instHasPullbacksType — Mathlib

English

The category of Type u has all pullbacks. For any f: X → S and g: Y → S, the pullback object is the subset of X × Y consisting of pairs (x,y) with f x = g y, equipped with projections to X and Y.

Русский

Категория Type u обладает всемиeter pullbacks. Пусть f: X → S и g: Y → S; тогда возвратный объект состоит из пар (x,y) ∈ X × Y с f(x) = g(y), с проекциями на X и Y.

LaTeX

$$$\\exists P = \\{(x,y) \\in X \\times Y \\mid f(x)=g(y)\\},\\; p_X: P\\to X,\\; p_Y: P\\to Y,\\; f\\circ p_X = g\\circ p_Y$$

Lean4

instance : HasPullbacks.{u} (Type u) :=
  -- FIXME does not work via `inferInstance` despite `#synth HasPullbacks.{u} (Type u)` succeeding.
    -- https://github.com/leanprover-community/mathlib4/issues/5752
    -- inferInstance

  hasPullbacks_of_hasWidePullbacks.{u} (Type u)

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