Small2

[pare@mathstat.dal.ca (Robert Pare)]

Thanks to all who replied to my posting either privately or on Categories.

I'd like to clarify a few points. Ross's interpretation of what I meant,
and which Vaughan agrees with, is not what I was trying to say. It is part
of the picture but not the one I was promoting.

Small categories play a different role in category theory than large
categories. Small categories are used for indexing things. Large
categories consist of the things we are indexing. Okay, syntactic was
not the right word. Combinatorial might be better although that has
finiteness overtones. Strict? Well, I'll just stick with small. Small
is not the same as essentially small. As Jeff Egger pointed out, the
category of finite sets is not small. It's not even clear what this category
is. The ZFCists would say that for each set A we get a finite set {A} and
another {{A}}, and Barwise might even wonder if these last two are distinct.
But I'm digressing.

Of course every small category can be considered as a large one (perhaps
large isn't the right word either). Then two equivalent ones would be
considered "the same". I don't think that this is how the working
categorician works. Sometimes equivalent categories are "the same" and
sometimes not. An equivalence relation is a category but we would lose
something (everything?) if we identified it with equality in the quotient
set. So I'm saying that, as a matter of "categorical hygiene", we could
be a bit more explicit about what kind "equivalence" we are allowing.
And without changing mathematical practice, I'm advocating that "small"
should imply that it's okay to talk of equality of objects.

As usual it helps put things in relief to generalize them.
Consider category theory in a world based on a topos S. A small category
is a category object in S. A large category like S or Group(S) is an
indexed category, given by a pseudo-functor S^op -> Cat, or a fibration
over S if you prefer. A small category C gives a large one by homming.
But something is lost in the process.

The object functor  Cat -> Set is not a 2-functor so if you compose it
with a pseudo-functor S^op -> Cat you get an assignment that doesn't
preserve composition in any sense. So a general indexed category doesn't
have a discrete category of objects. What you can do is take the groupoid
of isomorphisms 2-functor Cat -> Gpd, so that a large has a groupoid of
objects, thinking of a groupoid as a set with a 2-equality on it. An
indexed category gotten from a small category gives an actual functor
S^op -> Cat, so now you can compose with Ob : Cat -> Set so that a
small category has a discrete groupoid of objects.

And that's where the idea of considering categories with a specified
"equality groupoid" of isomorphisms, and equivalences defined to have
the isomorphisms from the corresponding groupoids. Small categories
have only identities, so equivalence means isomorphism, and large
categories have all isos. At the time there were no good examples
of intermediate groupoids (just products of small and large categories
and the like) but Hilbert spaces with unitary isos is a good one.

Well this is my second posting in a week bringing my life-time total
to three! Seems a good place to stop.

Best wishes for the new year. May your happiness be large and your
disappointments small.

Bob



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