Proposition 4.10.
For any small category \(C\), the inclusion \(C \hookrightarrow \An[C]\) exhibits \(\An[C]\) as the free logos generated by \(C\): for any other logos \(T\), restriction along the inclusion induces an equivalence of categories
\[\Fun_{\bbLog}(\An[C],T) \quad \simeq \quad \Fun(C,T).\]
Proof
Recall from [Lurie 2009, Proposition 5.3.6.2] that for a small category \(C\), the subcategory \(C^{\mathrm{rex}} \subseteq \PSh(C)\) generated by the representables under finite colimits is the free finite cocompletion of \(C\). It follows that \(C^{\lex} = ((C\catop)^{\mathrm{rex}})\catop\) is the free finite limit completion: for any other category \(D\) with finite limits, restriction along \(C \hookrightarrow C^{\lex}\) induces an equivalence
\[\Fun^{\lex}(C^{\lex},D) \quad \simeq \quad \Fun(C,D).\]
By the universal property of the presheaf category, we also see that restriction along \(C^{\lex} \hookrightarrow \PSh(C^{\lex})\) induces an equivalence \[\Fun^{\colim}(\PSh(C^{\lex}),T) \quad \simeq \quad \Fun(C^{\lex},T).\]
By Proposition 2.43, this functor restricts to the desired equivalence \[\Fun_{\bbLog}(\An[C],T) \quad = \quad \Fun^{\colim, \lex}(\PSh(C^{\lex}),T) \quad \simeq \quad \Fun^{\lex}(C^{\lex},T).\]
References
- Jacob Lurie. Higher topos theory. Ann. Math. Stud. 170, Princeton, NJ: Princeton University Press. 2009.