# Homotopy limit of a cosimplicial category

Consider the usual model structure on Cat (category of small categories). Which are the fibrations of the injective model structure on the category of cosimplicial categories $Fun( \Delta ,Cat )$?

Actually, I wonder how can I compute homotopy limits of cosimplicial categories (considering them as diagrams in Cat)...

Edit: I'm most interested in understanding how to compute the homotopy limit of a cosimplicial diagram in $Cat$

• I doubt that taking Cat as a target category may simplify the characterization of fibrations in the injective model structure on cosimplicial diagrams. I'd say you should proceed as in the general case. – Fernando Muro Feb 18 '14 at 16:53
• @Fernando Muro, Thank you (again) for your comment. Actually, I'm more interested in understanding how to compute homotopy limits of cosimplicial diagrams in $Cat$. Do you know if there is an easy way to do so? Thank you very much – Fernando Feb 18 '14 at 19:40
• The standard model structure on $\mathbf{Cat}$ is indeed simplicial, provided one uses as the nerve of the maximal subgroupoid of $[\mathcal{A}, \mathcal{B}]$ as the simplicial hom-space $\underline{\mathrm{Hom}}(\mathcal{A}, \mathcal{B})$: see this note of Rezk. Thus one can use the standard technique of bar and cobar constructions: see e.g. this preprint of Shulman). Note that the existence of injective/projective model structures is not needed. – Zhen Lin Feb 19 '14 at 10:25
• Maybe section 6.5 in pages.uoregon.edu/ddugger/hocolim.pdf can help. In general, this paper is a very nice account on hocolim (just dualize for holim). – Fernando Muro Feb 19 '14 at 11:33
• Lemma 3.4.11 in [Sketches of an elephant, Part B] shows that a certain non-full subcategory of $\mathbf{\Delta}$ is 2-coinitial. – Zhen Lin Feb 19 '14 at 21:30

I will answer the title question about computing homotopy limits. Let $A : \mathcal{C} \to \mathbf{Cat}$ be a small (strict) diagram. By expanding the definitions of the cobar construction, one eventually discovers that $\operatorname{holim} A$ can be computed as the end $$\int_{[n] : \mathbf{\Delta}} \left[ \mathbf{I} [n], \prod_{(c_0, \ldots, c_n)} \mathcal{C} (c_n, c_{n-1}) \times \cdots \times \mathcal{C} (c_1, c_0), A c_0 \right]$$ where $\mathbf{I} [n]$ is the contractible groupoid with $n + 1$ objects. In more familiar terms, this is just the hom-category of all morphisms between a certain pair of cosimplicial categories.

Here is a description of an object in $\operatorname{holim} A$:

• For each object $c$ in $\mathcal{C}$, we have an object $a_c$ in $A c$.
• For each morphism $f : c_1 \to c_0$ in $\mathcal{C}$, we have an isomorphism $\mu_f : f (a_{c_1}) \to a_{c_0}$ in $A c_0$; and $\mu_{\mathrm{id}_c} = \mathrm{id}_{a_c}$ for all objects $c$ in $\mathcal{C}$.
• For each composable pair $f_1 : c_2 \to c_1, f_0 : c_1 \to c_0$ in $\mathcal{C}$, we have a commutative triangle in $A c_0$, $$\begin{array}{ccc} f_0 (f_1 (a_{c_2})) & \rightarrow & f_0 (a_{c_1}) \\ & \searrow & \downarrow \\ && a_{c_0} \end{array}$$ i.e. $\mu_{f_0 \circ f_1} = \mu_{f_0} \circ f_0 (\mu_{f_1})$.
• For each composable triple in $\mathcal{C}$, we have a commutative tetrahedron in $A c_0$.
• etc.

In fact, the coherence conditions above degree 2 are automatic if all the triangles commute. Thus we see that an object in $\operatorname{holim} A$ is the same thing as a pseudocone over $A$ (where we regard $A$ as a pseudofunctor with the canonical coherence data). Unsurprisingly, the morphisms are the same, so $\operatorname{holim} A$ (as constructed above) is the pseudolimit of $A$.

• Thank you. Your answer helped me a lot. So, your conclusion was that the homotopy limit is the same concept of the pseudolimit of a strict diagram. I would like to ask, nevertheless, if there is a "end" formula like that of cobar construction for computing pseudolimits of pseudolimit of a "pseudodiagram" (non-strict diagram). – Fernando Mar 6 '14 at 3:07
• I do not know of such a formula. But there should be no real difference other than inserting coherence isomorphisms in the appropriate places. – Zhen Lin Mar 6 '14 at 8:14
• What is this notation inside the end, which consists of square brackets with two commas inside? $[ , , ]$? – Dean Young Feb 27 at 7:48