CatDat

pushouts

A category C\mathcal{C} has pushouts if every span of morphisms ASBA \leftarrow S \rightarrow B has a pushout ASBA \sqcup_S B. This is also known as a fiber coproduct. Equivalently, the coslice category S/CS/\mathcal{C} has binary coproducts.

Relevant implications

  • binary coproducts and  coequalizers implies   pushouts
    [dualized] The pullback of f:XSf : X \to S and g:YSg : Y \to S is the equalizer of p1f,p2g:X×YSp_1 \circ f, \, p_2 \circ g : X \times Y \rightrightarrows S.
  • binary coproducts and  pushouts implies   coequalizers
    [dualized] The equalizer of f,g:XYf,g : X \rightrightarrows Y is the pullback of (f,g):XY×Y(f,g) : X \to Y \times Y with the diagonal YY×YY \to Y \times Y.
  • pushouts and  initial object implies   binary coproducts
    [dualized] If 11 is a terminal object, then X×1Y=X×YX \times_1 Y = X \times Y.
  • wide pushouts is equivalent to   pushouts and  filtered colimits
    [dualized] To prove \Leftarrow, a wide pullback can be constructed as a filtered limit of finite pullbacks, and finite pullbacks can be reduced to binary products (the empty-indexed pullback always exists). Conversely, assume that wide pullbacks exist in C\mathcal{C}. For every object AA then the slice category C/A\mathcal{C} / A has wide pullbacks and a terminal object, hence is complete. Since a filtered limit can be finally reduced to such a slice, we are done.
  • groupoid implies   self-dual and  epi-regular and  pushouts and  filtered colimits and  right cancellative and  well-copowered
    [dualized] easy
  • self-dual and  pullbacks implies   pushouts
    trivial by self-duality
  • self-dual and  pushouts implies   pullbacks
    trivial by self-duality

Examples

Counterexamples

Unknown

For these categories the database has no info if they satisfy this property or not.