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    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeNov 16th 2018

    Added a link to Todd’s nice page on free cartesian category.

    diff, v25, current

  1. Add cocartesian monoidal category as a related concept

    Alexey Muranov

    diff, v27, current

    • CommentRowNumber3.
    • CommentAuthormattecapu
    • CommentTimeDec 8th 2022

    Stub about the free and cofree construction of cartesian categories

    diff, v34, current

    • CommentRowNumber4.
    • CommentAuthormattecapu
    • CommentTimeDec 8th 2022
    Fox's paper doesn't really explain what is the free construction of a cartesian structure over a monoidal one. Does anybody know a reference, or a simple explanation? My hunch is you freely add projections to your monoidal category, imposing naturality of the induced copy-delete structure.
    • CommentRowNumber5.
    • CommentAuthorTodd_Trimble
    • CommentTimeDec 28th 2022

    (Only seeing this now.)

    That sounds right, mattecapu. But I think a nice way of viewing it is by using distributive laws between spans. The rough idea is that morphisms in the free cartesian category can be presented in a “standard” form, as a “purely cartesian morphism” cc (a morphism in the free cartesian monoidal category on the underlying set of objects) followed by a morphism mm in the monoidal category. I’ll denote this by c|mc|m. When it comes time to compose two such morphisms, say c|mc|m followed by d|nd|n, you rewrite m;dm; d into a new form d;md' ; m' by using naturality of copy and delete, and then the composite will be c;d|m;nc; d|m' ; n. The distributive law is this rewriting process.

    Slightly more formally, I think it goes like this. Suppose given a monoidal category M\mathbf{M} whose underlying span is

    SdomMcodS.S \stackrel{dom}{\leftarrow} M \stackrel{cod}{\to} S.

    I think technically it might be easier dealing with strict monoidal categories here, so I’ll just assume that for now. By strictness, the monoidal structure on the object level gives a map S *SS^\ast \to S where the domain is the free monoid on SS. Now, the free cartesian monoidal category C\mathbf{C} on (the discrete category) SS has as its objects elements of S *S^\ast, and so its underlying span will look like

    S *srcCtarS *,S^\ast \stackrel{src}{\leftarrow} C \stackrel{tar}{\to} S^\ast,

    and now I think the underlying span of the free cartesian monoidal category on the monoidal category M\mathbf{M} is the span composite of

    SS *srcCtarS *SS \leftarrow S^\ast \stackrel{src}{\leftarrow} C \stackrel{tar}{\to} S^\ast \to S

    followed by SdomMcodSS \stackrel{dom}{\leftarrow} M \stackrel{cod}{\to} S. Abusing notation, let MC\mathbf{M} \circ \mathbf{C} denote this span composite.

    The distributive law will be a morphism of spans of the form CMMC\mathbf{C} \circ \mathbf{M} \to \mathbf{M} \circ \mathbf{C}. I’ll only give an example for how this goes. Suppose m:stm: s \to t is a morphism in M\mathbf{M}, and let δ t:ttt\delta_t: t \to t \otimes t be the copy morphism. Then m;δ tm; \delta_t is formally rewritten as δ s;(mm)\delta_s; (m \otimes m). Hopefully this makes the general idea clear.

    • CommentRowNumber6.
    • CommentAuthormattecapu
    • CommentTimeDec 28th 2022
    Thanks Todd, I really like this way of describing it!
    • CommentRowNumber7.
    • CommentAuthorJohn Baez
    • CommentTimeAug 23rd 2023

    Added a bit about how the comonoid structure of objects in a cartesian monoidal category is cocommutative and natural.

    diff, v36, current