Rebase infrastructure for pushouts to span diagrams #885
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| # Extensions of spans | ||||||
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| ```agda | ||||||
| module foundation-core.extensions-spans where | ||||||
| ``` | ||||||
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| <details><summary>Imports</summary> | ||||||
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| ```agda | ||||||
| open import foundation.dependent-pair-types | ||||||
| open import foundation.morphisms-arrows | ||||||
| open import foundation.spans | ||||||
| open import foundation.universe-levels | ||||||
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| open import foundation-core.function-types | ||||||
| ``` | ||||||
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| </details | ||||||
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| ## Idea | ||||||
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| Consider a [span](foundation.spans.md) `s` given by | ||||||
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| ```text | ||||||
| f g | ||||||
| A <----- S -----> B | ||||||
| ``` | ||||||
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| and maps `i : A → A'` and `j : B → B'`. The {{#concept "extension" Disambiguation="span"}} of `s` by `i` and `j` is the span | ||||||
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| ```text | ||||||
| i ∘ f j ∘ g | ||||||
| A' <------- S -------> B. | ||||||
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| ``` | ||||||
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| ## Definitions | ||||||
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| ### Extensions of spans with fixed domain and codomain | ||||||
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| #### Extensions on both sides | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 l5 : Level} | ||||||
| {A : UU l1} {A' : UU l2} | ||||||
| {B : UU l3} {B' : UU l4} | ||||||
| where | ||||||
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| extend-span-fixed-domain-codomain : | ||||||
| span-fixed-domain-codomain l5 A B → (A → A') → (B → B') → | ||||||
| span-fixed-domain-codomain l5 A' B' | ||||||
| pr1 (extend-span-fixed-domain-codomain s f g) = | ||||||
| spanning-type-span-fixed-domain-codomain s | ||||||
| pr1 (pr2 (extend-span-fixed-domain-codomain s f g)) = | ||||||
| f ∘ left-map-span-fixed-domain-codomain s | ||||||
| pr2 (pr2 (extend-span-fixed-domain-codomain s f g)) = | ||||||
| g ∘ right-map-span-fixed-domain-codomain s | ||||||
| ``` | ||||||
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| #### Extensions on the left | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 : Level} | ||||||
| {A : UU l1} {A' : UU l2} | ||||||
| {B : UU l3} | ||||||
| where | ||||||
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| left-extend-span-fixed-domain-codomain : | ||||||
| span-fixed-domain-codomain l4 A B → (A → A') → | ||||||
| span-fixed-domain-codomain l4 A' B | ||||||
| left-extend-span-fixed-domain-codomain s f = | ||||||
| extend-span-fixed-domain-codomain s f id | ||||||
| ``` | ||||||
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| #### Extensions on the right | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 : Level} | ||||||
| {A : UU l1} | ||||||
| {B : UU l3} {B' : UU l4} | ||||||
| where | ||||||
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| right-extend-span-fixed-domain-codomain : | ||||||
| span-fixed-domain-codomain l4 A B → (B → B') → | ||||||
| span-fixed-domain-codomain l4 A B' | ||||||
| right-extend-span-fixed-domain-codomain s g = | ||||||
| extend-span-fixed-domain-codomain s id g | ||||||
| ``` | ||||||
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| #### Extensions by morphisms of arrows on the left | ||||||
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| Consider a `s` with fixed domain and codomain given by | ||||||
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| ```text | ||||||
| f g | ||||||
| A <----- S -----> B | ||||||
| ``` | ||||||
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| and a [morphism of arrows](foundation.morphisms-arrows.md) `h : hom-arrow f' f` as indicated in the diagram | ||||||
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| ```text | ||||||
| h₀ g | ||||||
| S' -----> S -----> B | ||||||
| | | | ||||||
| f' | f | | ||||||
| V V | ||||||
| A' -----> A'. | ||||||
| h₁ | ||||||
| ``` | ||||||
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| Then we obtain a span `A' <- S' -> B`. | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 l5 : Level} {A : UU l1} {B : UU l2} | ||||||
| (s : span-fixed-domain-codomain l3 A B) | ||||||
| {S' : UU l4} {A' : UU l5} (f' : S' → A') | ||||||
| (h : hom-arrow f' (left-map-span-fixed-domain-codomain s)) | ||||||
| where | ||||||
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| spanning-type-left-extend-hom-arrow-span-fixed-domain-codomain : UU l4 | ||||||
| spanning-type-left-extend-hom-arrow-span-fixed-domain-codomain = S' | ||||||
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| left-map-left-extend-hom-arrow-span-fixed-domain-codomain : | ||||||
| spanning-type-left-extend-hom-arrow-span-fixed-domain-codomain → A' | ||||||
| left-map-left-extend-hom-arrow-span-fixed-domain-codomain = f' | ||||||
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| right-map-left-extend-hom-arrow-span-fixed-domain-codomain : | ||||||
| spanning-type-left-extend-hom-arrow-span-fixed-domain-codomain → B | ||||||
| right-map-left-extend-hom-arrow-span-fixed-domain-codomain = | ||||||
| ( right-map-span-fixed-domain-codomain s) ∘ | ||||||
| ( map-domain-hom-arrow f' (left-map-span-fixed-domain-codomain s) h) | ||||||
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| left-extend-hom-arrow-span-fixed-domain-codomain : | ||||||
| span-fixed-domain-codomain l4 A' B | ||||||
| pr1 left-extend-hom-arrow-span-fixed-domain-codomain = | ||||||
| spanning-type-left-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| pr1 (pr2 left-extend-hom-arrow-span-fixed-domain-codomain) = | ||||||
| left-map-left-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| pr2 (pr2 left-extend-hom-arrow-span-fixed-domain-codomain) = | ||||||
| right-map-left-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| ``` | ||||||
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| #### Extensions by morphisms of arrows on the right | ||||||
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| Consider a `s` with fixed domain and codomain given by | ||||||
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| ```text | ||||||
| f g | ||||||
| A <----- S -----> B | ||||||
| ``` | ||||||
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| and a [morphism of arrows](foundation.morphisms-arrows.md) `h : hom-arrow g' g` as indicated in the diagram | ||||||
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| ```text | ||||||
| g' | ||||||
| S' ----> B' | ||||||
| | | | ||||||
| h₀ | | h₁ | ||||||
| V V | ||||||
| S -----> B | ||||||
| | g | ||||||
| f | | ||||||
| V | ||||||
| A. | ||||||
| ``` | ||||||
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| Then we obtain a span `A <- S' -> B'`. | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 l5 : Level} {A : UU l1} {B : UU l2} | ||||||
| (s : span-fixed-domain-codomain l3 A B) | ||||||
| {S' : UU l4} {B' : UU l5} (g' : S' → B') | ||||||
| (h : hom-arrow g' (right-map-span-fixed-domain-codomain s)) | ||||||
| where | ||||||
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| spanning-type-right-extend-hom-arrow-span-fixed-domain-codomain : UU l4 | ||||||
| spanning-type-right-extend-hom-arrow-span-fixed-domain-codomain = S' | ||||||
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| left-map-right-extend-hom-arrow-span-fixed-domain-codomain : | ||||||
| spanning-type-right-extend-hom-arrow-span-fixed-domain-codomain → A | ||||||
| left-map-right-extend-hom-arrow-span-fixed-domain-codomain = | ||||||
| ( left-map-span-fixed-domain-codomain s) ∘ | ||||||
| ( map-domain-hom-arrow g' (right-map-span-fixed-domain-codomain s) h) | ||||||
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| right-map-right-extend-hom-arrow-span-fixed-domain-codomain : | ||||||
| spanning-type-right-extend-hom-arrow-span-fixed-domain-codomain → B' | ||||||
| right-map-right-extend-hom-arrow-span-fixed-domain-codomain = g' | ||||||
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| right-extend-hom-arrow-span-fixed-domain-codomain : | ||||||
| span-fixed-domain-codomain l4 A B' | ||||||
| pr1 right-extend-hom-arrow-span-fixed-domain-codomain = | ||||||
| spanning-type-right-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| pr1 (pr2 right-extend-hom-arrow-span-fixed-domain-codomain) = | ||||||
| left-map-right-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| pr2 (pr2 right-extend-hom-arrow-span-fixed-domain-codomain) = | ||||||
| right-map-right-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| ``` | ||||||
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| ### Extensions of spans | ||||||
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| #### Extensions on both sides | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 l5 : Level} | ||||||
| where | ||||||
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| extend-span : | ||||||
| (s : span l1 l2 l3) | ||||||
| {A' : UU l4} (f : domain-span s → A') | ||||||
| {B' : UU l5} (g : codomain-span s → B') → | ||||||
| span l4 l5 l3 | ||||||
| pr1 (extend-span s {A'} f {B'} g) = A' | ||||||
| pr1 (pr2 (extend-span s {A'} f {B'} g)) = B' | ||||||
| pr2 (pr2 (extend-span s {A'} f {B'} g)) = | ||||||
| extend-span-fixed-domain-codomain (span-fixed-domain-codomain-span s) f g | ||||||
| ``` | ||||||
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| #### Extensions on the left | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 : Level} | ||||||
| where | ||||||
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| left-extend-span : | ||||||
| (s : span l1 l2 l3) {A' : UU l4} (f : domain-span s → A') → span l4 l2 l3 | ||||||
| left-extend-span s f = extend-span s f id | ||||||
| ``` | ||||||
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| #### Extensions on the right | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 : Level} | ||||||
| where | ||||||
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| right-extend-span : | ||||||
| (s : span l1 l2 l3) {B' : UU l4} (g : codomain-span s → B') → span l1 l4 l3 | ||||||
| right-extend-span s g = extend-span s id g | ||||||
| ``` | ||||||
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| #### Extensions by morphisms of arrows on the left | ||||||
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| Consider a `s` with fixed domain and codomain given by | ||||||
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| ```text | ||||||
| f g | ||||||
| A <----- S -----> B | ||||||
| ``` | ||||||
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| and a [morphism of arrows](foundation.morphisms-arrows.md) `h : hom-arrow f' f` as indicated in the diagram | ||||||
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| ```text | ||||||
| h₀ g | ||||||
| S' -----> S -----> B | ||||||
| | | | ||||||
| f' | f | | ||||||
| V V | ||||||
| A' -----> A'. | ||||||
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| h₁ | ||||||
| ``` | ||||||
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| Then we obtain a span `A' <- S' -> B`. | ||||||
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 l5 : Level} (s : span l1 l2 l3) | ||||||
| {S' : UU l4} {A' : UU l5} (f' : S' → A') (h : hom-arrow f' (left-map-span s)) | ||||||
| where | ||||||
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| domain-left-extend-hom-arrow-span : UU l5 | ||||||
| domain-left-extend-hom-arrow-span = A' | ||||||
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| codomain-left-extend-hom-arrow-span : UU l2 | ||||||
| codomain-left-extend-hom-arrow-span = codomain-span s | ||||||
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| span-fixed-domain-codomain-left-extend-hom-arrow-span : | ||||||
| span-fixed-domain-codomain l4 | ||||||
| ( domain-left-extend-hom-arrow-span) | ||||||
| ( codomain-left-extend-hom-arrow-span) | ||||||
| span-fixed-domain-codomain-left-extend-hom-arrow-span = | ||||||
| left-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| ( span-fixed-domain-codomain-span s) | ||||||
| ( f') | ||||||
| ( h) | ||||||
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| left-extend-hom-arrow-span : span l5 l2 l4 | ||||||
| pr1 left-extend-hom-arrow-span = | ||||||
| domain-left-extend-hom-arrow-span | ||||||
| pr1 (pr2 left-extend-hom-arrow-span) = | ||||||
| codomain-left-extend-hom-arrow-span | ||||||
| pr2 (pr2 left-extend-hom-arrow-span) = | ||||||
| span-fixed-domain-codomain-left-extend-hom-arrow-span | ||||||
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| spanning-type-left-extend-hom-arrow-span : UU l4 | ||||||
| spanning-type-left-extend-hom-arrow-span = | ||||||
| spanning-type-span left-extend-hom-arrow-span | ||||||
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| left-map-left-extend-hom-arrow-span : | ||||||
| spanning-type-left-extend-hom-arrow-span → | ||||||
| domain-left-extend-hom-arrow-span | ||||||
| left-map-left-extend-hom-arrow-span = | ||||||
| left-map-span left-extend-hom-arrow-span | ||||||
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| right-map-left-extend-hom-arrow-span : | ||||||
| spanning-type-left-extend-hom-arrow-span → | ||||||
| codomain-left-extend-hom-arrow-span | ||||||
| right-map-left-extend-hom-arrow-span = | ||||||
| right-map-span left-extend-hom-arrow-span | ||||||
| ``` | ||||||
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| #### Extensions by morphisms of arrows on the left | ||||||
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| Consider a `s` with fixed domain and codomain given by | ||||||
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| ```text | ||||||
| f g | ||||||
| A <----- S -----> B | ||||||
| ``` | ||||||
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| and a [morphism of arrows](foundation.morphisms-arrows.md) `h : hom-arrow g' g` as indicated in the diagram | ||||||
|
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| ```text | ||||||
| g' | ||||||
| S' ----> B' | ||||||
| | | | ||||||
| h₀ | | h₁ | ||||||
| V V | ||||||
| S -----> B | ||||||
| | g | ||||||
| f | | ||||||
| V | ||||||
| A. | ||||||
| ``` | ||||||
|
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| Then we obtain a span `A <- S' -> B'`. | ||||||
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|
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| ```agda | ||||||
| module _ | ||||||
| {l1 l2 l3 l4 l5 : Level} (s : span l1 l2 l3) | ||||||
| {S' : UU l4} {B' : UU l5} (g' : S' → B') | ||||||
| (h : hom-arrow g' (right-map-span s)) | ||||||
| where | ||||||
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| domain-right-extend-hom-arrow-span : UU l1 | ||||||
| domain-right-extend-hom-arrow-span = domain-span s | ||||||
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| codomain-right-extend-hom-arrow-span : UU l5 | ||||||
| codomain-right-extend-hom-arrow-span = B' | ||||||
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| span-fixed-domain-codomain-right-extend-hom-arrow-span : | ||||||
| span-fixed-domain-codomain l4 | ||||||
| ( domain-right-extend-hom-arrow-span) | ||||||
| ( codomain-right-extend-hom-arrow-span) | ||||||
| span-fixed-domain-codomain-right-extend-hom-arrow-span = | ||||||
| right-extend-hom-arrow-span-fixed-domain-codomain | ||||||
| ( span-fixed-domain-codomain-span s) | ||||||
| ( g') | ||||||
| ( h) | ||||||
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| right-extend-hom-arrow-span : span l1 l5 l4 | ||||||
| pr1 right-extend-hom-arrow-span = | ||||||
| domain-right-extend-hom-arrow-span | ||||||
| pr1 (pr2 right-extend-hom-arrow-span) = | ||||||
| codomain-right-extend-hom-arrow-span | ||||||
| pr2 (pr2 right-extend-hom-arrow-span) = | ||||||
| span-fixed-domain-codomain-right-extend-hom-arrow-span | ||||||
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| spanning-type-right-extend-hom-arrow-span : UU l4 | ||||||
| spanning-type-right-extend-hom-arrow-span = | ||||||
| spanning-type-span right-extend-hom-arrow-span | ||||||
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| left-map-right-extend-hom-arrow-span : | ||||||
| spanning-type-right-extend-hom-arrow-span → | ||||||
| domain-right-extend-hom-arrow-span | ||||||
| left-map-right-extend-hom-arrow-span = | ||||||
| left-map-span right-extend-hom-arrow-span | ||||||
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| right-map-right-extend-hom-arrow-span : | ||||||
| spanning-type-right-extend-hom-arrow-span → | ||||||
| codomain-right-extend-hom-arrow-span | ||||||
| right-map-right-extend-hom-arrow-span = | ||||||
| right-map-span right-extend-hom-arrow-span | ||||||
| ``` | ||||||
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As commented before, the contents of this file read more like pastings or concatenations to me