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?

  • 1
    $\begingroup$ 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. $\endgroup$ – David Roberts Feb 16 '14 at 23:07
  • $\begingroup$ 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. $\endgroup$ – Fernando Feb 17 '14 at 6:38
  • $\begingroup$ @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)? $\endgroup$ – Fernando Feb 17 '14 at 6:47
  • $\begingroup$ 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) $\endgroup$ – Fernando Feb 17 '14 at 6:50
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    $\begingroup$ 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]) $\endgroup$ – David Roberts Feb 17 '14 at 7:10

The "descent category" can be understood as a limit from the point of view of enriched category theory. The paper considers a diagram E in the 2-category of categories. The latter as well as any 2-category can be considered as a Cat enriched category. Then the "descent category" is the weighted limit of E with an appropriate weight (which will be a functor from the truncated cosimplicial category diagram to Cat). In the same way, a descent object can be defined within any 2-category.

If you are interested in weighted limits you can look at Kelly's book on enriched categories. It is in TAC reprints.

As for why the descent category is the 2-dimensional analogue of an equalizer, there is a philosophical explanation. An equalizer of two set functions f and g is a set of those elements x for which f(x) = g(x). In the 2-dimensional situation you want to replace equalities by morphisms which satisfy coherence conditions. So an object of the 2-dimensional equalizer of functors f and g consists of an object x together with a morphism f(x) -> g(x) which satisfies two coherence equations.

  • $\begingroup$ Thank you! I'd already realized it, but I forgot to write here. The point that was not completely clear was the philosophical point you explained. But, now, by more than one reason, I got it: I understood well the analogy. Thank you for answering. $\endgroup$ – Fernando Feb 26 '14 at 1:28

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