Re: are sketches math objects?

["Zinovy Diskin" <zdiskin@swen.uwaterloo.ca>]

Okay, sketches are presentations of theories but Steve's claim was
that they are not mathematical objects. Michael's and mine
bewilderment is about why does the former imply the latter?  (at
least, why "of course" :)

Zinovy

On Fri, Sep 19, 2008 at 6:27 PM,  <Mark.Weber@pps.jussieu.fr> wrote:
> 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
>