category object

Let π’ž\mathcal{C} be an (∞,1)(\infty,1)-category.

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

In the left half of the below diagram constconst is a full and faithful functor. If π’ž\mathcal{C} has small colimits constconst has a left adjoint colim⊣constcolim\dashv const.

π’žβ†ͺconst←colimGrpd(π’ž)←Coreβ†ͺPreCat(π’ž) \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 π’ž\mathcal{C} be a presentable (∞,1)(\infty,1)-category. A choice of internal groupoids is a choice of a presentable full sub (∞,1)(\infty,1)-category Disc𝒳β†ͺπ’žDisc\mathcal{X}\hookrightarrow \mathcal{C} satisfying

  • DiscDisc has a left- and a right adjoint (Π⊣DiscβŠ£Ξ“)(\Pi\dashv Disc \dashv \Gamma)

  • For all f:Yβ†’Xf:Y\to X with Yβˆˆπ’žY\in \mathcal{C}, Xβˆˆπ’³X\in\mathcal{X} base change f *:𝒳/Xβ†’π’ž/Yf^*: \mathcal{X}/X\to \mathcal{C}/Y preserves colimits.

  • The codomain fibration of π’ž\mathcal{C} is an (∞,2)-sheaf when restricted to 𝒳\mathcal{X}: its classifying functor Ο‡:π’ž opβ†’\chi : \mathcal{C}^{op} \to (∞,1)Cat preserves (∞,1)-limits when restricted along 𝒳β†ͺπ’ž\mathcal{X} \hookrightarrow \mathcal{C}.