arXiv:1902.03831·math.CT·February 12, 2019·LICS
High-level methods for homotopy construction in associative $n$-categories
David Reutter, Jamie Vicary
TL;DR
This paper introduces a contraction operation for associative n-categories that algorithmically constructs homotopies, simplifying complex proofs and enabling implementation in proof assistants like homotopy.io.
Contribution
It presents a novel contraction procedure for building homotopies in associative n-categories, facilitating proof construction and automation.
Findings
01
The contraction operation reduces composite term lengths effectively.
02
Proof correctness is established via lifting connected colimits.
03
Homotopy.io implements the contraction mechanism for practical use.
Abstract
A combinatorial theory of associative -categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a combinatorial notion of homotopy with a natural geometrical interpretation. Such a theory has the potential to serve as an attractive foundation for a computer proof assistant for higher category theory, since it allows composites to be uniquely described, and relieves proofs from the bureaucracy of associators, unitors and their coherence. However, this basic theory lacks a high-level way to construct homotopies, which would be intractable to build directly in complex situations; it is not therefore immediately amenable to implementation. We tackle this problem by describing a contraction operation, which algorithmically constructs complex homotopies that reduce the lengths of composite terms. This…
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Figure 40Equations6
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Full text
High-level methods for homotopy
construction in associative -categories
David Reutter* 1*
Jamie Vicary* 1 2*
David Reutter* 1*
Jamie Vicary* 1 2*
(16 January 2019)
Abstract
A combinatorial theory of associative -categories has recently been proposed, with strictly associative and unital composition in all dimensions, and the weak structure arising as a notion of ‘homotopy’ with a natural geometrical interpretation. Such a theory has the potential to serve as an attractive foundation for a computer proof assistant for higher category theory, since it allows composites to be uniquely described, and relieves proofs from the bureaucracy of associators, unitors and their coherence. However, this basic theory lacks a high-level way to construct homotopies, which would be intractable to build directly in complex situations; it is not therefore immediately amenable to implementation.
We tackle this problem by describing a ‘contraction’ operation, which algorithmically constructs complex homotopies that reduce the lengths of composite terms. This contraction procedure allows building of nontrivial proofs by repeatedly contracting subterms, and also allows the contraction of those proofs themselves, yielding in some cases single-step witnesses for complex homotopies. We prove correctness of this procedure by showing that it lifts connected colimits from a base category to a category of zigzags, a procedure which is then iterated to yield a contraction mechanism in any dimension. We also present homotopy.io, an online proof assistant that implements the theory of associative -categories, and use it to construct a range of examples that illustrate this new contraction mechanism. 111Department of Computer Science, University of Oxford, UK222School of Computer Science, University of Birmingham, UK
I Introduction
The theory of associative -categories (ANCs) has recently been proposed [Dorn2018, Douglas2019]. As with the theory of strict -categories [Leinster2003], composition in this theory is strictly associative and unital in all dimensions. However, unlike the strict theory, ANCs retain a significant amount of weak structure—in the form of homotopies333Homotopy is a standard notion from algebraic topology, which can be understood informally to mean the continuous deformation of one topological structure into another. We use the term only in an informal sense, basing our formal mathematical development on the theory of associative -categories, which have a combinatorial foundation., with a natural geometric interpretation—making it reasonable to conjecture that every weak -category is equivalent to an ANC [Dorn2018, Conjecture I.5.0.4].444In dimension 3 the theory of ANCs agrees with the theory of Gray categories, a well-known model of 3-categories which is equivalent to the fully-weak theory [Gordon1995], but which has strict associators and unitors; for , ANCs are not expected to be equivalent to any previously-described theory.
It may therefore be possible for ANCs to serve as an attractive general language for calculations in higher category theory, if suitably encoded into a computer proof assistant. Strict associators and units would make composites unique, eliminating some of the bureaucracy of coherence, while the remaining weak structure—while still potentially of high complexity—could be reasoned about geometrically.
The major obstacle to realizing this goal is the difficulty of constructing nontrivial terms of the theory. In principle these can encode complex data, including not only the composites of generating types in arbitrary dimension, but also arbitrary homotopies of these composites. Each term has a dimension, and the -dimensional terms are called -diagrams.
As examples of such terms, consider Figures 1 and 2. Figure 1 shows two 2-diagrams, which can be interpreted as expressions in the string diagram calculus for a monoidal category [Selinger2011]; the arrow represents a homotopy of these 2-diagrams, and forms part of the data of a 3-diagram. Figure 2 shows two 3-diagrams, which can be interpreted as tangles [Baez1995] in the string diagram calculus for a braided monoidal category; the arrow represents a homotopy of these 3-diagrams, and forms part of the data of a 4-diagram.
The main mathematical contribution of this paper is the description of a contraction algorithm, which builds homotopies that reduce a given portion of the diagram in size, along with a full mathematical theory that demonstrates correctness of the procedure. This gives a high-level method for building nontrivial homotopies in an associative -category. Figures 1 and 2 both give examples; in each case, the second diagram was obtained by executing the contraction procedure on the first diagram.
Contraction can serve as the main workhorse for the construction of a range of nontrivial proofs in the theory. Given an initial composite -diagram, we produce our -dimensional proof object by contracting various -dimensional subdiagrams for to produce the content of the -dimensional proof object, as well as applying algebraic moves from the signature, and extending these recursively to the diagram as a whole using some further techniques described in Section LABEL:sec:additionalmethods. Once our proof is complete, we can then contract that proof term itself, to yield a shorter proof of the same logical statement. In Section LABEL:sec:homotopies we give two fully-worked examples of this entire proof construction workflow. Indeed, we conjecture that contractions, together the associated recursive techniques that we discuss in Section LABEL:sec:additionalmethods, yield a universal toolkit which can in principle construct any homotopy in the theory.