Hi,

I was thinking about presenting categories with nothing but equations over morphisms. I wondered about products. The definition of a product has its genesis in the following diagram shape

A->B

A->C

You would say that whenever you have this shape, then for any D such that

D->A->B

and

D->A->C

blah blah...the axioms for the product. I thought about how to do this with just words in the arrows without reference to either source and target or objects. You can start with words like

$dab$ and $dac$

where a:D->A and and b: A->B and c:A->C and d:E->D.

You might say: if, the existence of the equations

$dab=e$ and

$dac=f$

implies that $a$ is unique in that if there exists words

$dxb$ $dyc$

then $x=y=a$

then this constitutes a product.

Could something like this work? I mean, can you present a category as just words over morphisms along with equations and a bit of language with quantifiers? Second, could this kind of definition work for products?

justwords and equations. For example, here are two categories presented normally:Chas two objects x and y, and Hom(x,y) has two arrows f and g;Dhas four objects a, b, x, y and Hom(a,b) and Hom(x,y) each have a single arrow f, g respectively. In your language, there are only two letters, namely f and g, no other words, and no equations in either category. So how do you tellCfromD? – Ryan Reich Jun 25 '11 at 13:32