Spahn category object (Rev #2, changes)

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

A groupoid object in $\mathcal{C}$ 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 $const$ is a full and faithful functor. If $\mathcal{C}$ has small colimits $const$ has a left adjoint $colim\dashv const$ .

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

(…)

Definition (choice of internal groupoids)

Let $\mathcal{C}$ be a presentable $(\infty,1)$ -category. A choice of internal groupoids is a choice of a presentable full sub $(\infty,1)$ -category $Disc\mathcal{X}\hookrightarrow \mathcal{C}$ satisfying

$Disc$ has a left- and a right adjoint $(\Pi\dashv Disc \dashv \Gamma)$
For all $f:Y\to X$ with $Y\in \mathcal{C}$ , $X\in\mathcal{X}$ base change $f^*: \mathcal{X}/X\to \mathcal{C}/Y$ preserves colimits.
The codomain fibration of $\mathcal{C}$ is an (∞,2)-sheaf when restricted to $\mathcal{X}$ : its classifying functor $\chi : \mathcal{C}^{op} \to$ (∞,1)Cat preserves (∞,1)-limits when restricted along $\mathcal{X} \hookrightarrow \mathcal{C}$ .

Revision on November 2, 2012 at 00:23:49 by Stephan Alexander Spahn?. See the history of this page for a list of all contributions to it.