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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). 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??

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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 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.)

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  • $\begingroup$ The fact that every algebra has a canonical presentation by free ones is usually presented as part of the proof of the adjoint functor theorem, e.g. section VI.7 of CWM. $\endgroup$
    – Mike Shulman
    Commented Jun 20, 2010 at 16:40
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    $\begingroup$ I've seen this characterization of algebras (as presheaves on the kleisli category) attributed (here: pps.jussieu.fr/~mellies/papers/segal-lics-2010.pdf) to Linton, "Relative functorial semantics: adjointness results", Lecture Notes in Mathematics, vol. 99, 1969. $\endgroup$
    – Noam Zeilberger
    Commented Jun 20, 2010 at 20:43
  • $\begingroup$ @M.S.: (I presume you mean Beck's theorem.) Yes, thanks, that's where I'd seen this before. @N.Z.: Street doesn't attribute this result directly to Linton, but now that I look closely I find he calls it (in the introduction) 'the usual interpretation' of algebras. $\endgroup$
    – Finn Lawler
    Commented Jun 20, 2010 at 21:32
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    $\begingroup$ The pullback characterisation is due to Linton, and appears as Observation 1.1 of the 1969 paper An Outline of Functorial Semantics. $\endgroup$
    – varkor
    Commented Nov 28, 2021 at 18:44

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