# The universal property of pullbacks

```agda
module foundation-core.universal-property-pullbacks where
```

<details><summary>Imports</summary>

```agda
open import foundation.action-on-identifications-functions
open import foundation.cones-over-cospan-diagrams
open import foundation.dependent-pair-types
open import foundation.postcomposition-functions
open import foundation.universe-levels

open import foundation-core.contractible-maps
open import foundation-core.contractible-types
open import foundation-core.equivalences
open import foundation-core.function-types
open import foundation-core.functoriality-dependent-pair-types
open import foundation-core.homotopies
open import foundation-core.identity-types
```

</details>

## Definition

### The universal property of pullbacks

```agda
module _
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3}
  (f : A → X) (g : B → X) {C : UU l4} (c : cone f g C)
  where

  universal-property-pullback : UUω
  universal-property-pullback =
    {l : Level} (C' : UU l) → is-equiv (cone-map f g c {C'})

module _
  {l1 l2 l3 l4 l5 : Level} {A : UU l1} {B : UU l2} {X : UU l3}
  (f : A → X) (g : B → X) {C : UU l4} (c : cone f g C)
  where

  map-universal-property-pullback :
    universal-property-pullback f g c →
    {C' : UU l5} (c' : cone f g C') → C' → C
  map-universal-property-pullback up-c {C'} =
    map-inv-is-equiv (up-c C')

  compute-map-universal-property-pullback :
    (up-c : universal-property-pullback f g c) →
    {C' : UU l5} (c' : cone f g C') →
    cone-map f g c (map-universal-property-pullback up-c c') ＝ c'
  compute-map-universal-property-pullback up-c {C'} =
    is-section-map-inv-is-equiv (up-c C')
```

## Properties

### The homotopy of cones obtained from the universal property of pullbacks

```agda
module _
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3}
  (f : A → X) (g : B → X) {C : UU l4}
  (c : cone f g C) (up : universal-property-pullback f g c)
  {l5 : Level} {C' : UU l5} (c' : cone f g C')
  where

  htpy-cone-map-universal-property-pullback :
    htpy-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c'))
      ( c')
  htpy-cone-map-universal-property-pullback =
    htpy-eq-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c'))
      ( c')
      ( compute-map-universal-property-pullback f g c up c')

  htpy-vertical-map-map-universal-property-pullback :
    vertical-map-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c')) ~
      vertical-map-cone f g c'
  htpy-vertical-map-map-universal-property-pullback =
    htpy-vertical-map-htpy-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c'))
      ( c')
      ( htpy-cone-map-universal-property-pullback)

  htpy-horizontal-map-map-universal-property-pullback :
    horizontal-map-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c')) ~
      horizontal-map-cone f g c'
  htpy-horizontal-map-map-universal-property-pullback =
    htpy-horizontal-map-htpy-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c'))
      ( c')
      ( htpy-cone-map-universal-property-pullback)

  coh-htpy-cone-map-universal-property-pullback :
    coherence-htpy-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c'))
      ( c')
      ( htpy-vertical-map-map-universal-property-pullback)
      ( htpy-horizontal-map-map-universal-property-pullback)
  coh-htpy-cone-map-universal-property-pullback =
    coh-htpy-cone f g
      ( cone-map f g c (map-universal-property-pullback f g c up c'))
      ( c')
      ( htpy-cone-map-universal-property-pullback)
```

### 3-for-2 property of pullbacks

Given two cones `c` and `c'` over a common cospan `f : A → X ← B : g`, and a map
between them `h` such that the diagram

```text
              h
          C ----> C'
        /   \   /   \
      /      / \      \
    /   /          \    \
   ∨ ∨                 ∨ ∨
  A --------> X <-------- B
        f           g
```

is coherent, then if two out of the three conditions

- `c` is a pullback cone
- `c'` is a pullback cone
- `h` is an equivalence

hold, then so does the third.

```agda
module _
  {l1 l2 l3 l4 l5 : Level}
  {A : UU l1} {B : UU l2} {X : UU l3} {C : UU l4} {C' : UU l5}
  {f : A → X} {g : B → X} (c : cone f g C) (c' : cone f g C')
  (h : C' → C) (KLM : htpy-cone f g (cone-map f g c h) c')
  where

  inv-triangle-cone-cone :
    {l6 : Level} (D : UU l6) →
    cone-map f g c ∘ postcomp D h ~ cone-map f g c'
  inv-triangle-cone-cone D k =
    ap
      ( λ t → cone-map f g t k)
      ( eq-htpy-cone f g (cone-map f g c h) c' KLM)

  triangle-cone-cone :
    {l6 : Level} (D : UU l6) → cone-map f g c' ~ cone-map f g c ∘ postcomp D h
  triangle-cone-cone D k = inv (inv-triangle-cone-cone D k)

  abstract
    is-equiv-up-pullback-up-pullback :
      universal-property-pullback f g c →
      universal-property-pullback f g c' →
      is-equiv h
    is-equiv-up-pullback-up-pullback up up' =
      is-equiv-is-equiv-postcomp h
        ( λ D →
          is-equiv-top-map-triangle
            ( cone-map f g c')
            ( cone-map f g c)
            ( postcomp D h)
            ( triangle-cone-cone D)
            ( up D)
            ( up' D))

  abstract
    up-pullback-up-pullback-is-equiv :
      is-equiv h →
      universal-property-pullback f g c →
      universal-property-pullback f g c'
    up-pullback-up-pullback-is-equiv is-equiv-h up D =
      is-equiv-left-map-triangle
        ( cone-map f g c')
        ( cone-map f g c)
        ( postcomp D h)
        ( triangle-cone-cone D)
        ( is-equiv-postcomp-is-equiv h is-equiv-h D)
        ( up D)

  abstract
    up-pullback-is-equiv-up-pullback :
      universal-property-pullback f g c' →
      is-equiv h →
      universal-property-pullback f g c
    up-pullback-is-equiv-up-pullback up' is-equiv-h D =
      is-equiv-right-map-triangle
        ( cone-map f g c')
        ( cone-map f g c)
        ( postcomp D h)
        ( triangle-cone-cone D)
        ( up' D)
        ( is-equiv-postcomp-is-equiv h is-equiv-h D)
```

### Uniqueness of maps obtained via the universal property of pullbacks

```agda
module _
  {l1 l2 l3 l4 : Level} {A : UU l1} {B : UU l2} {X : UU l3}
  (f : A → X) (g : B → X) {C : UU l4} (c : cone f g C)
  where

  abstract
    uniqueness-universal-property-pullback :
      universal-property-pullback f g c →
      {l5 : Level} (C' : UU l5) (c' : cone f g C') →
      is-contr (Σ (C' → C) (λ h → htpy-cone f g (cone-map f g c h) c'))
    uniqueness-universal-property-pullback up C' c' =
      is-contr-equiv'
        ( Σ (C' → C) (λ h → cone-map f g c h ＝ c'))
        ( equiv-tot (λ h → extensionality-cone f g (cone-map f g c h) c'))
        ( is-contr-map-is-equiv (up C') c')
```

## Table of files about pullbacks

The following table lists files that are about pullbacks as a general concept.

{{#include tables/pullbacks.md}}