Subcategories (Pat Donaly)

Subcategories (Pat Donaly)

[Jpdonaly]

Categorists:

Here is something which bothers me.

It seems to be common for textbook writers to prove that the composite of a
pair of monomorphisms is a monomorphism, and it may be taken for granted that
everyone knows that every object is a monomorphism. These two facts imply that
the monomorphisms in a category form a subcategory of it. A similar remark
applies to pullbacks: On page 16 of "Sheaves in Geometry and Logic", Mac Lane and
Moerdijk prove among other things that, in the category of commutative
squares of a category, the composite of a pair of pullbacks is a pullback, which is
a good start toward establishing that the pullbacks form a subcategory of the
commutative squares, but Mac Lane and Moerdijk are satisfied with calling the
multiplicativity of pullbacks a "pasting lemma" (the quotes are theirs). In
proposition 18.16. on page 121 of their 1973 book, "Category Theory", Herrlich
and Strecker do this sort of thing wholesale, leaving me to wonder, is there
something wrong with the subcategory concept?

To be honest, I have noticed that the habit of naming categories after
their objects to the extent possible makes it difficult to speak of a
subcategory which has the same objects as its parent, but it nevertheless
seems strange that, after generalizing the subobject notion, category
theory would terminologically orphan its own subobjects. Any
clarifications or corrections of these impressions?

Pat Donaly