walking commutative square

 category theory
Notation Square\Square Objects four objects a,b,c,da,b,c,d Morphisms morphisms a→ba \to b, b→db \to d, a→ca \to c, c→dc \to d, identities, and one morphism a→da \to d Related Fork\Fork, II, ForkSquare\ForkSquare External nLab Link

This category consists of a commutative square: a→b↓↓c→d\begin{CD} a @>>> b \\ @VVV @VVV \\ c @>>> d \end{CD} Its name comes from the fact that a functor Square→C\Square \to \C is the same as a commutative square in C\C. Notice that the category is isomorphic to the product category I×II \times I of the walking morphism with itself. Hence, most (but not all) properties are inherited from it.

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: dd
  • initial object: aa
  • products: b×c=ab \times c = a, x×x=xx \times x = x, a×x=aa \times x = a, d×x=xd \times x = x
  • coproducts: b⊔c=db \sqcup c = d, a⊔x=xa \sqcup x = x, d⊔x=dd \sqcup x = d, x⊔x=xx \sqcup x = x

Special morphisms

  • isomorphisms: only the identities
  • monomorphisms: every morphism
  • epimorphisms: every morphism
  • regular monomorphisms: same as isomorphisms
  • regular epimorphisms: same as isomorphisms