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Computing Critical Pairs in 2Dimensional Rewriting Systems
, 2010
"... Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for ..."
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Cited by 5 (2 self)
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Rewriting systems on words are very useful in the study of monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative for the elements of the presented monoid. Polygraphs are a higherdimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of ncategories. Here, we are interested in proving confluence for polygraphs presenting 2categories, which can be seen as a generalization of termrewriting systems. For this purpose, we propose an adaptation of the usual algorithm for computing critical pairs. Interestingly, this framework is much richer than term rewriting systems and requires the elaboration of a new theoretical framework for representing critical pairs, based on contexts in compact 2categories. Term rewriting systems have proven very useful to reason about terms modulo equations. In some cases, the equations can be oriented and completed in a way giving rise to a normalizing (i.e. confluent and terminating) rewriting system, thus providing a notion of canonical representative of equivalence classes of terms. Usually, terms are freely generated by a signature (Σn)n∈N, which consists of a family of sets Σn of generators of arity n, and one considers equational theories on such a signature, which are formalized by sets of pairs of terms called equations. For example, the equational theory of monoids contains two generators m and e, whose arities are respectively 2 and 0, and three equations
A folk model structure on omegacat
, 2009
"... The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generat ..."
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Cited by 1 (0 self)
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The primary aim of this work is an intrinsic homotopy theory of strict ωcategories. We establish a model structure on ωCat, the category of strict ωcategories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All object are fibrant while free objects are cofibrant. We further exhibit model structures of this type on ncategories for arbitrary n ∈ N, as specialisations of the ωcategorical one along right adjoints. In particular, known cases for n = 1 and n = 2 nicely fit into the scheme.
Polygraphic resolutions and homology of monoids
, 2007
"... We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1 ..."
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We prove that for any monoid M, the homology defined by the second author by means of polygraphic resolutions coincides with the homology classically defined by means of resolutions by free ZMmodules. 1
TOWARDS HIGHERDIMENSIONAL REWRITING THEORY
"... Abstract. String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical represent ..."
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Abstract. String rewriting systems have proved very useful to study monoids. In good cases, they give finite presentations of the monoids, allowing their manipulation by a computer. Even better, when the presentation is confluent and terminating, they provide one with a notion of canonical representative of the elements of the presented monoid. Polygraphs are a higherdimensional generalization of this notion of presentation, from the setting of monoids to the much more general setting of ncategories. One of the main purposes of this article is to give a progressive introduction to the notion of higherdimensional rewriting system provided by polygraphs, and describe its links with standard rewriting theory (in particular string and term rewriting systems). After introducing the general setting, we will be interested in proving local confluence for polygraphs presenting 2categories and introduce a framework in which a 2dimensional rewriting system admits a finite number of critical pairs. Recent developments in category theory have established higherdimensional categories as a fundamental theoretical setting in order to study situations arising in various areas of mathematics, physics and computer science. A nice survey of these can be found in [2],