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Let CatAdj be the 1-category whose objects are small categories which are finitely complete, and whose morphisms are adjoint pairs of functors whose inverse image is exact. The composition is strict composition of adjoint functors, of course. The direction of a morphism is by the definition in my question below the direction of the INVERSE image functor.
1) Is CatAdj finitely complete ?
I know that finite products exist but do not know if equalizers exist. In fact that construction is used in reducing some situations with descent for families to comonadic descent.
2) If it is what is the construction of equalizers in that category and arrows guaranteed by universality ? In other words given a parallel pair in CatAdj, a cone in CatAdj, what is the unique arrow from such that the composition equals a cone in CatAdj ?
I am interested in many variants of the above questions.
Your CatAdj is not finitely complete as a 1-category. Let C be the terminal category, and let D be the indiscrete category on two objects, which is of course equivalent to C; both are complete and cocomplete. There are two functors f,g: C → D which preserve all limits and colimits and are equivalences of categories, hence both have adjoints. But there is no finitely complete category E with a functor h:E→C such that fh=gh, since there is no object of C such that f(x)=g(x) and E cannot be empty (since it must have a terminal object).
Wow, thanks a lot. This kills the way I wanted, but maybe it gives a hint (I have some vague idea) how to correct the setup to achieve what I wanted in some sense.
In my application I start with a cone over the diagram in CatAdj and want (for certain descent purposes), to replace that cone over a diagram with a single morphism to a limiting cone. If there is no cone whatsoever as in your counterexample, then I have no input to my problem so I do not really care. But if I have a cone I need a limiting cone to reduce the cone with a single morphism and then to look for comonadic descent for that singke morphism. If one has a discrete diagram then the issues about it are well known, when it is not, one would like to have some sort of a descent with constraints. The constraints are what makes me wanting equalizers. But if I have no cone whatsoever than I do not care. If I have some cone I want to have a limiting cone and then I care about the map from my original cone to the limiting cone; this one I use then to formulate my constrained descent problem as a usual comonadic descent. Sorry for being vague a bit.
I’m a little puzzled by what application, especially descent-related, would require 1-categorical equalizers rather than 2-categorical descent objects.
Surely, the descent data are constructed using 2-cells in the game, but a reformulation of a problem involving several adjoint pairs in terms of single adjoint pair is strict, as is seen from the non-constrained case. Look at gluing categories from localizations (zoranskoda), the construction of thick Q-s from the family in the section “From a family of localizations to a comonad”. I am interested when I can make similar construction involving constraints, not just a cone over a discrete diagram of categories. The equalizers here have nothing to do with the common equalizers in the descent data construction, they are just constraints among the local data, before I reassemble them with a single comonad. I am just replacing a cover by a family with cover by a single map; but now with constraint. The descent problem is still well defined: which objects in category correspond to a family of objects in each of the categories below, plus additional data and what are the additional data…but now the constraints taken into account.
Okay, I think I get the idea. Well, good luck! (-:
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