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Concentration structures on categories and horizontal categorification

Yangxiao Luo, Shunyu Wan

TL;DR

The paper introduces concentration structures as equivalence relations on morphisms to formalize horizontal categorification and decategorification, producing a concentration monoid that encodes the horizontal decategorification. It establishes a robust functorial framework, showing that concentration monoids interact well with substructures, quotients, and semidirect products, and that horizontal categorification/decategorification correspond to 2-lifting phenomena. It further develops G-equivariant direct limits, revealing a semidirect product decomposition with the acting group, and provides concrete examples such as R-braid groups. A key application links concentration structures to fundamental groupoids and fibrations, showing how $\pi_1(X)$ arises as a concentration group and that pullbacks along fibrations recover groups from trivial categories, thereby connecting algebraic and geometric perspectives. Overall, the framework unifies horizontal categorification with classical algebraic constructions and topological invariants, offering systematic tools to realize groups and other algebraic objects as concentration monoids of categorical structures.

Abstract

We introduce a theory for encoding and manipulating algebraic data on categories via $\textit{concentration structures}$, which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a $\textit{concentration monoid}$, which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence.

Concentration structures on categories and horizontal categorification

TL;DR

The paper introduces concentration structures as equivalence relations on morphisms to formalize horizontal categorification and decategorification, producing a concentration monoid that encodes the horizontal decategorification. It establishes a robust functorial framework, showing that concentration monoids interact well with substructures, quotients, and semidirect products, and that horizontal categorification/decategorification correspond to 2-lifting phenomena. It further develops G-equivariant direct limits, revealing a semidirect product decomposition with the acting group, and provides concrete examples such as R-braid groups. A key application links concentration structures to fundamental groupoids and fibrations, showing how arises as a concentration group and that pullbacks along fibrations recover groups from trivial categories, thereby connecting algebraic and geometric perspectives. Overall, the framework unifies horizontal categorification with classical algebraic constructions and topological invariants, offering systematic tools to realize groups and other algebraic objects as concentration monoids of categorical structures.

Abstract

We introduce a theory for encoding and manipulating algebraic data on categories via , which are equivalence relations on morphisms that satisfy certain axioms. For any category with a concentration structure we can functorially construct a , which can be used to give a precise definition of horizontal categorification and decategorification. Moreover, by studying concentration structures on fundamental groupoids, we show that every group arises as the concentration monoid of a trivial category, up to category equivalence.

Paper Structure

This paper contains 26 sections, 37 theorems, 26 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Concentration structures and concentration preserving functors
  3. Concentration monoids
  4. Horizontal categorification and decategorification
  5. $G$-equivariant direct limits
  6. Concentration structures on fundamental groupoids
  7. Organization
  8. General properties of categories with concentration
  9. Well-definedness and functoriality of taking concentration monoid
  10. $n$-concentration structures
  11. Pullbacks of concentration structures
  12. $2$-lifting functors and horizontal decategorification
  13. Concentration monoid and adjoint functors
  14. Sub-concentrations and quotient concentrations
  15. More examples of concentration structures
  16. ...and 11 more sections

Key Result

Theorem 1.4

Taking concentration monoid is a functor $\mathbf{M}: \mathscr{C}at_{\sim} \to \mathscr{M}on$. When restricted on $\mathscr{G}rpd_{\sim}$, it is a functor $\mathbf{M}: \mathscr{G}rpd_{\sim} \to \mathscr{G}rp$.

Figures (6)

  • Figure 1: The category in Example \ref{['ex: Z/2 and Z/4 category with concentration']}.
  • Figure 2: The category in Example \ref{['ex: surjective but not 2-lifting functor']}.
  • Figure 3: The category in Example \ref{['ex: Z2 Z4 with one morphism']}.
  • Figure 4: (A) The category in Example \ref{['ex: z2 times z2']}. (B) The category in Example \ref{['ex: not 3-concentration']}.
  • Figure 5: (a) An element $\alpha \in Br_{|A|}$. (b) The group homomorphism $F(\iota_A^B)$ adds backside trivial strands (marked in red) to $\alpha$. The endpoints of those trivial strands are on $B \setminus A$.
  • ...and 1 more figures

Theorems & Definitions (104)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Definition 1.8: Horizontal categorification
  • Definition 1.9
  • ...and 94 more