nLab monad retromorphism

Idea

A retromorphism of monads is a retrocell between the carriers of the monads, which preserves the unit and multiplication.

Monads in double categories

Recall that a monad $(A, t, \eta, \mu)$ in a double category consists of a loose morphism $t \colon A \nrightarrow A$ and cells called the unit and multiplication, respectively, subject to the following axioms.

1. Left unitality: $\mu \circ (\eta \odot 1_{t}) = 1_t$
2. Right unitality: $\mu \circ (1_{t} \odot \eta) = 1_{t}$
3. Associativity: $\mu \circ (\mu \odot 1_{t}) = \mu \circ (1_{t} \odot \mu)$

Definition

Let $\mathbb{D}$ be a double category equipped with a choice of companions denoted $(-)_{\ast}$.

Definition

A monad retromorphism $(f, \varphi) \colon (A, t) \nrightarrow (B, s)$ consists of a tight morphism $f \colon A \to B$ and a cell in $\mathbb{D}$ subject to the following axioms.

1. Respects units: $\varphi \circ (1_{f_{\ast}} \odot \eta_{s}) = \eta_{t} \odot 1_{f_{\ast}}$
2. Respects multiplication: $(\mu_{t} \odot 1_{f_{\ast}}) \circ (1_{t} \odot \varphi) \circ (\varphi \odot 1_{s}) = \varphi \circ (1_{f_{\ast}} \odot \mu_{s})$

Note that the cell $\varphi$ in the definition corresponds to a retrocell in $\mathbb{D}$ of the following shape.

Properties

Proposition

A monad retromorphism $(f, \varphi) \colon A \nrightarrow B$ induces a module $(A, t) \nrightarrow (B, s)$ consisting of the loose morphism $t \odot f_{\ast} \colon A \nrightarrow B$ and the cells

determining left action and right action, respectively.

Examples

• A monad internal to the double category $\mathbb{S}\mathrm{pan}$ is a small category. A monad retromorphism is then a retrofunctor. That each retrofunctor determines a profunctor is a corollary of Proposition .

• A monad internal the double category of $\mathcal{V}$-matrices for a distributive monoidal category $(\mathcal{V}, \otimes, I)$ is an enriched category. A monad retromorphism is then an enriched retrofunctor, and each enriched retrofunctor determines an enriched profunctor.

References

The terminology is introduced in the following thesis, where the definition is alluded to:

• Matthew Di Meglio, The category of asymmetric lenses and its proxy pullbacks. Master’s thesis. Macquarie University, 2021. doi: 10.25949/20236449, pdf

A definition is introduced in:

• Bryce Clarke, The double category of lenses, PhD thesis, Macquarie University, 2022. (doi:10.25949/22045073.v1)

Monad retromorphisms are further studied in:

• Robert Paré, Retrocells, arXiv:2306.06436