Re: Yoneda Lemma when there is a monad

[Tom Leinster <Tom.Leinster@glasgow.ac.uk>]

Dear Chris,

> C^T embeds in [(C_T)^op,Set]; i.e. any category of algebras embeds in
> the presheaf category of the Kleisli category.

This part is morally clear, I think, via the notion of density.

A functor F: A --> B is called dense if the induced functor

     Hom(F, -): B --> [A^op, Set]

is full and faithful.  An equivalent condition (stated loosely) is that
every object of B is a colimit of objects of the form F(a) (with a in A)
in a canonical way.  (See e.g. Categories for the Working Mathematician.)

Now take a monad T on a category C.  Every T-algebra is canonically a
coequalizer of free T-algebras.  So every object of C^T is canonically a
colimit of objects of the subcategory C_T.

As long as "canonically" means what I assume it does (and I haven't
checked the details), this tells us that the inclusion C_T --> C^T is
dense.  Hence the induced functor C^T ---> [(C_T)^op, Set] is full and
faithful - which I guess is what you mean by an embedding.

All the best,
Tom


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