walking coreflexive pair

 category theory
Notation Δ≤1\Delta^{\leq 1} Objects two objects [0][0] and [1][1] Morphisms the identities, two morphisms i,j:[0]⇉[1]i,j : [0] \rightrightarrows [1], a morphism p:[1]→[0]p : [1] \to [0] with pi=pj=id⁡[0]p i = p j = \id_{[0]}, and the two idempotent morphisms ip,jp:[1]⇉[1]ip, jp : [1] \rightrightarrows [1]. Related Δ\Delta, Pair\Pair, Split\Split

This category is equal to the truncated simplex category Δ≤1\Delta^{\leq 1}, i.e. the full subcategory of Δ\Delta spanned by [0]={0}[0] = \{0\} and [1]={0<1}[1] = \{0 < 1\}; this also explains our notation of the category and its objects. [0]↪  i  ↞  p  ↪  j  [1][0] \begin{array}{c} \xhookrightarrow{~~i~~} \\ \xtwoheadleftarrow{~~p~~} \\ \xhookrightarrow{~~j~~} \end{array} [1] The morphisms i,ji,j are the two inclusions, pp is their unique retraction, and ip,jp:[1]⇉[1]ip,jp : [1] \rightrightarrows [1] are the two constant maps. The name of this category comes from the fact that a functor Δ≤1→C\Delta^{\leq 1} \to \C is the same as a coreflexive pair in C\C. Its dual is therefore the walking reflexive pair.

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: [0][0]

Special morphisms

  • isomorphisms: only the identities
  • monomorphisms: the identities and ii, jj
  • epimorphisms: the identities and pp
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms