walking commutative square

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 out of it is the same as a commutative square in the target category. Notice that the category is isomorphic to the product category {0→1}×{0→1}\{0 \to 1\} \times \{0 \to 1\} of the walking morphism with itself. Hence, most (but not all) properties are inherited from it. It is also isomorphic to the partial order of positive divisors of 66.

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