High-level methods for homotopy construction in associative n-categories

Abstract

A combinatorial theory of associative n-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 n-categories, and use it to construct a range of examples that illustrate this new contraction mechanism. 12 1 arXiv:1902.03831v1 [math.CT]

Semantics for two-dimensional type theory

“…Benjamin, Finster, and Mimram [7] (see also [13]) design a type theory whose models are precisely ω-categories à la Grothendieck-Maltsiniotis. There are also computer tools implementing them, see, e. g., [5,30].…”

The smash product of monoidal theories

“…We briefly mention other potential developments. Christoph Dorn, David Reutter, and Jamie Vicary have defined a semistrict algebraic model of n-categories, called associative n-categories, which is equivalent to Gray-categories for n = 3 [Dor18,RV19]. It is conceivable that the adjunction of Section 4.2 relating diagrammatic sets to Gray-categories may generalise to associative n-categories for n > 3.…”

Operads for complex system design specification, analysis and synthesis

“…4] of physical systems employing practical semantic models and algorithms [21,22,87]. Software tools are increasingly focused on scaling up computation, for example [24,49,51], as opposed to software to augment human calculation, as in [10,53,67], and managing complex domains with commercial-grade tools [19,63,82]. Recent work on quantum circuit optimization leverages such developments [29,52] and the use of wiring diagrams to improve computational efficiency via normal forms is also being explored [64].…”