cocompletion of a discrete–pair join

 category theory
Notation C^\widehat{\C} Objects small presheaves on the join of a large discrete category with the walking parallel pair Morphisms natural transformations Related Sp\Sp, [CRing,Set][\CRing, \Set], 2\2, Pair\Pair

This rather technical category has been added as an example of a cocomplete category that does not have equalizers, but it also satisfies many other interesting property combinations.
To construct it, we start with the category C\C that has two objects A,BA,B and every set XX as an object. (Instead of the collection of sets, we can take any other large collection.) Apart from the identities, there two morphisms f,g:A⇉Bf,g : A \rightrightarrows B and, for every set XX, a unique morphism uX:X→Au_X : X \to A and a unique morphism vX:X→Bv_X : X \to B. The composition is defined by f∘uX=g∘uX=vXf \circ u_X = g \circ u_X = v_X. There are no morphisms between distinct sets. XuX ⁣ ⁣↙↘ ⁣ ⁣vXA    →f→g    B\begin{array}{c} X \\[0.25ex] {\raisebox{1ex}{$\scriptstyle u_X$}} \!\! \swarrow \qquad \searrow \!\! {\raisebox{1ex}{$\scriptstyle v_X$}} \\ A \;\; \begin{array}{c} \xrightarrow{\quad f \quad }\\[-1.25ex] \xrightarrow[\quad g \quad ]{} \end{array} \;\; B \end{array} In other words, C\C is the join of a large discrete category and the walking parallel pair. The category in this entry is the free cocompletion C^\widehat{\C} consisting of small presheaves Cop→Set\C^{\op} \to \Set, i.e. those presheaves that can be written as a small colimit of representable functors. See here for a couple of equivalent characterizations that hold for general C\C, as well as general results on free cocompletions, and see here for a concrete description of small presheaves in this specific setting.

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: constant presheaf with value 11
  • initial object: constant presheaf with value 00
  • coproducts: objectwise defined disjoint union of presheaves

Special morphisms

  • isomorphisms: natural isomorphisms
  • monomorphisms: natural transformations that are injective at every object
  • epimorphisms: natural transformations that are surjective at every object
  • regular monomorphisms: same as monomorphisms
  • regular epimorphisms: same as epimorphisms

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