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Stream: learning: questions

Topic: a construction on commutative monoids

Jules Hedges (Jul 30 2026 at 13:26):

This is arguably an abstract algebra question but pretty sure the answer will involve enough category theory to make it on topic. Write $F : \mathbf{Set} \to \mathbf{CMon}$ for the free functor, which takes a set to the set of finite multisets on it. (Commutative monoids is the case I'm interested in but I think the question works for any class of algebras for which finite direct sums are a biproduct)

I have a commutative monoid $A$, a set $B$, a monoid homomorphism $f : A \to F (B)$ and a $B$-indexed family of commutative monoids $C_b$. I'm interested in the set of pairs
$\displaystyle \left\{ (a, c) \mid a \in A, c \in \bigoplus_{b \in f (a)} C_b \right\}$
I'm pretty sure that this set can easily be given the structure of a commutative monoid. I wonder what this construction should be called, and what universal property it has

Jules Hedges (Jul 30 2026 at 13:33):

For example, the neutral element is $(0, *)$ where $0 \in A$, and then $f (0) = \varnothing$ and so $\bigoplus_{b \in \varnothing} C_b = \{ * \}$

El Mehdi Cherradi (Jul 30 2026 at 14:56):

At first glance, this looks like the pullback along $f$ of the Grothendieck construction $G(C_b) \to F(B)$ associated to the family $F(B) \to \mathbf{CMon} \subset \mathbf{Cat}$ obtained by left Kan extension along the unit of free-forgetful adjunction from the original family $B \to \mathbf{CMon}$.

edit: Not too sure about what the extension should be, but it should be somewhat close to that?

Todd Trimble (Jul 30 2026 at 19:31):

When you write $b \in f(a)$, are you taking multiplicities into account, i.e., treating $f(a)$ as a multi-set?

Jules Hedges (Jul 31 2026 at 11:02):

I was taking multiplicites into account yes, but if it turns out that it only works if you replace $M$ with finite powerset (aka free commutative idempotent monoid monad) then I won't be at all disappointed

Jules Hedges (Jul 31 2026 at 14:54):

Right, the hard part of this is turning a set-indexed family of monoids into a "fibration"/"bundle" of monoids. In the category of sets this is the fact that $\mathbf{Set}^X \cong \mathbf{Set} / X$. Wouldn't it be nice if it was the case that (where $X$ remains a set) $\mathbf{CMon}^X \cong \mathbf{CMon} / FX$ ? Could this be true? Is it true if you replace $F$ with finite powerset (ie ignore multiplicites)?

James Deikun (Jul 31 2026 at 15:28):

I think this is a special case of the monoidal Grothendieck construction (for monoidal fibrations) described in Framed bicategories and monoidal fibrations ...