Suppose I know what the category of free algebras for a particular monad look like. Can I then describe what the category of Eilenberg–Moore algebras look like? E.g. Suppose that I have a good handle on a category $D$ and that I have two functors $F$ and $G$, with $G:D \to C$ and $F$ left-adjoint to $G$ and $F$ essentially surjective- this guarantees that this adjunction exhibits $D$ as the Kleisli-category for the monad $T:=GF$ on $C$ (see [Characterization of Kleisli adjunctions](https://mathoverflow.net/questions/26075/characterization-of-kleisli-adjunctions)). Is there a way to exhibit the Eilenberg–Moore category $C^T$ as "generalized objects of $D$"? This is somewhat vague of a question I realize, but I'm not sure how to make it more precise. (Feel free to help me do so).


EDIT: To be slightly more specific, in "Toposes, Triples, and Theories" it is said that the Eilenberg-Moore category "is in efect all the possible quotients of objects in Kleisli's category". Can anyone make this precise in a way that answers my question??



Eilenberg–Moore algebras in terms of Kleisli ones
One nice result is Street's theorem 14 in *The formal theory of monads*, generalized in *Elementary cosmoi*, which says that $C^T$ is isomorphic to the full subcategory of $[(C\_T)^{\mathrm{op}}, \mathrm{Set}]$ containing those presheaves that become representable when precomposed with the inclusion $C \to C\_T$. That is, $C^T$ is the pullback of $[F^{\mathrm{op}},\mathrm{Set}]$ along the Yoneda embedding. So at least informally you can think of T-algebras as being represented by certain [formal colimits](http://ncatlab.org/nlab/show/free+cocompletion) of free algebras.


I think it's standard, too, that every algebra has a canonical presentation -- its (2-truncated) bar resolution -- as the coequalizer of maps between free objects. Can't think of a good reference at the moment, but Mac Lane has a section on the bar construction, and there's some good stuff on nlab. Maybe try Kelly & Power, *Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads*, JPAA 89 (1993).


I don't know if there's a relationship between these two ideas, but I'd be very interested to find out! (Can't do it myself, I'm supposed to be on holiday.)


