On small generators of an infinity category

2012-05-19T01:30:24Z

Suppose that $\mathcal{C}$ is an $\infty$-category with pullbacks and small coproducts. Suppose that $\mathcal{D}$ is a small subcategory for every object $C,$ the canonical map $$f_C:\coprod\limits_{\left(D\to C\right) \in \left(\mathcal{D}/C\right)_0} D \to C$$ is an effective epimorphism.

(Reminder: This means that if $U^+_*:N(\Delta_+)^{op} \to \mathcal{C}$ is the augmented simplicial object which consists of iterative pullbacks of $f_C$ against itself (and with $U^+_*\left([-1]\right)=C$) then $C$ is the colimit of $N(f_C):=U^+_*|N(\Delta)^{op}.$ i.e. since $N(\Delta_+)^{op}$ is the right cone on $N(\Delta)^{op},$ this is demanding that $U^+_*$ is a colimiting cococone for $N(f_C)$)

I would like to show that that every object $C \in \mathcal{C}_0$ is the colimit of the canonical functor $$\pi_C:\mathcal{D}/C \to \mathcal{C}.$$ The corresponding result for $1$-categories is classical.

(Remark: If $\mathcal{C}$ is an $\infty$-topos then Lemma 6.2.3.13 of HTT guarentees the converse, namely if $C$ is the colimit of $\pi_C$ then the map $f_C$ is an effective epimorphism.)

The idea of the proof is simple enough:

Given a cocone $\rho$ for $N(f_C)$ on $C'$ one can make an assignemnt $$\left(f_\alpha:D_\alpha \to C\right) \mapsto D_\alpha \hookrightarrow \coprod\limits_{\left(D\to C\right) \in \left(\mathcal{D}/C\right)_0} D \stackrel{\rho(0)}{\longrightarrow} C'$$ and (by making certain choices) one can extend it to a cocone for $\pi_C$. Conversely, given a cocone $\mu$ for $\pi_C$ on $C',$ the morphism $$\coprod\limits_{\left(f_\alpha: D_\alpha\to C\right) \in \left(\mathcal{D}/C\right)_0} D_\alpha \stackrel{\coprod\left(\mu(f_\alpha)\right)}{\longrightarrow} C'$$ can (likewise with choices) be extended to a cocone for $N(f_C)$.

However, the problem with this is the choices. I would like (for instance) to show that the $\infty$-categories $\mathcal{C}_{\pi_C/ \cdot}$ and $\mathcal{C}_{N(f_C)/ \cdot}$ are equivalent, but any map of simplicial sets I try to cook up level-wise between them involves so many chocices I am not sure how to extend it functorially to a map of simplicial sets.

On small generators of an infinity category - MathOverflow

On small generators of an infinity category