A completeness criterion for $\infty$-categories

We all know that for ordinary categories $\mathscr{C}, \mathscr{D}$ (with $\mathscr{C}$ small) the limit of a functor $F:\mathscr{C} \to \mathscr{D}$, if it exists, can be constructed by using products and equalizers.

This gives us a criterion, stating that a category is complete if it admits all products and equalizers, and Lurie proves the analogue for quasicategories in Proposition 4.4.2.6 in his "Higher Topos Theory".

I was wondering whether a similar statement could hold in Riehl and Verity's $\infty$-cosmoi setting (http://www.math.jhu.edu/~eriehl/scratch.pdf). Consider an $\infty$-cosmos $\mathcal{K}$, and let us focus on simplicial sets- shaped diagrams, i.e. $\infty$-functors $f:1 \to K^J$, where $1$ is the (simplicially enriched) terminal object of $\mathcal{K}$, $K$ is an object of $\mathcal{K}$ and $J\in \textbf{sSet}$.

A limit of $f$ is defined as pair $(l:1 \to K, \alpha: \Delta \circ l \Rightarrow f)$, where $\alpha$ is a 2-cell in the underlying homotopy category and $\Delta: K \to K^J$ is the constant diagram (i.e $\Delta=K^{J \to 1}$), such that $\alpha$ is an absolute right lifting diagram (with a slight abuse of language, see Def.2.2.5 and Def.2.2.9 in the abovementioned link).

It possible to prove that this notion of limit is representably defined, in the sense that such data define a limit iff the corresponding diagrams obtained by applying $map_{\mathcal{K}}(X,-):\mathcal{K} \to \textbf{qCat}$ are such, for any $K\in \mathcal{K}$, and we know that for quasicategories we can construct such limits by using products and equalizers. I would like, with a suitable use of the 2-Yoneda lemma, to transfer it back to the $\infty$-cosmos $\mathcal{K}$.

Now, in the case of ordinary categories I think I can prove that given any two categories $\mathscr{C}, \mathscr{D}$, it possible to construct a category $\widetilde{\mathscr{C}}$, depending only on $\mathscr{C}$, and a functor $F:\mathscr{D}^\mathscr{C} \to \mathscr{D}^\mathscr{\widetilde{C}}$ such that $\lim_{\mathscr{C}} \cong \lim_{\widetilde{\mathscr{C}}}\circ F$, where $\widetilde{\mathscr{C}}$ captures the fact that the limit is constructed just by using products and equalizers. While trying to generalize this fact to simplicial sets I am facing a problem: I had to use the "join along a profunctor" construction in order to define the $\tilde{(-)}$ endofunctor, which does not seem to have a handy counterpart in simplicial sets.

Finding an analogous construction for generale simplicial sets would translate the purpose of this question, but I'd be happy (probably happier, actually) if another "neater" construction turns out to be possible.

• I think you are looking for the Bousfield–Kan formula. Aaron Mazel-Gee has a preprint (to appear) discussing that. – Zhen Lin Oct 13 '15 at 11:49
• Do you mind expanding a little? – Edoardo Lanari Oct 13 '15 at 11:59
• The homotopy-theoretic analogue of the coproduct–coequaliser formula is the Bousfield–Kan formula, which involves coproducts and so-called "geometric realisation". This is well known. – Zhen Lin Oct 13 '15 at 12:14