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Stream: community: our work

Topic: Day convolution for algebraic patterns

Jaco Ruit (Apr 01 2026 at 19:38):

Hi all, 



Thomas Blom, Félix Loubaton, and I posted a paper on the arXiv today about (in particular) exponentiability in the context of virtual double ∞-categories (https://arxiv.org/abs/2603.29815). We wanted to share this with the community on this Zulip as a follow-up to the discussion of @Nathanael Arkor's very nice article in https://categorytheory.zulipchat.com/#narrow/channel/274877-community.3A-our-work/topic/Exponentiable.20virtual.20double.20categories.20and.20presheaves. In the paper, we provide necessary and sufficient criteria for the exponentiability of morphisms in ∞-categories of ∞-operad-like structures including ∞-operads (analogous to Pisani's characterization for ordinary operads), virtual double ∞-categories (similar to the result for ordinary VDCs that was announced by Ea T in the thread about Nathanael’s paper), and iterated versions of these.



--- Thomas, Félix, and Jaco

Matteo Capucci (he/him) (Apr 01 2026 at 19:44):

I just skimmed the intro an it's very interesting! I will read further eventually. For now I find it really cool that the criterion you get it's basically Conduchè's for plain functors in Cat. It's a testament to the great abstraction that algebraic patterns and algebrads (great name) are.

John Onstead (Apr 02 2026 at 22:14):

I also found the paper interesting, as I was one of the participants in the conversation about Nathanael Arkor's article.

I hadn't encountered algebraic patterns or algebrads previously, so this paper was a nice introduction to that framework. That said, I am more familiar with the framework of generalized multicategories (which also include operads and virtual double categories). This made me wonder if there might be some way to "translate" between these perspectives.

John Onstead (Apr 02 2026 at 22:16):

For instance here might be some sort of sketch (of course as a warning, I'm not an expert, so this could be completely wrong!) Construction 6.2 of the paper presents the idea of the "underlying graph" of an algebrad. Given some algebraic pattern $\mathcal{O}$, the category of underlying graphs of algebrads for this algebraic pattern is $[\mathcal{O}^{el}, \mathrm{Cat}]$. Perhaps we could then equip this category with loose/horizontal arrows that encode some sort of tree data for $\mathcal{O}$ on the underlying graphs, forming a virtual double category. Then (some of) the monads in this VDC could be the algebrads for the algebraic pattern.

As an example, using the example 6.14 of the virtual double category algebraic pattern, the category of underlying graphs is just $\mathrm{Graph(Cat)}$. A subcategory of this (on graph objects on discrete categories) is the usual category $\mathrm{Graph}$. Indeed, this is the underlying vertical category of the VDC $\mathrm{HKleisli}(\mathrm{Graph}, fc)$ whose monads are VDCs.

John Onstead (Apr 04 2026 at 02:51):

I had a bit more time to think on this and I've found another interesting thing about algebrads; I might as well write them here for anyone interested (apologies for the lots of text, just a lot of thoughts I wanted to put somewhere!)
It centers on the idea of Segal $\mathcal{O}$-categories for some algebraic pattern $\mathcal{O}$, which fit into the analogy "Segal $\mathcal{O}$-categories are to algebrads as representable multicategories are to generalized multicategories" (Example 2.9). First, let $\Gamma: \mathrm{Algad}(\mathcal{O}) \to [\mathcal{O}^{el}, \mathrm{Cat}]$ be the underlying graph on an algebrad. We can compose this with the (non-full) inclusion $\mathrm{SegalCat}(\mathcal{O}) \to \mathrm{Algad}(\mathcal{O})$ to get an "underlying graph of a segal category" $U: \mathrm{SegalCat}(\mathcal{O}) \to [\mathcal{O}^{el}, \mathrm{Cat}]$. The question to then ask is when $U$ is monadic. As it happens, it's monadic in many of the interesting cases, in which case the monad on the graph category whose algebras are the Segal categories has the algebrads for $\mathcal{O}$ as its "normalized" generalized multicategories.

Here's two examples. First, let $\mathcal{O} = \mathbb{F}_*$. The Segal categories are the symmetric monoidal categories. The composed underlying graph functor is just $U: \mathrm{SymMon} \to \mathrm{Cat}$ the usual forgetful functor. Indeed, this is monadic, for the free symmetric monoidal category monad on Cat. The normalized generalized multicategories for this monad are symmetric multicategories, which are what the paper calls "operads" (as in colored operad), the algebrads for the pattern $\mathbb{F}_*$. Second, let $\mathcal{O} = \Delta^{op, \sharp}$. The Segal categories are the double categories, and the composed underlying graph functor is $U: \mathrm{DoubCat} \to \mathrm{CatGraph}$. This is also monadic, in particular for the fdc (free double category) monad. The normalized fdc-multicategories are the fc-multicategories, that is, the virtual double categories, which are also the algebrads for the pattern $\Delta^{op, \sharp}$.

John Onstead (Apr 04 2026 at 02:51):

Also, as an unrelated thought to the above, I'm quite looking forward to the future work from Section 1.6 on doing formal category theory with algebrads. For instance, I'm especially curious to know how they'll define a "profunctor" between O-algebrads, which would yield the virtual equipment of O-algebrads that would act as the formal category theory setting (unless you can do better than a virtual equipment, IE, a proarrow equipment or even Yoneda structure- that in itself would be interesting to learn about!)

Jaco Ruit (Apr 07 2026 at 08:47):

Dear John,

Thanks for your comments! It would definitely be nice to see a comparison between generalized multicategories and algebrads. We don’t know a lot about how this would be done; but one can find a small discussion already in https://arxiv.org/pdf/2401.04704, Remark 6.5.2 (which seems closely related to the strategy you pointed out).

As a reply to your latest comment: It turns out that profunctors between algebrads for suitable patterns indeed assemble to a normal virtual double $\infty$-category. Suitable means that the functor $p^* : \mathrm{PSh}(\Delta) \to \mathrm{PSh}(\Omega[O])$ induced by the unique map $O \to \ast$ of patterns, preserves complete Segal objects (i.e.\ carries $\infty$-categories to $O$-algebrads). This happens if the elementary slices of $O$ are weakly contractible (for instance, if $O$ is the pattern for VDCs or generalized symmetric operads). A profunctor $X \to Y$ between algebrads can then be defined as a map $E \to p^*[1]$ whose fiber above 0 and 1 are $Y$ and $X$ respectively. We will prove a Yoneda lemma for general $O$, that can be used to show that representable $O$-algebrads admit presheaf objects in the resulting NVDC of $O$-algebrads. This can be applied to a range of examples, in particular, to the case that $O=\Delta^{\mathrm{op},\natural}$, recovering a normal virtual double $\infty$-category of $\infty$-VDC's with presheaf objects for double $\infty$-categories.