Kleisli categories for monads on presheaves

[Uwe Egbert Wolter <Uwe.Wolter@uib.no>]

Dear all,

many thanks for the very useful replies concerning my question about
Grothendieck-Yoneda-Colimits. Now another question on top of it:

I'm more on the "applied side" and interested in syntactic
representation of things. For a many-sorted algebraic signature \Sigma
with a finite set (discrete category) S of sorts the construction of
\Sigma-terms gives us a monad T_\Sigma:Set^S -> Set^S. The syntactic
category with S^* as set of objects, finite tuples of terms as morphisms
and "composition by substitution" (Lawvere) can be seen as a subcategory
of the Kleisli category of this monad.

We generalized recently the concept of algebraic signatures and algebras
to graphs: input and out put arities of operations are graphs as well as
the carriers of algebras are graphs. We describe the construction of
"graph terms" and get a monad on Set^B with B the category given by two
parallel arrows s,t:E->V. What we would like to have is a nice
generalization of the construction of syntactic Lawvere categories to
this case.

I learned now that "the category [C^op,Set] is the free colimit
completion of C". My question is, if there are similar results for the
Kleisli category of a monad on [C^op,Set]?

Best regards

Uwe


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