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Stream: learning: reading & references

Topic: Monoidal Closure of tensor in Poly

Ian Price (Oct 24 2026 at 17:59):

Hello,

In the poly book https://toposinstitute.github.io/poly/poly-book.pdf it is proven in section 4.5 that there is an adjoint to the parallel/dirichlet product in the category of polynomial functors. I was wondering where this was first proven.

I can see that it is in Spivak's "functorial aggregation" paper as Proposition 2.1.12 paper. The first draft of that is from 2021. https://arxiv.org/abs/2111.10968v1 However, it's not clear if that's the origin.

Jacques Carette (Oct 24 2026 at 18:45):

I would not be surprised if the core of that proof was already in Joyal's work on Species, and the case of Poly is a specialization of that.

John Baez (Oct 24 2026 at 20:49):

I don't know about Poly, but for the category of species, which are presheaves on the groupoid of finite sets, the Dirichlet product is obtained via [[Day convolution]] from a monoidal structure on the groupoid of finite sets. Thanks to how Day convolution works, the Dirichlet product thus preserves colimits in each argument, and it follows (via an adjoint functor theorem) that it has an adjoint in each argument.

Ian Price (Oct 25 2026 at 11:58):

Thanks Jacques and John,
I hadn't thought to explicitly look in the species literature. The story using Day convolution makes a lot of sense.

Orestis Melkonian (Oct 26 2026 at 11:16):

We call tensor the "Hancock" product in Scotland, so I felt obliged to point you to Hyvernat-Setzer-Hancock's work on interactions structures (a.k.a. indexed containers, a.k.a. dependent polynomial functors). So the Poly case is precisely the special case where the index set I = 1.

• I learned it from Programming interfaces and basic topology, 2006, where the actual proof is left as an exercise to the reader, although IMHO if you have the definition you already have the proof :)
• You can actually find the proof in Hyvernat's thesis, §3.4, 2005, which is probably the earliest explicit origin, and phrased in the language of containers/polynomial-functors.
• The cartesian case is tackled in Higher-order Containers, 2010 by Altenkirch-Levy-Staton.

Chaitanya Leena Subramaniam (Nov 03 2026 at 13:24):

Apologies in advance for rehashing this. I'm not yet convinced that the category $\mathrm{Poly}$ in question is closed for the symmetric monoidal Dirichlet product. If it is true, then the argument involves something subtle that I'm not seeing. It doesn't seem to follow immediately from the closure of the Dirichlet product on species.

Consider the composite $k=ji$ of the bijective-on-objects inclusions $\mathbb{N} \to \mathrm{core(Fin)}\to \mathrm{Fin}$, where $\mathbb{N}$ is the set of natural numbers, $\mathrm{Fin}$ is a skeleton of the category of finite sets and functions, and $\mathrm{core(Fin)}$ is its groupoid core.

By left Kan extension, we have a composite $k_! = j_! i_! : \mathrm{Set}^{\mathbb{N}} \to \mathrm{Species} \to \mathrm{Set^{Fin}}$, where the third category is just the category of finitary endofunctors on Set. Then the category $\mathrm{Poly}$ is the full image of $k_!$. (I understand that people consider the category $\mathrm{POLY}$ of polynomial functors of _all_ arities---and not just finitary ones---but that doesn't change what follows.)

Both $i$ and $j$ are monoidal for $(\mathbb{N},\times)$, $(\mathrm{core(Fin)},\times)$ and $(\mathrm{Fin},\times)$. Hence so are $i_!$ and $j_!$ (for Day convolution). This shows that the Dirichlet product in $\mathrm{Poly}$ is the image of the Dirichlet product of polynomial endofunctors qua species.

By presentability and bi-cocontinuity of Day convolution, $\mathrm{Set^{Fin}}$ is closed symmetric monoidal for the "Dirichlet" product $\otimes$ of finitary endofunctors on Set. If we denote the adjoint to $(-\otimes F)$ by $(F\multimap-)$, then we need to show that $k_!P\multimap k_!Q$ is in the image of $k_!$.

But I can't see why this is the case. We have the formula $k_! R = \Sigma_n \left(R_n \times \mathrm{Fin}(n,-)\right)$. I can't reduce $k_!P\multimap k_!Q$ to this form. The formula provided in Proposition 2.1.12 of https://arxiv.org/abs/2111.10968v1 claims that this should hold, at least when $P$ has finitely many operations. I'd be grateful if someone could explain this formula to me, hopefully in a conceptual way :slight_smile:. Maybe I need to bash my head against it some more.