# n-limits and the Descent Category

Probably, this question could be at https://math.stackexchange.com/, but, for some reason, I can't access that site right now. So, I apologize in advance.

At this article http://arxiv.org/ftp/math/papers/0303/0303175.pdf , page 4, last diagram, professor Ross Street says that the Descent Category (as defined by him) of the truncated cosimplicial category is, in someway, analogue to the equalizers.

I assumed that he is talking about 2-limits (since, if he were talking about 1-limits, the limit analogue to pullback would be the pulback itself). Am I right?

(1) If I am right, I wonder if the Descent Category, defined by him, is the 2-limit of that truncated cosimplicial category. If it is not true, why is the descent category defined by him related to the 2-limits/limits?

(2) I would like to know where I can find texts on n-limits. Which would be the best way to understand it?

• If you're interested in $n$-limits, these are more easily treated using homotopy colimits, as we don't understand the $n$-category of $(n-1)$-categories very well, for $n\gt 3$, say. A coequaliser is the homotopy colimit (in a 1-category!) of a simplicial diagram truncated all the way down to a pair of parallel arrows. The "2-" version of this is a slightly less truncated simplicial diagram. Mapping out of these (in an appropriate model structure, in the absence of a good higher category structure) gives the cosimplicial version Street has, and replaces colimits by limits. – David Roberts Feb 16 '14 at 23:07
• David Roberts, thank you for your comment. Why didn't you posted your comment as an answer? I would at least give you an "upvote" (actually, I would accept your comment as an answer). If you don't mind, I would like to better understand what you said and I'm going to post some commentaries asking some details. – Fernando Feb 17 '14 at 6:38
• @David Roberts , I do understand the basics of homotopy colimits, at least in the context of model categories. But I got some doubts. Firstly, when you say that the coequalizer in a 1-category are the coequalizers themselves, are you talking about 1-category with the trivial model structure (which means that the weak equivaleces are the isomorphisms)? – Fernando Feb 17 '14 at 6:47
• Secondly, where I can find this view point you talked about? Which would be the basic reference to understand weak n limits? Finally, is it easy to describe the 2-limit of that truncated cosimplicial category defined by Street? If it so, what would be this 2-limit? Thank you very much. (By the way, as I said, you may post your comment as an answer) – Fernando Feb 17 '14 at 6:50
• Regarding homotopy limits in a 1-category C, I'm thinking of something like that category being that of the constant objects in the category of simplicial presheaves, so the model structure is 'in the oo-category direction', rather than among the objects of C. If you like, think of homotopy colimits as n-colimits (n in [1,oo]) – David Roberts Feb 17 '14 at 7:10