# [FOM] The Category of Categories, and Sketches

Laurent Delattre nouvid-fom at yahoo.fr
Mon Mar 6 03:58:59 EST 2006

```Hello all,

I am interested in the axiomatic definition of the
category of categories as a foundation for
mathematics.

1) Can anyone give me a possibly simple example of a
(small) category which is not definable by a sketch?

2) As I understand it, a sketch defines a small
category by giving sets of objects and morphisms, a
set of equations, and sets of limiting cones and
cocones. Do we really need to specify cones and
cocones? Let me clarify my question on a example. To
state that two objects A and B have a product A*B with
projections p1 and p2, I can state it by a limiting
cone. But instead I can specify it by three equations:

p1 o <f,g> = f
p2 o <f,g> = g
<p1 o h, p2 o h> = h

where f:X->A, g:X->B, h:X->A*B and 'o' is composition.
So do we really need cones and cocones in the
definition of a sketch? If a theory is definable by a
sketch, isn't it simply definable by sets of objets,
morphisms and equations?

3) Is there a sketch which defines the category of
sets and functions?

4) Is there a sketch which defines the category of
small (respectively, locally small) categories?

Thanks!

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