Rigidification of Algebras Over Essentially Algebraic Theories

Rosický, Jiřı́

doi:10.1007/s10485-013-9329-4

Cited by 4 publications

(5 citation statements)

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“…The aim of the present paper is to show, in Theorem 8, that this is the case. With this result in place Rosický's theorems' 3.3 and 5.1 of [22] yield rigidification results for homotopy algebras of finite product theories in Cat-his Theorem 5.1 now asserts that, in Cat, each homotopy algebra of a finite product theory is weakly equivalent to a strict algebra, a direct analogue of the results in the simplicial setting described above.…”

Section: Introductionmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

“…In [22] Rosický has investigated the possibility of extending these rigidification results to other settings, by allowing his base of enrichment V to be a monoidal model category other than simplicial sets, and by considering weighted limit theories more general than finite product theories. One of his rigidification results, Theorem 3.3 of [22], requires that each cofibrant weight, or cofibrant object in [J, V] with its projective model structure, admits a certain kind of colimit decomposition. He asked the author whether such a colimit decomposition exists in the case that V = Cat with weak equivalences the equivalences of categories, and when the theories under consideration are just finite product theories-the special nature of the colimit decomposition now requiring that each cofibrant object of [J, Cat] can be presented as a sifted colimit of finite coproducts of representables, in which moreover each colimit involved is homotopically well behaved in a manner described in Section 4.2.…”

Section: Introductionmentioning

confidence: 99%

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A Colimit Decomposition for Homotopy Algebras in Cat

Bourke

2012

Appl Categor Struct

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Badzioch showed that in the category of simplicial sets each homotopy algebra of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to extend this result to other contexts Rosický observed a key point to be that each homotopy colimit in SSet admits a decomposition into a homotopy sifted colimit of finite coproducts, and asked the author whether a similar decomposition holds in the 2-category of categories Cat. Our purpose in the present paper is to show that this is the case.

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Section: Introductionmentioning

confidence: 72%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Colimit Decomposition for Homotopy Algebras in Cat

Bourke

2012

Appl Categor Struct

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“…cit., Theorem 6.4) for an arbitrary one-sorted algebraic theory (not only the theory of groups). It was extended to all multi-sorted theories in [17], and further to limit theories and to diagrams in model categories other than sSet in [61].…”

Section: Representation and Cyclic Homology Of Homotopy Simplicial Gr...mentioning

confidence: 99%

Derived Character Maps of Groups Representations

Berest¹,

Ramadoss²

2022

Preprint

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In this paper, we construct and study derived character maps of finite-dimensional representations of ∞-groups. As models for ∞-groups we take homotopy simplicial groups, i.e. homotopy simplicial G op -algebras over the algebraic theory of groups (in the sense of [4]). We define cyclic, symmetric and representation homology for 'group algebras' k[Γ] over such groups and construct canonical trace maps relating these homology theories. In the case of one-dimensional representations, we show that our trace maps are of topological origin: they are induced by natural maps of (iterated) loop spaces that are well studied in homotopy theory. Using this topological interpretation, we deduce some algebraic results about representation homology: in particular, we prove that the symmetric homology of group algebras and one-dimensional representation homology are naturally isomorphic, provided the base ring k is a field of characteristic zero. We also study the behavior of the derived character maps of n-dimensional representations in the stable limit as n → ∞, in which case we show that they 'converge' to become isomorphisms.A comparison of these constructions can be found in [12, Appendix]. The relation to our present work is briefly discussed in the end of Section 3.2.

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“…Homotopy limit sketches were proposed by Rosický [Ro13] with the purpose of extending rigidification results of Badzioch [Ba02] and Bergner [Be05] to finite limit theories. Lack and Rosický in [LR13] proved that the V-categories of homotopy models of homotopy limit V-sketches can be characterized as the homotopy locally presentable V-categories.…”

Section: Sketchesmentioning

confidence: 99%