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Coherence in monoidal track categories
"... Abstract – We introduce homotopical methods based on rewriting on higherdimensional 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 use rewriting ..."
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Abstract – We introduce homotopical methods based on rewriting on higherdimensional 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 use rewriting methods on polygraphs to solve it. The setting is generalized to more general coherence problems, seen as 3dimensional word problems in a track category. We prove general results that, in the case of braided monoidal categories, yield the coherence theorem for braided monoidal categories.
Twisted differential nonabelian cohomology Twisted (n−1)brane nbundles and their ChernSimons (n+1)bundles with characteristic (n + 2)classes
, 2008
"... We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shif ..."
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We introduce nonabelian differential cohomology classifying ∞bundles with smooth connection and their higher gerbes of sections, generalizing [138]. We construct classes of examples of these from lifts, twisted lifts and obstructions to lifts through shifted central extensions of groups by the shifted abelian ngroup B n−1 U(1). Notable examples are String 2bundles [9] and Fivebrane 6bundles [133]. The obstructions to lifting ordinary principal bundles to these, hence in particular the obstructions to lifting Spinstructures to Stringstructures [13] and further to Fivebranestructures [133, 52], are abelian ChernSimons 3 and 7bundles with characteristic class the first and second fractional Pontryagin class, whose abelian cocycles have been constructed explicitly by Brylinski and McLaughlin [35, 36]. We realize their construction as an abelian component of obstruction theory in nonabelian cohomology by ∞Lieintegrating the L∞algebraic data in [132]. As a result, even if the lift fails, we obtain twisted String 2 and twisted Fivebrane 6bundles classified in twisted nonabelian (differential) cohomology and generalizing the twisted bundles appearing in twisted Ktheory. We explain the GreenSchwarz mechanism in heterotic string theory in terms of twisted String 2bundles and its magnetic dual version – according to [133] – in terms of twisted Fivebrane 6bundles. We close by transgressing differential cocycles to mapping
HIGHERDIMENSIONAL NORMALISATION STRATEGIES FOR ACYCLICITY
"... Abstract – We introduce acyclic track polygraphs, a notion of complete categorical cellular models for small categories: they are polygraphs containing generators, with additional invertible cells for relations and higherdimensional globular syzygies. We give a rewriting method to realise such a mo ..."
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Abstract – We introduce acyclic track polygraphs, a notion of complete categorical cellular models for small categories: they are polygraphs containing generators, with additional invertible cells for relations and higherdimensional globular syzygies. We give a rewriting method to realise such a model by proving that a convergent presentation canonically extends to an acyclic track polygraph. For that, we introduce normalising strategies, defined as homotopically coherent ways to relate each cell of a track polygraph to its normal form, and we prove that acyclicity is equivalent to the existence of a normalisation strategy. Using track polygraphs, we extend to every dimension the homotopical finiteness condition of finite derivation type, introduced by Squier in string rewriting theory, and we prove that it implies a new homological finiteness condition that we introduce here. The proof is based on normalisation strategies and relates acyclic track polygraphs to free abelian resolutions of the small categories they present.
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, 2004
"... Abstract. A functor is constructed from the category of globular CWcomplexes to that of flows. It allows to compare the Shomotopy equivalences (resp. the Thomotopy equivalences) of globular complexes with the Shomotopy equivalences (resp. the Thomotopy equivalences) of flows. Moreover, one prov ..."
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Abstract. A functor is constructed from the category of globular CWcomplexes to that of flows. It allows to compare the Shomotopy equivalences (resp. the Thomotopy equivalences) of globular complexes with the Shomotopy equivalences (resp. the Thomotopy equivalences) of flows. Moreover, one proves that this functor induces an equivalence of categories from the localization of the category of globular CWcomplexes with respect to Shomotopy equivalences to the localization of the category of flows with respect to weak