Every monad is induced canonically by two universal adjunctions, introduced respectively by Kleisli and by Eilenberg and Moore. Since neither paper introduced names for the corresponding categories, they came to be known as the Kleisli category and the Eilenberg–Moore category.
There are various reasons why it is preferable to prefer descriptive names for mathematical concepts than eponymic names (which is not to say that there are not points in favour of either, but for the purpose of this question, we make the assumption that it may be desirable to have alternatives to eponyms).
The Eilenberg–Moore category has an established, descriptive synonym: the category of algebras. However, the Kleisli category does not appear to have a similarly widespread synonym. I am aware of two candidates, neither of which is particularly satisfactory:
- The Kleisli category embeds into the Eilenberg–Moore category as the full subcategory of free algebras, and so it is accurate to call it the category of free algebras. However, this suffers from two defects. The first is that the Kleisli presentation (in terms of morphisms $X \to T(Y)$) and the category of free algebras carry different intuitions, and it is helpful to have two different terms to state the non-tautology "The Kleisli category is equivalent to the category of free algebras.". The second is that, in contexts other than ordinary category theory in which there are two universal adjunctions inducing a monad (such as in formal category theory), there may exist a Kleisli object and Eilenberg–Moore object, but for which the canonical comparison map between them is not fully faithful. That is, there are settings in which the two concepts are distinct.
- From a two-dimensional perspective, the Eilenberg–Moore category is an algebra object for a monad, whereas the Kleisli category is an opalgebra object (i.e. an algebra object in $\mathcal K^{\text{op}}$). From this perspective, one might be tempted to call the Kleisli category the opalgebra category or category of opalgebras. However, the objects of the Kleisli category are not opalgebras (since the opalgebra object has a mapping-out property, in contrast to the algebra object). So this name is potentially misleading.
Question. What are some alternative terms for the Kleisli category that are: (A) descriptive; (B) do not carry misleading intuition?
I would be particularly happy to hear examples that already occur in the literature (perhaps in fields like programming language theory, where the Kleisli category occurs commonly). However, I am also willing to accept suggestions that do not yet occur in the literature, when accompanied with some justification for the aptness of the suggested term.
