The free bifibration on a functor

2025-11-10

AI Summary

This paper addresses the construction of the free bifibration generated by a functor $p : D o C$. We develop a proof-theoretic approach grounded in bifibrational logic and cut-free sequent calculus, introducing a focusing strategy to define normal forms—thereby enabling decidability of term equivalence and duplicate-free enumeration of relative homomorphism sets. Our key contributions are threefold: (i) establishing an intrinsic connection between free bifibrations and adjoint structures (left/right adjoints); (ii) incorporating permutation equivalence into categorical semantics via preorder factorization and zigzag bicategory techniques; and (iii) instantiating the framework with the identity functor to recover the lattice of noncrossing partitions as a quotient structure, thereby categorifying planar trees and increasing forests as combinatorial objects.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Decentralized Web and Fediverse systems

Abstract

We consider the problem of constructing the free bifibration generated by a functor of categories $p : D o C$. This problem was previously considered by Lamarche, and is closely related to the problem, considered by Dawson, Par'e, and Pronk, of"freely adjoining adjoints"to a category. We develop a proof-theoretic approach to the problem, beginning with a construction of the free bifibration $Lambda_p : mathcal{B}mathrm{if}(p) o C$ in which objects of $mathcal{B}mathrm{if}(p)$ are formulas of a primitive"bifibrational logic", and arrows are derivations in a cut-free sequent calculus modulo a notion of permutation equivalence. We show that instantiating the construction to the identity functor generates a _zigzag double category_ $mathbb{Z}(C)$, which is also the free double category with companions and conjoints (or fibrant double category) on $C$. The approach adapts smoothly to the more general task of building $(P,N)$-fibrations, where one only asks for pushforwards along arrows in $P$ and pullbacks along arrows in $N$ for some subsets of arrows, which encompasses Kock and Joyal's notion of _ambifibration_ when $(P,N)$ form a factorization system. We establish a series of progressively stronger normal forms, guided by ideas of _focusing_ from proof theory, and obtain a canonicity result under assumption that the base category is factorization preordered relative to $P$ and $N$. This canonicity result allows us to decide the word problem and to enumerate relative homsets without duplicates. Finally, we describe several examples of a combinatorial nature, including a category of plane trees generated as a free bifibration over $omega$, and a category of increasing forests generated as a free ambifibration over $Delta$, which contains the lattices of noncrossing partitions as quotients of its fibers by the Beck-Chevalley condition.

Problem

• Constructing free bifibrations generated by functors between categories
• Developing proof-theoretic approaches using bifibrational logic systems
• Establishing canonicity results for word problems and homsets enumeration

Innovation

• Constructs free bifibration using proof-theoretic logic formulas
• Develops cut-free sequent calculus with permutation equivalence
• Adapts to ambifibrations via focusing-inspired normal forms