Re: Teaching Category Theory

[Jeff Egger <jeffegger@yahoo.ca>]

--- Peter LeFanu Lumsdaine <plumsdai@andrew.cmu.edu> wrote:

> >> [In my experience, non-category-theorists, when asked to
> >> provide a definition of category, almost uniformly supply (what amounts
> >> to) the definition of an enriched category, in the case V=Set---which I
> >> find quite intriguing.] 
> 
> Surely the intriguing thing here is not (as I understand you to be
> suggesting) the set-centricity that they're imposing, but rather that they're
> not imposing it as far as usual? 

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.  

The issue at stake may seem minor: do we think of a class of arrows 
(which can later be partitioned into homsets), or do we think of the 
homsets first (and take their disjoint union later)?  But perhaps 
the fact that one group of people prefers one approach and everyone
else the other is symptomatic of a psychological divide? 

It's also worth noting, perhaps, how flukey it is that in the case 
V=Set, V-internal and small V-enriched categories happen to coincide.
Consider V=Cat, for example.  Or, note how different the requirements 
on V are, for V-internal and V-enriched categories to be defined. 

>When asked to define pretty much any
> algebraic gadget, most mathematicians will define a model of that algebraic
> gadget in Set (see e.g. en.wikipedia.org/wiki/Group_%28mathematics%29 ).

It is true that one would expect set-theoretic conservatives to deal with
small categories (~internal categories in the case V=Set), and more flexible
mathematicians to use arbitrary large categories (~internal categories, where 
V is a category of "large sets", or classes).  This only re-inforces the 
points made above.

Cheers,
Jeff.