nLab Mealy morphism




When we define internal categories of some category 𝒞\mathcal{C} as monads in the category of spans Span(𝒞)\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 Span(𝒞)\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(𝒞)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.


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

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

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

satisfying the following commutative diagrams.



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

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

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

Some talk slides which mention Mealy morphisms:

Last revised on January 10, 2024 at 10:41:46. See the history of this page for a list of all contributions to it.