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? 



n-limits and the Descent Category
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. 


