Kleisli categories for monads on presheaves

Kleisli categories for monads on presheaves

[Uwe Egbert Wolter]

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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Re: Kleisli categories for monads on presheaves

[Tom Hirschowitz]

Hi Uwe,

Not quite what you're asking for, but not too far either is the fact
that for any monad T on ℂ, Kl(T) arises when factoring ℂ → T-Alg as
identity-on-objects / fully faithful:

ℂ → Kl(T) → T-alg.

This is used in the literature on abstract nerve theorems, see, e.g.,
Familial 2-functors and parametric right adjoints by Mark Weber, or
Polynomial functors and trees by Joachim Kock. This may be relevant
to your ideas about presenting monads on graphs.

The following papers might also be relevant.

  - Albert Burroni. Algèbres graphiques. Diagrammes, tome 7 (1982).

  - Dubuc and Kelly. A Presentation of Topoi as Algebraic Relative to
    Categories or Graphs. J. of Algebra 81 (1983).

  - Kelly and Power. Adjunctions whose counits are coequalizers, and
    presentations of finitary enriched monads. JPAA 89 (1993). 

Best wishes,
Tom


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