-Precategory",
"id",
"is-reflexive-Globular-Type",
"is-transitive-Globular-Type"
] | https://github.com/UniMath/agda-unimath | c85d7fb834778f96a66576318cdc4ef3d4b80a26 |
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