Koszul Resolutions

Priddy, Stewart

doi:10.2307/1995637

Cited by 101 publications

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“…These definitions generalize the bar-cobar duality for algebras and coalgebras discovered by Moore [23], as well as Priddy's notion of Koszul duality for algebras [24]. They also illuminate the relationship between Quillen's models for rational homotopy theory [25].…”

Section: Introductionmentioning

confidence: 59%

“…It should be pointed out that 'Koszul' dual is not really an appropriate name for what we are calling KP . As originally described by Priddy [24] for algebras, and Ginzburg-Kapranov [15] for operads, the Koszul dual in the algebraic setting is a certain minimal model for the (dual of the) bar construction. It is much smaller than, but quasi-isomorphic to, the full bar construction and, for example, helps us write down explicit resolutions for algebras over operads.…”

Section: Koszul Duality For Termwise-finite Operadsmentioning

confidence: 99%

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Bar-cobar duality for operads in stable homotopy theory

Ching

2012

Journal of Topology

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We extend bar–cobar duality, defined for operads of chain complexes by Getzler and Jones, to operads of spectra in the sense of stable homotopy theory. Our main result is the existence of a Quillen equivalence between the category of reduced operads of spectra (with the projective model structure) and a new model for the homotopy theory of cooperads of spectra. The crucial construction is of a weak equivalence of operads between the Boardman–Vogt W‐construction for an operad P, and the cobar–bar construction on P. This weak equivalence generalizes a theorem of Berger and Moerdijk that says the W‐ and cobar–bar constructions are isomorphic for operads of chain complexes. Our model for the homotopy theory of cooperads is based on ‘pre‐cooperads’. These can be viewed as cooperads in which the structure maps are zigzags of maps of spectra that satisfy coherence conditions. Our model structure on pre‐cooperads is such that every object is weakly equivalent to an actual cooperad, and weak equivalences between cooperads are detected in the underlying symmetric sequences. We also interpret our results in terms of a ‘derived Koszul dual’ for operads of spectra, which is analogous to the Ginzburg–Kapranov dg‐dual. We show that the double derived Koszul dual of an operad P is equivalent to P (under some finiteness hypotheses) and that the derived Koszul construction preserves homotopy colimits, finite homotopy limits and derived mapping spaces for operads.

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Section: Introductionmentioning

confidence: 59%

Section: Koszul Duality For Termwise-finite Operadsmentioning

confidence: 99%

Bar-cobar duality for operads in stable homotopy theory

Ching

2012

Journal of Topology

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“…Remark 5.12. We interpret Theorem 5.11 in terms of Koszul duality, after [21] and [37]. Given the calculation of the Koszul dual operad DE n E n [−n], computed at the level of homology by Getzler and Jones [18] and computed in chain complexes by Fresse [17], these functors should be equivalent to restriction and induction along the Koszul dual of the map E n−1 → E n .…”

Section: Factorization Homology With Coefficients In Free N-disk Algementioning

confidence: 99%

Factorization homology of topological manifolds

2015

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Factorization homology theories of topological manifolds, after Beilinson, Drinfeld, and Lurie, are homology-type theories for topological n-manifolds whose coefficient systems are n-disk algebras or n-disk stacks. In this work, we prove a precise formulation of this idea, giving an axiomatic characterization of factorization homology with coefficients in n-disk algebras in terms of a generalization of the Eilenberg-Steenrod axioms for singular homology. Each such theory gives rise to a kind of topological quantum field theory, for which observables can be defined on general n-manifolds and not only closed n-manifolds. For n-disk algebra coefficients, these field theories are characterized by the condition that global observables are determined by local observables in a strong sense. Our axiomatic point of view has a number of applications. In particular, we give a concise proof of the non-abelian Poincaré duality of Salvatore, Segal, and Lurie. We present some essential classes of calculations of factorization homology, such as for free n-disk algebras and enveloping algebras of Lie algebras, several of which have a conceptual meaning in terms of Koszul duality.

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“…In differential graded algebra, derived Koszul duality (or bar duality) concerns the contravariant adjunction between the category of augmented differential graded algebras and itself [16,23] (named for the special case of Koszul algebras [1,2,25]). The dual of an augmented differential graded k-algebra A is an augmented differential graded k-algebra E that models the A-module endomorphisms of k, End A (k, k).…”

Section: Introductionmentioning

confidence: 99%

Derived Koszul duality and involutions in the algebraic K -theory of spaces

Blumberg

Mandell

2011

Journal of Topology

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We interpret different constructions of the algebraic K‐theory of spaces as an instance of derived Koszul (or bar) duality and also as an instance of Morita equivalence. We relate the interplay between these two descriptions to the homotopy involution. We define a geometric analog of the Swan theory Gℤ(ℤ[π]) in terms of Σ+∞ Ω X and show that it is the algebraic K‐theory of the E∞ ring spectrum DX=SX+.

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Koszul Resolutions

Cited by 101 publications

References 0 publications

Bar-cobar duality for operads in stable homotopy theory

Bar-cobar duality for operads in stable homotopy theory

Factorization homology of topological manifolds

Derived Koszul duality and involutions in the algebraic K -theory of spaces

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