7,027 research outputs found
Coherence of Gray Categories via Rewriting
Over the recent years, the theory of rewriting has been extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to low-dimensional weak categories, and consider in details the first non-trivial case: presentations of tricategories. By a general result, those are equivalent to the stricter Gray categories, for which we introduce a notion of rewriting system, as well as associated tools: critical pairs, termination orders, etc. We show that a finite rewriting system admits a finite number of critical pairs and, as a variant of Newman\u27s lemma in our context, that a convergent rewriting system is coherent, meaning that two parallel 3-cells are necessarily equal. This is illustrated on rewriting systems corresponding to various well-known structures in the context of Gray categories (monoids, adjunctions, Frobenius monoids). Finally, we discuss generalizations in arbitrary dimension
Globular: an online proof assistant for higher-dimensional rewriting
This article introduces Globular, an online proof assistant for the
formalization and verification of proofs in higher-dimensional category theory.
The tool produces graphical visualizations of higher-dimensional proofs,
assists in their construction with a point-and- click interface, and performs
type checking to prevent incorrect rewrites. Hosted on the web, it has a low
barrier to use, and allows hyperlinking of formalized proofs directly from
research papers. It allows the formalization of proofs from logic, topology and
algebra which are not formalizable by other methods, and we give several
examples
Normalization for planar string diagrams and a quadratic equivalence algorithm
In the graphical calculus of planar string diagrams, equality is generated by
exchange moves, which swap the heights of adjacent vertices. We show that left-
and right-handed exchanges each give strongly normalizing rewrite strategies
for connected string diagrams. We use this result to give a linear-time
solution to the equivalence problem in the connected case, and a quadratic
solution in the general case. We also give a stronger proof of the Joyal-Street
coherence theorem, settling Selinger's conjecture on recumbent isotopy
Weak Cat-Operads
An operad (this paper deals with non-symmetric operads)may be conceived as a
partial algebra with a family of insertion operations, Gerstenhaber's circle-i
products, which satisfy two kinds of associativity, one of them involving
commutativity. A Cat-operad is an operad enriched over the category Cat of
small categories, as a 2-category with small hom-categories is a category
enriched over Cat. The notion of weak Cat-operad is to the notion of Cat-operad
what the notion of bicategory is to the notion of 2-category. The equations of
operads like associativity of insertions are replaced by isomorphisms in a
category. The goal of this paper is to formulate conditions concerning these
isomorphisms that ensure coherence, in the sense that all diagrams of canonical
arrows commute. This is the sense in which the notions of monoidal category and
bicategory are coherent. The coherence proof in the paper is much simplified by
indexing the insertion operations in a context-independent way, and not in the
usual manner. This proof, which is in the style of term rewriting, involves an
argument with normal forms that generalizes what is established with the
completeness proof for the standard presentation of symmetric groups. This
generalization may be of an independent interest, and related to matters other
than those studied in this paper. Some of the coherence conditions for weak
Cat-operads lead to the hemiassociahedron, which is a polyhedron related to,
but different from, the three-dimensional associahedron and permutohedron.Comment: 38 pages, version prepared for publication in Logical Methods in
Computer Science, the authors' last version is v
Biequivalences in tricategories
We show that every internal biequivalence in a tricategory T is part of a
biadjoint biequivalence. We give two applications of this result, one for
transporting monoidal structures and one for equipping a monoidal bicategory
with invertible objects with a coherent choice of those inverses.Comment: Accepted for publication, to appear in Theory and Applications of
Categorie
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