walking splitting

  • notation: Split\mathrm{Split}
  • objects: two objects 00 and 11
  • morphisms: the identities, morphisms i:0→1i : 0 \to 1, p:1→0p : 1 \to 0 satisfying pi=id0pi = \mathrm{id}_0, and the idempotent ip:1→1ip : 1 \to 1.
  • Related categories: Idem\mathrm{Idem}, {0⇄1}\{0 \rightleftarrows 1\}

This category could also be called the "walking split idempotent" (or "walking section", "walking retraction"), but we chose a name that emphasizes that the splitting belongs to the data. Notice that the 55 given morphisms are indeed closed under composition. For example, p∘ip=pp \circ ip = p and ip∘i=iip \circ i = i.
The walking splitting can be interpreted as a skeleton of the category of F2\mathbb{F}_2-vector spaces of dimension ≤1\leq 1.

Satisfied Properties

Properties from the database

Deduced properties

Unsatisfied Properties

Properties from the database

Deduced properties*

*This also uses the deduced satisfied properties.

Unknown properties

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Special objects

  • terminal object: 00
  • initial object: 00

Special morphisms

  • isomorphisms: the two identities
  • monomorphisms: the identities and ii
  • epimorphisms: the identities and pp
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms