2
$\begingroup$

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?

Improve this question
$\endgroup$
5
  • 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, 2014 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, 2014 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, 2014 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, 2014 at 6:50
  • 1
    $\begingroup$ 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, 2014 at 7:10

1 Answer 1

Reset to default
6
$\begingroup$

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.

Improve this answer
$\endgroup$
1
  • $\begingroup$ 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, 2014 at 1:28

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.