Re: Teaching Category Theory

["Ronnie Brown" <ronnie.profbrown@btinternet.com>]

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.




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