Re: Bourbaki and Categories

[Mark.Weber@pps.jussieu.fr]

Dear Michael,

Semantically, as Lawvere observed long ago, a monad gives rise not just to
a category of algebras but also to a forgetful functor into the category
on which the monad acts. For any category C the functor

"semantics" : Mnd(C)^op --> CAT/C

whose object map sends a monad on C to its associated forgetful functor is
full and faithful. Thus a pair of monads on C giving rise to isomorphic
forgetful functors must necessarily be isomorphic. So your observations
about different monads giving rise to the same algebras, while correct, do
not tell the whole story on the semantic side.

The situation is of course different for sketches: they too give rise to
forgetful functors (into Set), but this does not suffice to determine a
given sketch up to isomorphism in the same way, and this justifies Steve
Lack's perspective of "sketches as presentations of theories".

Mark Weber

> Previously, Michael Barr wrote:
>
> Of course sketches are mathematical objects in their own right.  Of
> course, the functor that assigns to each sketch the corresponding theory
> is not full or faithful.  But the definition is precise, the notion of
> model is also precise, so I have no idea what, if any, content there is in
> the claim.  Incidentally, you might with equal justice claim that triples
> are not mathematical objects since two distinct triples can have
> isomorphic categories of Eilenberg-Moore algebras.  In fact there are
> triples (or theories) on Set that have infinitary operations, yet whose
> category of models is isomorphic to Set.
>
> Michael
>