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Register a new userRewriting in Gray categories with applications to coherence
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Over the recent years, the theory of rewriting has been used and extended in order to provide systematic techniques to show coherence results for strict higher categories. Here, we investigate a further generalization to Gray categories, which are known to be equivalent to tricategories. This requires us to develop the theory of rewriting in the setting of precategories, which include Gray categories as particular cases, and are adapted to mechanized computations. We show that a finite rewriting system in precategories admits a finite number of critical pairs, which can be efficiently computed. We also extend Squiers theorem to our context, showing that a convergent rewriting system is coherent, which means that any two parallel 3-cells are necessarily equal. This allows us to prove coherence results for several well-known structures in the context of Gray categories: monoids, adjunctions, Frobenius monoids.
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In this paper, we study rewriting modulo a set of algebraic axioms in categories enriched in linear categories, called linear~$(2,2)$-categories. We introduce the structure of linear~$(3,2)$-polygraph modulo as a presentation of a linear~$(2,2)$-category by a rewriting system modulo algebraic axioms. We introduce a symbolic computation method in order to compute linear bases for the vector spaces of $2$-cells of these categories. In particular, we study the case of pivotal $2$-categories using the isotopy relations given by biadjunctions on $1$-cells and cyclicity conditions on $2$-cells as axioms for which we rewrite modulo. By this constructive method, we recover the bases of normally ordered dotted oriented Brauer diagrams in te affine oriented Brauer linear~$(2,2)$-category.
We introduce homotopical methods based on rewriting on higher-dimensional categories to prove coherence results in categories with an algebraic structure. We express the coherence problem for (symmetric) monoidal categories as an asphericity problem for a track category and we use rewriting methods on polygraphs to solve it. The setting is extended to more general coherence problems, seen as 3-dimensional word problems in a track category, including the case of braided monoidal categories.
This paper presents the proof of the coherence theorem for Ann-categories whose set of axioms and original basic properties were given in [9]. Let $$A=(A,{Ah},c,(0,g,d),a,(1,l,r),{Lh},{Rh})$$ be an Ann-category. The coherence theorem states that in the category $ A$, any morphism built from the above isomorphisms and the identification by composition and the two operations $tx$, $ts$ only depends on its source and its target. The first coherence theorems were built for monoidal and symmetric monoidal categories by Mac Lane [7]. After that, as shown in the References, there are many results relating to the coherence problem for certain classes of categories. For Ann-categories, applying Hoang Xuan Sinhs ideas used for Gr-categories in [2], the proof of the coherence theorem is constructed by faithfully ``embedding each arbitrary Ann-category into a quite strict Ann-category. Here, a {it quite strict} Ann-categogy is an Ann-category whose all constraints are strict, except for the commutativity and left distributivity ones. This paper is the work continuing from [9]. If there is no explanation, the terminologies and notations in this paper mean as in [9].
Indexed symmetric monoidal categories are an important refinement of bicategories -- this structure underlies several familiar bicategories, including the homotopy bicategory of parametrized spectra, and its equivariant and fiberwise generalizations. In this paper, we extend existing coherence theorems to the setting of indexed symmetric monoidal categories. The most central theorem states that a large family of operations on a bicategory defined from an indexed symmetric monoidal category are all canonically isomorphic. As a part of this theorem, we introduce a rigorous graphical calculus that specifies when two such operations admit a canonical isomorphism.
Restriction categories were introduced to provide an axiomatic setting for the study of partially defined mappings; they are categories equipped with an operation called restriction which assigns to every morphism an endomorphism of its domain, to be thought of as the partial identity that is defined to just the same degree as the original map. In this paper, we show that restriction categories can be identified with emph{enriched categories} in the sense of Kelly for a suitable enrichment base. By varying that base appropriately, we are also able to capture the notions of join and range restriction category in terms of enriched category theory.
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