# Is there a “properly categorical” description of Eilenberg-Moore algebras on a relative monad?

A relative monad is defined in Altenkirch et al. essentially as a pair of functors $J,F:\mathcal{C}\to\mathcal{D}$, a natural transformation $\eta:J\to F$, and an operation $(-)^{*}:\mathcal{D}(J-,F-)\to\mathcal{D}(F-,F-)$ with conditions that give a composition rule $\mathcal{D}(JB,FC)\times\mathcal{D}(JA,FB)\to\mathcal{D}(JA,FC)$ in which the components of $\eta$ are the identities. (I'm paraphrasing a bit; the paper doesn't start with the assumption that $F$ is functorial or $\eta,(-)^{*}$ natural, but derives these as theorems.) That is, the correct notion of the Kleisli category of a relative monad seems to be what the definition of a relative monad was tailor-made to give.

An Eilenberg-Moore algebra on a relative monad consists of an object $X\in|\mathcal{D}|$ and an operation $\chi:\mathcal{D}(J-,X)\to\mathcal{D}(F-,X)$ such that for any $f:JA\to X$ $f=\chi(f)\circ\eta_{A}$ and for any $g:JB\to FA$ we have $\chi(\chi(f)\circ g)=\chi(f)\circ g^{*}$. As with the definition of a relative monad itself, the authors of the paper proposing this definition don't phrase what's happening in what I would think of as "categorical terms", and aren't really interested in delving into the high level consequences of this definition, so I'm wondering if there's a more conceptual (i.e. friendly to extremely abstract categorical types) way of phrasing what's happening with this version of Eilenberg-Moore algebras. I would like to set out criteria for a good answer here, but I don't know how to do this without reference to the very vague notion of what a "properly categorical" description of something is. Perhaps the following paragraph of my thinking on the topic can suggest the flavor I'm looking for.

I'm aware that the definition as given in the cited paper gives one that an algebra is at least partially a cone $\chi$ under $F\circ Q:(J\downarrow X)\to\mathcal{C}\to\mathcal{D}$ (where $Q$ is the obvious forgetful functor) such that $\chi\circ\eta Q$ is equal to the "canonical cone" under $J\circ Q$ with vertex $X$, which is likewise the case in an ordinary monad where $J=id_{\mathcal{C}}$. The $\chi(\chi(f)\circ g)=\chi(f)\circ g^{*}$ condition must say something more than this, though, as I see no way of pulling this back out from only the description in terms of cones just given. If relative monads always arose from a (lax) monoidal structure on $\mathcal{D^C}$ I would look for the extra structure there, but the conditions for this given in Altenkirch et al. don't hold generally enough (and in particular don't hold in the cases I'm studying).

• It seems to me that a good start would be to get a more "properly categorical" description of relative monads themselves. You say that sometimes a relative monad is a monoid in a lax monoidal category; after a glance at the paper, I think the reason this fails is because certain Kan extensions don't exist. There's an answer for this problem -- when monoidal products fail to exist for "representability reasons", you can probably replace the monoidal category you don't quite have with a multicategory. So I'll guess that relative monads are monoids in a suitable multicategory. – tcamps May 6 '15 at 1:50
• In the formal theory of monads, the Kleisli and Eilenberg-Moore constructions are characterized 2-categorically. If you try to generalize naively, your 1-cell composition won't be globally defined; I'm not sure how to deal with this except to look for the structure of a virtual double category hanging around. As far as I know, the formal theory of monads in virtual double categories hasn't been fully worked out or at least written down. – tcamps May 6 '15 at 2:06
• I'll look into those! I'm generally comfortable with the category-ness of relative monads, since in essence all it says is that the initial splitting of $F$ into a $J$-relative adjunction exists; and relative adjunctions make sense in nice 2-categorical terms. It's when it comes to the EM algebras that I find even the picture of a composite of relative adjoints unhelpful. – Malice Vidrine May 6 '15 at 11:19