nLab Mealy morphism

Idea

When we define internal categories of some category $\mathcal{C}$ as monads in the category of spans $\mathrm{Span}(\mathcal{C})$, there is a subtlety concerning the definition of internal functors. It is not the case that the morphisms of monads (namely colax monad morphisms) give us internal functors directly: we need to ask that the 1-cell of the morphism is a left adjoint in $\mathrm{Span}(\mathcal{C})$ (which corresponds to asking that the left leg of the span be the identity, or an isomorphism). (This is related to the behaviour of profunctors in that they correspond to functors (via the Yoneda embedding) exactly when they are left adjoints.) If in considering morphisms of monads in $Span(\mathcal{C})$ we don’t require this left-adjoint condition then we end up constructing Mealy morphisms between internal categories.

‘Mealy morphisms’ are named after Mealy machines, which, in turn, were named after George H. Mealy.

Definition

Let $\mathbf{A}$ and $\mathbf{B}$ be categories enriched in a monoidal category $(\mathbf{V}, I, \otimes)$.

A Mealy morphism $(F, \phi) \colon \mathbf{A} \rightarrow \mathbf{B}$ consists of a pair of functions $F \colon Ob\mathbf{A} \rightarrow Ob\mathbf{B}$ and $\phi \colon \mathbf{A} \rightarrow \mathbf{V}$, and for each pair of objects $A, A'$ in $\mathbf{A}$ a function

$$\phi(A, A') \colon \mathbf{A}(A, A') \otimes \phi(A) \longrightarrow \phi(A') \otimes \mathbf{B}(FA, FA')$$

satisfying the following commutative diagrams.

Properties

• There is a bicategory of small $V$-enriched categories, Mealy morphisms, and Mealy cells. It is the Kleisli bicategory for the free cocompletion of a small $V$-category under copowers. This is analogous to the universal property of the bicategory Prof of profunctors as the Kleisli bicategory for the free cocompletion of a small category under small colimits. This is proven in Paré 2012.

References

The notion of Mealy morphism between enriched categories was introduced in the paper:

• Robert Paré, Mealy Morphisms of Enriched Categories, Applied Categorical Structures 20 (2012) pp 251–273, doi:10.1007/s10485-010-9238-8.

Mealy morphisms are mentioned in Example 16.8 and Example 16.27 in the paper:

• Richard Garner, Michael Shulman, Enriched categories as a free cocompletion, Advances in Mathematics, 289, 2016 (arXiv:1301.3191, doi:10.1016/j.aim.2015.11.012)

Mealy morphisms are also considered under the name “two-dimensional partial map” (a notion attributed to Lawvere) in the Appendix of the paper:

• Stephen Lack, Ross Street, The formal theory of monads II, Journal of Pure and Applied Algebra, 175, 2002 ( doi:10.1016/S0022-4049(02)00137-8)

Some talk slides which mention Mealy morphisms:

• Robert Paré, Mealy Morphisms for Tensor Categories