A cocomma square from the Kleisli category of a monad

Question

$\require{AMScd}$Let $T : X\to X$ be a monad on a category $X$; define the category $X §_T X$ as follows: the objects are $X_0 + X_0'$, i.e. two disjoint copies of the objects of $X$, and the morphisms are specified as follows:

• $X §_T X(x,y)=X(x,y)$ for $x,y$ both in the first summand;
• $X §_T X(x',y')=X(x',y')$ for $x',y'$ both in the second summand;
• $X §_T X(x,y') = Kl(T)(x,y')=X(x, Ty')$;
• $X §_T X(x',y) = \varnothing$.

The definition is an unwinding of the following cocomma construction: the square $$\begin{CD} X @= X \\ @VTVV \Leftarrow @VVin_1V \\ X @>>in_2> X §_T X \end{CD}$$ is a cocomma, filled by an evident choice of natural transformation $in_1\Rightarrow in_2 T$.

Does this category have a name? Does it have a particular significance as an object that can be attached to $T$?

Answer 1

The construction you describe is precisely the collage of the representable distributor (i.e. profunctor) $X(1, T) \colon X \not\to X$, defined as $(x', x) \mapsto X(x', Tx)$. Abstractly, the collage is the lax colimit of the distributor, viewed as a 1-cell in the bicategory $\mathbf{Dist}$ of distributors.

However, there is a more natural colimit construction associated to this distributor. In particular, since $T$ is a monad, the distributor $X(1, T)$ is a monad in the bicategory $\mathbf{Dist}$, and so it is more natural to take a colimit that respects this monad structure: this is precisely the notion of Kleisli object in a bicategory. Every monad $(P, \mu, \eta)$ in $\mathbf{Dist}$ has a Kleisli object: the underlying category has hom-sets given by $P$, with identities and composition given by the monad structure. The Kleisli inclusion is always representable, and is given by the identity-on-objects functor sending $f \colon x' \to x$ to $\eta f \in P(x', x)$. In particular, the Kleisli object of the representable distributor $X(1, T)$ is precisely the Kleisli category of $T$. That is: the Kleisli object of a monad $T$ in $\mathbf{Cat}$ is preserved by taking representable distributors.

Finally, note that, by the universal property of the collage, for any monad $P$ in $\mathbf{Dist}$, there is a canonical comparison distributor $\mathbf{Kl}(P) \not\to \mathbf{Coll}(P)$, whose action is given by $P$, which explains why the collage construction you descirbe appears to "contain" the Kleisli category.

In summary, the construction you describe is close to being a natural construction on $T$, except that we ought also to take the monad structure into account when we take the colimit, which is precisely what the Kleisli object does.