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Profunctorial algebras

Quentin Aristote, Umberto Tarantino

TL;DR

This work develops a 2D Barr-style theory by converting relations into two-sided discrete fibrations and extending Set-based monads to profunctorial contexts. An exact-squares criterion governs when a pseudomonad on a base bicategory extends to its TSDF bicategory, enabling broad construction of profunctorial extensions, in particular to $ extbf{PROF}$ from $ extbf{CAT}$. The ultrafilter/ultracompletion case yields the ultracategory framework, and the profunctorial extension $ig\bbbeta$ geometrizes ultraconvergence spaces as normalized lax algebras, with ultracategories appearing as representable instances; this provides a robust algebraic lens on topology and related categorifications. The paper further reveals distributive-law perspectives, shows presheaf stability for ultracategories, and discusses variations using filters, pointing to 2D domain-theoretic and logic applications and future avenues for 2D semantics.

Abstract

We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first characterize, in terms of exact squares, when pseudomonads on a bicategory extend to its bicategory of two-sided discrete fibrations. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories satisfying our criterion and thus extending to profunctors. Among these, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.

Profunctorial algebras

TL;DR

This work develops a 2D Barr-style theory by converting relations into two-sided discrete fibrations and extending Set-based monads to profunctorial contexts. An exact-squares criterion governs when a pseudomonad on a base bicategory extends to its TSDF bicategory, enabling broad construction of profunctorial extensions, in particular to from . The ultrafilter/ultracompletion case yields the ultracategory framework, and the profunctorial extension geometrizes ultraconvergence spaces as normalized lax algebras, with ultracategories appearing as representable instances; this provides a robust algebraic lens on topology and related categorifications. The paper further reveals distributive-law perspectives, shows presheaf stability for ultracategories, and discusses variations using filters, pointing to 2D domain-theoretic and logic applications and future avenues for 2D semantics.

Abstract

We provide a bicategorical generalization of Barr's landmark 1970 paper, in which he describes how to extend Set-monads to relations and uses this to characterize topological spaces as the relational algebras of the ultrafilter monad. With two-sided discrete fibrations playing the role of relations in a bicategory, we first characterize, in terms of exact squares, when pseudomonads on a bicategory extend to its bicategory of two-sided discrete fibrations. As a wide class of examples, we show that every Set-monad induces a pseudomonad on the 2-category of categories satisfying our criterion and thus extending to profunctors. Among these, we then focus on the ultracompletion pseudomonad, whose pseudoalgebras are ultracategories: we characterize the normalized lax algebras of its profunctorial extension as ultraconvergence spaces, a recently-introduced categorification of topological spaces.
Paper Structure (29 sections, 35 theorems, 61 equations, 4 figures)

This paper contains 29 sections, 35 theorems, 61 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Relations in a bicategory
  3. Two-sided discrete fibrations
  4. The bicategory of two-sided discrete fibrations
  5. Extending to two-sided discrete fibrations
  6. Exact squares
  7. Extension theorems
  8. Extensions of pseudomonads
  9. Pseudomonads on CAT, extended to PROF
  10. A class of examples
  11. Quotient pseudomonads
  12. Ultracategories and distributive laws
  13. Extensions to PROF and pseudodistributive laws
  14. Ultracategories
  15. Ultraconvergence spaces, algebraically
  16. ...and 14 more sections

Key Result

Theorem 2.13

Let $\mathsf{K}$ be a bicategory with finite products, pseudopullbacks and comma objects, and suppose given a factorization system $(\mathcal{E},\mathcal{M})$ on $\mathsf{K}$ such that: The above composition defines a bicategory $\mathbf{TSDF}(\mathsf{K})$ with objects those of $\mathsf{K}$ and hom-categories the $\mathbf{TSDF}(\mathsf{K})(A,B)$.

Figures (4)

  • Figure 1: Algebraic presentations of topological spaces
  • Figure :
  • Figure :
  • Figure :

Theorems & Definitions (107)

  • Definition 2.1: benabouIntroductionBicategories1967
  • Example 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.5
  • Definition 2.6
  • Example 2.7
  • Definition 2.8: carboniModulatedBicategories1994
  • Example 2.9
  • Definition 2.10
  • ...and 97 more