category of set functions and commutative squares

 set theory
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 Set\Set, Set×Set\Set \times \Set, Sh(X)\Sh(X), Mono\Mono, [(N,≤),Set][(\IN,\leq),\Set] External 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 Sierpiński topos, since it is equivalent to the category of sheaves on the Sierpiński space.

Satisfied properties

Assigned properties

Deduced properties

Unsatisfied properties

Assigned properties

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

—

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

Indistinguishable categories

These categories in the database currently have exactly the same properties as the category of set functions and commutative squares. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.