forked commutative square

  • Notation: ForkSquare\ForkSquare
  • Objects: A,B,C,D,EA,B,C,D,E
  • Morphisms: The morphisms are generated by e:A→Be : A \to B, f:A→Cf : A \to C, g:B→Dg : B \to D, m:C→Dm : C \to D and u,v:D⇉Eu,v : D \rightrightarrows E, subject to the relations g∘e=m∘fg \circ e = m \circ f, u∘g=v∘gu \circ g = v \circ g, and u∘m=v∘mu \circ m = v \circ m. In total, there are 1515 morphisms.
  • Related categories: Square\Square, Fork\Fork

This finite category is generated by the graph A→eBf↓      ↓gC→mD→u→vE\begin{array}{ccccc} A & \xrightarrow{\hspace{1em} e \hspace{1em}} & B & & \\ \text{\scriptsize $f$}\bigg\downarrow\;\, && \;\,\bigg\downarrow\text{\scriptsize $g$} && \\ C & \xrightarrow{\hspace{1em} m \hspace{1em}} & D & \begin{array}{c} \xrightarrow{\hspace{1em} u \hspace{1em}}\\ \xrightarrow{\hspace{1em} v \hspace{1em}} \end{array} & E \end{array} and the evident relations: the square commutes, and the parallel pair u,vu,v is equalized by both gg and mm. We have added this category to the database solely as an example of an extremal monomorphism (namely mm) that is not a strong monomorphism. There is probably no common name for this category, but "forked commutative square" seems like a good fit.

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

  • initial object: AA

Special morphisms

  • isomorphisms: only the identities
  • monomorphisms: every morphism
  • epimorphisms: all morphisms except for the three non-identity morphisms with codomain DD, namely gg, mm, and the diagonal g∘eg \circ e
  • regular monomorphisms: same as isomorphisms
  • regular epimorphisms: same as isomorphisms

Morphisms

The database stores 1 morphism based on the forked commutative square.