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Stream: community: our work

Topic: Ea E Thompson

Ea E T (they/she) (May 24 2026 at 18:24):

@Kevin Carlson and I just posted our paper on characterizations of exponentiable virtual double categories last week: https://arxiv.org/abs/2605.20586

This work arose late last summer out of a desire to understand higher morphisms and enrichments for virtual double categories (VDCs). We initially discovered a characterization of exponentiable VDCs, and morphisms of such more generally, in terms of a Conduche-like condition for decompositions of multicells, as described back in this talk in the Topos Institute Berkeley Seminar. This characterization led to simple constructions of a number of families of exponentiable VDCs, including those representable by pseudo-double categories and those arising as cospans in arbitrary categories.

Since this initial characterization we've expanded our classification to address a conjecture in a recent paper of Arkor's (https://arxiv.org/abs/2508.11611v2), while also providing a characterization in terms of decompositions involving only binary, unary, and nullary multicells. This most recent characterization provides an effective framework for testing and constructing exponentiable VDCs.

At the end of the paper we dig deeper into this theory of exponentiable VDCs to begin understanding when the resulting exponentials are representable by pseudo-double categories, or more generally when they have weak composites, using a similar treatment to that of Paré in the case of pseudo-double categories.

As an example, these characterizations allowed us to show that for any VDC $\mathbb{D}$, we have a yoneda type embedding $\mathbb{D}\hookrightarrow \mathbb{S}\mathsf{pan}^{\mathsf{fc}(\mathbb{D})^{op_t}}$ into a locally cocomplete VDC with non-nullary composites, where $\mathsf{fc}(\mathbb{D})$ is the free strict double category on the VDC $\mathbb{D}$. Additionally, if $\mathbb{D}$ has loose units, then a choice of such provides an embedding into a locally cocomplete pseudo-double category $\mathbb{D}\hookrightarrow \mathbb{V}\mathsf{df}(\mathsf{fc}(\mathbb{D})^{op_t},\mathbb{S}\mathsf{pan})$. In particular, we can freely add loose units to an arbitrary VDC $\mathbb{D}$, giving us the ability to embed arbitrary VDCs into locally cocomplete pseudo-double categories $\mathbb{D}\hookrightarrow\mathbb{F}_u(\mathbb{D})\hookrightarrow \mathbb{V}\mathsf{df}(\mathsf{fc}(\mathbb{F}_u(\mathbb{D}))^{op_t},\mathbb{S}\mathsf{pan})$. We're hoping to investigate the universal property of such embeddings in future work.

James Deikun (May 26 2026 at 01:59):

Found a few typos:

• Corollary 3.5: wrong parentheses and arrow in the introduction of $\lambda$.
• Definition 3.6: "first for" -> "first four".

James Deikun (May 26 2026 at 02:06):

It seems like the techniques can be extended to algebras of other accessible monads on presheaf categories, as they are based on limit sketches. I'm not sure how much the nice theorem on limit sketches would be mangled by pulling it back across the interpretation of accessible monads on presheaf categories as limit sketches though.

James Deikun (May 26 2026 at 02:08):

(In particular, it seems like Burroni/Leinster-style $T$-categories on p.r.a. monads $T$ probably could be treated pretty uniformly, though, as the Leinster "plus" construction heavily constrains its resulting monad.

Ea E T (they/she) (May 27 2026 at 16:47):

James Deikun said:

(In particular, it seems like Burroni/Leinster-style $T$-categories on p.r.a. monads $T$ probably could be treated pretty uniformly, though, as the Leinster "plus" construction heavily constrains its resulting monad.

Yes! The main challenging step in this process is understanding what the colimits in models look like. In the case of $T$-multicategories one can use that the theory category is fibered over the simplex category to perform a similar argument to that in our paper to construct an explicit pre-sheaf model for the colimit using left Kan extensions of pre-sheaves. We originally didn't pursue this since we were told that Lack and Fujii were currently working on the $T$-multicategory setting, and our applications of interest were primarily in the case of VDCs.

However, this approach was used in the setting of quasi-categories to give precisely this kind of characterization in the recent pre-print https://arxiv.org/abs/2603.29815v2 (although binary-nullary reductions aren't considered there). We are hoping to perform a similar characterization for certain special classes of limit sketches, such as those for $T$-algebras when $T$ is a p.r.a monad on a pre-sheaf category, since the natural embedding of $T$-algebras in $T$-multicategories does not reflect exponentiable morphisms.

James Deikun (May 27 2026 at 19:58):

In fact it makes all the morphisms exponentiable! Can't really complain about it, since it's nice having a small embedding that adds exponentials to all the existing morphisms!