regular, geometric and coherent categories

regular, geometric and coherent categories

[John Baez]

Dear Categorists -

In my quest to understand how various flavors of monoidal
category can be seen as having various flavors of "internal
language" and "internal logic", I've been enjoying the section
in Johnstone's "Sketches of an Elephant" where he discusses
different fragments of first-order logic and how they can
be interpreted in categories with different properties.

Of course, this being a book on topos theory, none of this
deals with monoidal categories where the tensor product
is not cartesian - my main interest, for applications
to quantum logic.  But, it's still lots of fun.

I'd like to get a better feel for some of these things.
For example, he talks about

"cartesian categories"
"regular categories",
"geometric categories",
"coherent categories"

and describes which fragment of first-order logic can be
interpreted in each of these things:

"cartesian logic",
"regular logic",
"geometric logic"
"coherent logic".

Here's some stuff I think I know.  I know the definitions of
the above concepts, as long as I have the book open to the
right page... but I left it at home, so these could be wrong!

Cartesian categories have finite limits.  Regular categories
are cartesian categories with regular epi/mono factorizations,
which must be stable under pullbacks.  Geometric categories are
regular categories admitting arbitrary unions of subobjects,
which must be stable under pullbacks.   Coherent categories
are geometric categories where pullback of subobjects has a
right adjoint (which plays the role of "for all").

I have a fairly good feel for categories with finite limits and
"finite limits theories"; the others seem more mysterious to me,
since I don't know enough examples illustrating the distinctions.
Categories monadic over Set are regular, so AbGp is regular - but
it's not coherent, since in a coherent category every morphism to
the initial object is an isomorphism.  What are some other examples
of all these things?

regular, geometric, and coherent categories

[John Baez]

I wrote:

> Here's some stuff I think I know.  I know the definitions of
> the above concepts, as long as I have the book open to the
> right page... but I left it at home, so these could be wrong!

Some were.

> Cartesian categories have finite limits.  Regular categories
> are cartesian categories with regular epi/mono factorizations,
> which must be stable under pullbacks.


Fine.

> Geometric categories are
> regular categories admitting arbitrary unions of subobjects,
> which must be stable under pullbacks.   Coherent categories
> are geometric categories where pullback of subobjects has a
> right adjoint (which plays the role of "for all").

These were mixed up.