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Expanded the examples section at end (prodded by discussion in the blog), stating Kan extension and geometric realization.
Also added a toc.
I’ve (tidied up a bit and) added a new subsection at end explaining why the hom-weighted (co)limit definition implies the (co)equalizer one.
I wanted to add labels, à la AMSLatex’s equation
environment, to a couple of displayed diagrams, but I can’t remember/never knew if iTeX can do that. Anyone know?
Incidentally, the hom-weighted limit definition of ends follows, at least when V=Set, from the universal-extranatural one once you know that there is a Yoneda-type lemma for extranaturals giving a bijection between Exnat(pt, F) and Nat(hom_C, F) (maths rendering doesn’t seem to be working here, at least in the preview). Has anyone seen this extranatural-Yoneda written down elsewhere? I came up with it myself a while ago, but it must have been noticed before.
To add numbers to a displayed equation, use \[ ... \label{foo} \]
. Then (eq:foo)
will produce ‘(1)’ (or whatever number it is) and an internal link.
That works perfectly, thanks!
Another couple of minor edits/additions.
Entry end requires that the definition of an end of -enriched functor requires that is symmetric. Is this really necessary? (I think I know some examples at least where it is not, but maybe I misunderstand) On the other hand, Kelly’s book starts requiring that is symmetric precisely before defining the tensor product of enriched categories (entry not existing yet; the notation of tensor product of enriched categories is freely used at end without warning). What is the problem (if any) with defining the tensor product of -enriched categories if is not symmetric ?
I see that Matsuhiro Takeuchi is in his paper introducing what is now Takeuchi product used the notation for end and coend interchanged, what is unfortunately followed later by modern writers in Hopf algebras, for that context. This comes in fact from earlier paper by Sweedler who assigns the notation to MacLane. See page 87 in
Worse, e.g. the label would suggest in category community that is an object in some category and a functor. For Sweedler and Takeuchi will be a morphism in category theoretic interpretation. In fact, as MacLane explains in his section on coends, the usual coend reproduces the tensor product over rings, where ring is understood as a category with one object and worth of morphisms, and modules are understood as functors ; and is understood as a bifunctor .
I believe you can get away with a braiding, but I’m not sure how you’d get away with less. I suppose that if for example the homs of live in the center of the monoidal category , you can get away with strictly less and speak of a -enriched functor .
Maybe the best way of answering the question is to ask another. If and are -enriched, how do you define as a -category, and in particular how are you supposed to define the composition law? If , then some sort of interchange seems to be necessary to get the composition.
As Todd says, you do need some sort of symmetry or braiding in order to define the tensor product of -categories in general. However, you can define the notion of -profunctor without any symmetry or braiding, by using separate left and right actions. I wonder whether you could define the end or coend of a -profunctor directly without needing to regard it as a -functor ?
Thank you guys, I will have to extensively think of you suggestions.
Let me mention just the trivial case of a functor of the form where are just -enriched. If this end of gives the usual “inner end” of a functor which is in that case a monoid. Instead of one usually by adjunction considers map , which become actions in the special case . The universal property is that for every object in and every -natural transformation there is a unique morphism in such that . This special case, in this adjoint form, makes sense for biclosed but not symmetric , it seems to me (and one does not care how the domain category of be called, as at least enriched hom makes sense always by definition). Though I am not sure, if I translated fully the conditions for universal wedges of morphisms and along another line .
In the case of a tensor product of a functor and , where is merely monoidal biclosed and cocomplete, we have a coequalizer diagram
with not a trace of symmetry in sight.
We can do something similar for the hom of two functors , phrased in terms of actions . If we are careful and say is left adjoint to (so we are just assuming closed monoidal here), then one can carefully check there is a canonical isomorphism
and we can also curry the action on as an extranatural family of maps . In that case we have a family
(using the action on ) and a family
(using the curryed action on ), and take the equalizer
using these two families to define the parallel arrows.
I haven’t investigated the situation for more general ends and coends but without symmetry.
An even more general statement is that if is a locally cocomplete bicategory whose composition has right adjoints in each variable (i.e. all right extensions and right liftings exist), then there is another such bicategory whose objects are -enriched categories and whose morphisms are bimodules. An advantage of phrasing and proving it like that is that it doesn’t even make sense to talk about a bicategory being “symmetric”, so there’s less danger that you’re using symmetry without realizing it.
Amazing answers :)
Added to end a pointer to Fosco’s beautiful introduction which was also discussed here.
Ingo, thanks for adding the pointer! I could have sworn that I had added this earlier, but clearly I didn’t.
There is some strange wording at end:
[where is customary notation for the enriched hom of in ]
What was intended? It sort of looks like a half completed edit.
Added an explicit definition of ends in terms of wedges; natural transformations as ends, and an example of (co)end calculus.
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