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category of set functions and commutative squares

  • notation: Set→\Set^{\rightarrow}
  • objects: triples (X,Y,f)(X, Y, f) where XX and YY are sets, and f:X→Yf : X \to Y is a function
  • morphisms: a morphism (X,Y,f)→(X′,Y′,f′)(X, Y, f) \to (X', Y', f') is a pair of functions ℓ:X→X′\ell : X \to X' and r:Y→Y′r : Y \to Y' making a commutative square X→fYℓ↓↓rX′→f′Y′\begin{CD} X @>{f}>> Y \\ @V{\ell}VV @VV{r}V \\ X' @>>{f'}> Y' \end{CD}
  • Related categories: Set×Set\Set \times \Set, Set\Set, Mono\Mono, Sh(X)\Sh(X)
  • nLab Link

This category is an example of the arrow category Arr(C)\Arr(\C), where C\C is the category of sets. It is also known as the Sierpinski topos, since it is equivalent to the category of sheaves on the Sierpinski space.

Satisfied Properties

Assigned properties

Deduced properties

Unsatisfied Properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: the unique function 1→11 \to 1
  • initial object: the unique function 0→00 \to 0
  • products: component-wise defined cartesian product, equipped with the product function
  • coproducts: component-wise defined disjoint union, equipped with the disjoint union of the functions

Special morphisms

  • isomorphisms: pairs (ℓ,r)(\ell, r) where ℓ\ell and rr are both bijections
  • monomorphisms: pairs (ℓ,r)(\ell, r) where ℓ\ell and rr are both injective
  • epimorphisms: pairs (ℓ,r)(\ell, r) where ℓ\ell and rr are both surjective
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms