Lax Monoidal Functor Terminology

Lax Monoidal Functor Terminology

[Ross Tate]

Hello,

I've identified some useful non-standard properties of lax monoidal functors
on Cartesian-like monoidal categories, and I am curious as to whether these
properties already have names.

Suppose I have a monoidal category with diagonals (diag : A -> A * A). Some
lax monoidal functors (F, merge: FA * FB -> F(A * B)) have the property that
diag;merge : FA -> F(A * A) equals F(diag). Is there a name for this
property for either the special Cartesian case or the general case? What
about the 2-categorical case where there's a 2-cell from diag;merge to
F(diag) or vice-versa?

Now suppose I have a monoidal category with terminators (term : A -> I).
Some lax monoidal functors (F, unit: I -> FI) have the property that
term;unit : FA -> FI equals F(term). Again, is there a name for this for the
Cartesian, general, or 2-categorical cases?

While I'm at it, is there a name for Cartesian-like monoidal categories with
diagonals, terminators, and projections? The best I've found is Cartesian
structures on bicategories, but my monoidal categories do not have the
required 2-cells/identities. At best, when there are 2-cells, the 2-cells
are not isomorphisms, which I guess makes it a lax Cartesian structure.

Thanks for any help you can provide,
Ross

P.S. This is my first time using this mailing list, so please tell me if
there's anything I should know about.


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