[Note from moderator: a response to this item from Jeff Egger was posted earlier, but the original was not... ] On 8/31/07, Jeff Egger <jeffegger@yahoo.ca> wrote:

Actually, what I find intriguing is that it is the definition of enriched category which seems to have priority over the definition of internal category. There are, I suppose, historical reasons for this (pre-1960 the focus tended to be on AbGp-enriched categories) ---but I think it fair to say that (for as long as I can remember, which obviously isn't that long from a "historical" perspective) the majority of category theorists tend to adopt the internal category style of definition (of category) as more primitive.

My guess would be that it's because for non-category theorists, many (perhaps most) categories which arise in practice are enriched (over something more exotic than Set), while few are internal (to something more exotic than Set). Even when working over Set, I think it's fair to say that the vast majority of categories arising in mathematical practice are locally small. Since in general, neither enriched nor internal category theory is a special case of the other, it doesn't seem justified to me to consider either one as "more primitive". However, it's worth pointing out that both are a special case of categories enriched in a bicategory, or in a double category. Actually, currently my favorite level of generality is something I call a "monoidal fibration". Roughly, the idea is that you have two different "base" categories, S and V, such that the object-of-objects comes from S while the object-of-morphisms comes from V. When S and V are the same, you get internal categories, and when S=Set, you get classical enriched categories. This could be regarded as "explaining" the coincidence of internal and enriched categories for V=Set. I wrote a bit about this at the end of "Framed Bicategories and Monoidal Fibrations" (arXiv:0706.1286), but I intend to say more in a forthcoming paper. Mike

"Jeff Egger" <jeffegger@yahoo.ca> wrote:

--- Michael Shulman <shulman@math.uchicago.edu> wrote:

[I don't see the original post, so I'm responding here]

Actually, currently my favorite level of generality is something I call a "monoidal fibration". Roughly, the idea is that you have two different "base" categories, S and V, such that the object-of-objects comes from S while the object-of-morphisms comes from V. When S and V are the same, you get internal categories, and when S=Set, you get classical enriched categories. This could be regarded as "explaining" the coincidence of internal and enriched categories for V=Set.

Heh... I'm studying the same problem as a part of my Master Thesis (under supervision of prof. Andrzej Tarlecki), but fortunately :-) in a bit different framework. The chief concept of my work is a definition of a category ("elementary category") in a fibred monoidal category (i.e. each fibre is monoidal and reindexing functors preserve the monoidal structure) over a base category with binary products. Than, roughly speaking, for a category C with finite limits, C-enriched categories are just "Fam : Fam(C) -> Set"-elementary categories, and C-internal categories are just "Cod : C^{->} -> C"-elementary categories. It turns out (if I didn't make mistakes :-)), that when C has Set-indexed coproducts, than there is an adjunction between the global section functor C(1, -) : Cod -> Fam and the "coproduct functor" \coprod_{-}(1) : Fam -> Cod. Furthermore, if the coproducts are universal, than these functors are fibred and preserves the monoidal structures, and if additionally all global sections in C are disjoint (i.e. the pullback of two different global section is an initial object) than this adjunction is an equivalence of categories (these results give us approximations C-internal categories ---> C-enriched categories and in the other direction). Best regards, Michal R. Przybylek

One would like to leave students with a very positive attitude. The following quotation , from the Stanford Encyclopedia of Philosophy, might help: " Category theory has come to occupy a central position in contemporary mathematics and theoretical computer science, and is also applied to mathematical physics. ......." http://plato.stanford.edu/entries/category-theory/ I am writing as someone who has come into category theory from algebraic topology, and been struck by the utility of the language and results for what I needed over the years (and by the welcome). A separate matter, but intriguingly related, is categories and groupoids as sources of useful algebraic structures. This seems connected with the notion of partial operations, and so my notion of `higher dimensional group theory' and `higher dimensional algebra' is that of studying algebraic structures with partial operations defined under geometric conditions. This concurs with the vision in Higgins, Philip J. Algebras with a scheme of operators. Math. Nachr. 27 1963 115--132. In this view, the objects of a category play a key role. This has covered the developments I had in mind for modelling some underlying structures in homotopy theory, and which were developed with Philip Higgins, and later with Loday. Relevant was Philip's reporting of the view of Philip Hall that one should study the algebra that arises naturally from the geometry without trying to force the algebra into a preconceived mould. In the late 1960s, when Bill Cockcroft and I received notes from Saunders of lectures on category theory for our comments, Bill and I replied that what we really wanted was `Categories for the working mathematician'. I still hold to that. To me this means general theory with specific examples which show how the general theory makes life easier, even controls the calculations. Eilenberg insisted a construction should be defined, and its properties developed, in terms of the universal property, which should also explain existence. So when dealing with structures at various levels it is very useful to know left adjoints commute with colimits, right adjoints commute with limits, and this can tell one how to compute colimits and limits. This also leads to induced constructions (change of base). I have recently found uses (to me!) of fibrations of categories: the inclusion of a fibre preserves connected colimits. Simple examples of the use of this are: Ob: Groupoids \to Sets, forget: (groupoid modules) \to groupoids; forget: (2-Cat) \to Cat and compositions of these. Of course it was generalisations of the van Kampen theorem to higher dimensions, and the (previously rare) use in homotopy theory of colimits of algebraic structures, that made it useful to do such computations. I have only recently really understood the notion of dense subcategory, and its use for representing an object as a coend. What I have not done is use the theory of monads. Is this ignorance on my part? I am happy to be enlightened! One of the points of a course for the students might be `need to know'. Hence the need for explicit and varied examples. How to balance this with theory? Ronnie www.bangor.ac.uk/r.brown ----- Original Message ----- From: "Jeff Egger" <jeffegger@yahoo.ca> To: <categories@mta.ca> Sent: Tuesday, September 04, 2007 5:30 PM Subject: categories: Re: Teaching Category Theory --- Michael Shulman <shulman@math.uchicago.edu> wrote:

My guess would be that it's because for non-category theorists, many (perhaps most) categories which arise in practice are enriched (over something more exotic than Set), while few are internal (to something more exotic than Set).

I'm not sure I agree with that: internal groupoids, at the very least, show up in a variety of situations which non-category theorists can be, and are, interested in. Perhaps one of the reasons why some people try to deal with groupoids as if they weren't a special case of categories is because they never thought of categories in any other way than as a mass of hom-sets.

Even when working over Set, I think it's fair to say that the vast majority of categories arising in mathematical practice are locally small.

Now I do think there is a good reason for this, which is the fact that in functorial semantics (by which I don't just mean the original, universal-algebraic, case), the domain category is typically small. Raising to a small power does not destroy local smallness.

Since in general, neither enriched nor internal category theory is a special case of the other, it doesn't seem justified to me to consider either one as "more primitive".

I agree with this entirely, of course. It follows that, in a first course on category theory, one should present both styles of definition as soon as possible. This, in turn, suggests (but does not prove) that one should not sweep size distinctions under the carpet.

Actually, currently my favorite level of generality is something I call a "monoidal fibration". Roughly, the idea is that you have two different "base" categories, S and V, such that the object-of-objects comes from S while the object-of-morphisms comes from V. When S and V are the same, you get internal categories, and when S=Set, you get classical enriched categories. This could be regarded as "explaining" the coincidence of internal and enriched categories for V=Set. I wrote a bit about this at the end of "Framed Bicategories and Monoidal Fibrations" (arXiv:0706.1286), but I intend to say more in a forthcoming paper.

I look forward to it! Cheers, Jeff. -- No virus found in this incoming message. Checked by AVG Free Edition. Version: 7.5.484 / Virus Database: 269.13.2 - Release Date: 01/09/2007 00:00