Profunctorial algebras

Quentin Aristote

Joint work with: Umberto Tarantino

The aim of this talk is to present a $2$ -dimensional version of Barr’s landmark Relational algebras paper [1]. In this paper, Barr first noticed that, viewing relations $R\subseteq X\times Y$ as spans of their projections $X\leftarrow R\rightarrow Y$ , there is a natural way to extend the action of a functor $F\colon\mathbf{Set}\to\mathbf{Set}$ to $\mathbf{Rel}$ — namely, by having $F$ act on the two projections. Such an extension preserves the natural order of relations; it also preserves their composition precisely when $F$ preserves weak pullbacks, in which case $F$ extends to a 2-functor $\underline{F}\colon\mathbf{Rel}\to\mathbf{Rel}$ . Similarly, a monad structure on $F$ extends to that of a lax monad, and it can be characterized when this extension is strict. Barr’s leading application of this result was to characterize the category $\mathbf{Top}$ of topological spaces as the relational algebras for the ultrafilter monad $\beta$ , that is, the lax algebras of its extension $\underline{\beta}\colon\mathbf{Rel}\to\mathbf{Rel}$ .

We extend both results to the setting of bicategories. Building on [3, 4, 5], relations in a bicategory $\mathsf{K}$ can be identified with two-sided discrete fibrations, which determine a bicategory $\mathbf{TSDFib}(\mathsf{K})$ if $\mathsf{K}$ satisfies some conditions akin to $1$ -dimensional regularity. Our first main result, formulated in terms of exactness à la Guitart [6], reads as follows.

Theorem. A pseudomonad $\langle\mathsf{T},\eta,\mu\rangle$ on a regular bicategory $\mathsf{K}$ extends to a pseudomonad $\langle\underline{\mathsf{T}},\underline{\eta},\underline{\mu}\rangle$ on $\mathbf{TSDFib}(\mathsf{K})$ if and only if:

1.

$\mathsf{T}\colon\mathsf{K}\to\mathsf{K}$ preserves exact squares, and
2.

the naturality squares of its unit $\eta\colon\operatorname{id}\Rightarrow\mathsf{T}$ and multiplication $\mu\colon\mathsf{T}^{2}\Rightarrow\mathsf{T}$ are exact.

We then focus on the case of $\mathbf{CAT}$ , the 2-category of (locally small) categories, so that $\mathbf{TSDFib}(\mathsf{K})$ can be identified with $\mathbf{PROF}$ , the bicategory of categories and small profunctors. In his talk at CT24, Rosolini introduced the ultracompletion pseudomonad on $\mathbf{CAT}$ to study Makkai’s ultracategories [7] as its pseudoalgebras. In this talk, we will make a profunctorial version of ultracategories emerge as algebras for the extension of the ultracompletion pseudomonad to $\mathbf{PROF}$ : the resulting notion will recover ultraconvergence spaces, the categorification of topological spaces recently introduced in [8, 9] to extend Makkai’s Stone duality for first order logic [7] to geometric logic.

[1] M. Barr, Relational algebras, Reports of the Midwest Category Seminar IV. Ed. by S. Mac Lane et al., Springer (1970), 39–55.
[2] Q. Aristote and U. Tarantino, Profunctorial algebras, in preparation, 2026.
[3] A. Carboni, S. Johnson, R. Street, and D. Verity, Modulated bicategories, Journal of Pure and Applied Algebra, vol. 94, no. 3, pp. 229–282, 1994.
[4] R. Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle, vol. 21, no. 2, pp. 111–160, 1980.
[5] F. Loregian and E. Riehl, Categorical notions of fibration, Expositiones Mathematicae, vol. 38, no. 4, pp. 496–514, 2020.
[6] R. Guitart, Relations et carrés exacts, Annales des sciences mathématiques du Québec, vol. 4, no. 2, pp. 103–125, 1980.
[7] M. Makkai, Stone duality for first order logic, Advances in Mathematics 65.2 (1987), 97–170.
[8] G. Saadia, Extending conceptual completeness via virtual ultracategories, preprint arXiv:2506.23935, 2025.
[9] S. van Gool, J. Marquès and U. Tarantino, Toposes with enough points as categories of étale spaces, preprint arXiv:2508.09604, 2025.