Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
created category of operators
fixed (by removing) the nonsensical clause that Mike pointed out in a query box (i had been thinking of something else)
Hey guys, regarding this page, you point out that the category of operators construction is left 2-adjoint to the endomorphism operad (underlying multicategory) construction. This seems to be shown in theorem 7.2 of Hermida’s paper “Representable Multicategories.” However, the category of operators is clearly NOT the free monoidal category on a multicategory (right?) It is the free semi-cartesian monoidal category on the free semi-cartesian multicategory of that multicategory. But shouldn’t the “underlying multicategory” functor from monoidal categories to multicategories be adjoint to the free monoidal category functor? Or am I not understanding something?
Note, this is not pointing out an error in the page, but pointing out an error in my thinking. Just wondering if anyone can help.
By composing adjunctions, the free semi-cartesian monoidal category on the free semi-cartesian multicategory of a multicategory is left adjoint to the underlying multicategory of the underlying semi-cartesian multicategory of a semi-cartesian monoidal category. The latter is equivalently the underlying multicategory of (the underlying monoidal category of) a semi-cartesian monoidal category. Apparently the underlying multicategory of a monoidal category is sometimes (mis)called the “endomorphism operad”, so in that sense the category of operators is left adjoint to it, but only when the “underlying multicategory” functor has as its domain the category of semi-cartesian monoidal categories. This ought to be clarified on our pages. I have not checked what Hermida wrote, but I strongly suspect that he was talking not about the category of operators but about the prop generated by a multicategory, which doesn’t incorporate semicartesianness.
Ok, great, this makes a lot of sense. I was also thinking that maybe Hermida was talking about the PROP rather than the category of operators, in which case an edit should be made on category of operators. Hermida’s presentation of monoidal categories (as category objects in monoids) was a little opaque to me, so I wasn’t totally sure whether it was the PROP of the COO.
I made an edit.
1 to 6 of 6