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6

Day convolution is a categorification of the monoid algebra construction. There is a formal analogy between the two, but one is not a literal generalisation of the other. So to address your question 3, we should not expect to recover the usual convolution from Day convolution. Let's develop the following analogy: \begin{array}{|c|c|} \hline \textbf{monoid ...


1

I don't know if this can be of help, but what I found really useful to gain an intuition behind Day convolution is the correspondence between convolution products and promonoidal structures; the two things can be identified, as every convolution arises from a single promonoidal structure (this dates back to the work of Day himself, and I stated the result in ...


2

Hmm, a logical view of presheaves is as categorified predicates. If we choose the source category as discrete, then we can interpret the coend formula as a categorification of an existential quantification (if our presheaves only return {} or {$*$} then it will be exactly existential quantification.) A discrete monoidal category is a monoid. The coend ...


3

The functor $-\otimes n$ (or $-\otimes n$) has no right (or left) adjoint for any $n>0$. Indeed, every morphism in the braid category is and endomorphism, so there exists a map $i\otimes n\to j$ iff $i+n=j$. So if an adjoint existed, $[n,j]$ would have to satisfy $[n,j]=i$ iff $i+n=j$, but this is impossible for any $j<n$.


1

Your first construction is a category of sets a set-monoid acts on. Its objects are called monoid actions, and are exactly what you wrote down - either sets equipped with a nice operation $A × M → M$, or a way to interpret elements of $A$ as transformations of $M$ (ie. a nice function $A → \mathrm{End}(M))$. These things certainly don't come automatically ...


9

Here's a simpler example: in ring theory, picking a basis of your ring (as a module over the ground commutative ring $k$) is extra structure. But in category theory there is sometimes a "distinguished basis" (e.g. the simple objects in an abelian category), which will pass to just a basis after decategorifying. For example, Hecke algebras have a famous ...


2

You're asking two basically independent questions here, so they really should be separated, but yes, taking duals is monoidal. The argument generalizes to the observation that taking adjoints of 1-morphisms in 2-categories respects composition. As for braidings, one way to define a braided monoidal category is as a "monoidal monoidal category" (or "$E_2$ ...


1

You're right about the first thing. The easiest way to see his is to. It's the monoidal category as a bicategorywith a single object. Then a dual is just an adjoint, and it's well known that adjoint said are functorial in this way. On the other hand, to call the braiding a monoidal natural transformation would be the wrong approach. Try to write out what ...



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