category object (Rev #1, changes)

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Let 𝒞\mathcal{C} be an (,1)(\infty,1)-category.

A groupoid object in 𝒞\mathcal{C} is defined to be a simplicial object satifying the groupoidal Segal condition (we could call this condition also ‘’horn partition condition’’. We will see, that a precategory object reps. a category object will be defined to satisfy a weaker ‘’pair horn condition’‘ resp. a ‘’inner horn partition condition’’.)

In the left half of the below diagram constconst is a full and faithful functor. If 𝒞\mathcal{C} has small colimits constconst has a left adjoint colimconstcolim\dashv const.

𝒞constcolimGrpd(𝒞)CorePreCat(𝒞) \mathcal{C} \stackrel{\overset{colim}{\leftarrow}}{\underset{const}{\hookrightarrow}} Grpd(\mathcal{C}) \stackrel{\hookrightarrow}{\underset{Core}{\leftarrow}} PreCat(\mathcal{C})

Revision on November 1, 2012 at 22:31:17 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.