category object (Rev #1, 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 $colim\dashv const$.

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