Bar constructions and Quillen homology of modules over operads. (English) Zbl 1197.18002
In the context of symmetric spectra and unbounded chain complexes over a commutative ring, the author proves that a total left derived functor arising from an operad map can be expressed in terms of a realization of a simplicial bar construction, assuming a cofibrancy condition. A special case of this is the main theorem of the paper: the topological Quillen homology of an algebra (resp. module) over an operad in symmetric spectra or unbounded chain complexes can be expressed in terms of the realization of a simplicial bar construction, provided the simplicial bar construction is objectwise cofibrant.
More precisely, for the general statement, consider a morphism \(f:{\mathcal O} \rightarrow {\mathcal O}'\) of operads in symmetric spectra or unbounded chain complexes over a commutative ring. The category of \({\mathcal O}\)-algebras (respectively left \({\mathcal O}\)-modules) can be equipped with an appropriate model structure. If \(X\) is an \({\mathcal O}\)-algebra (respectively \({\mathcal O}\)-module) and the simplicial bar construction \(B({\mathcal O},{\mathcal O},X)\) is objectwise cofibrant in \({\mathcal A}lg_{\mathcal O}\), then there is a zigzag of weak equivalences \(Lf_*(X) \simeq |B({\mathcal O},\mathcal{O'},X)|\). For example, if \({\mathcal O}\) is a cofibrant operad and \(X\) is a cofibrant \({\mathcal O}\)-algebra, then \(B({\mathcal O},{\mathcal O},X)\) is object wise cofibrant.
For the main theorem on topological Quillen homology, consider an augmented operad \({\mathcal O}\) in symmetric spectra or unbounded chain complexes over a commutative ring. If \(X\) is an \({\mathcal O}\)-algebra and the simplicial bar construction \(B({\mathcal O},{\mathcal O},X)\) is objectwise cofibrant, then there is a zigzag of weak equivalences \(I \circ^{\text{L}}_{\mathcal O} X \simeq |B(I,{\mathcal O},X)|\). A similar statement holds for left \({\mathcal O}\)-modules.
An auxiliary result is that the forgetful functor from \({\mathcal O}\)-algebras to symmetric spectra or unbounded chain complexes preserves, up to weak equivalence, homotopy colimits indexed by \(\Delta^{\text{op}}\).
The author compares his main theorem with a similar result by M. Basterra on the total left derived “indecomposables” functor in the context of \(S\)-modules [“André-Quillen cohomology of commutative \(S\)-algebras”, J. Pure Appl. Algebra 144, No. 2, 111–143 (1999; Zbl 0937.55006)] and a result by B. Fresse on the total left derived “indecomposables” functor in the context of non-negative chain complexes [“Koszul duality of operads and homology of partition posets”, Contemp. Math. 346, 115–215 (2004; Zbl 1077.18007)].
More precisely, for the general statement, consider a morphism \(f:{\mathcal O} \rightarrow {\mathcal O}'\) of operads in symmetric spectra or unbounded chain complexes over a commutative ring. The category of \({\mathcal O}\)-algebras (respectively left \({\mathcal O}\)-modules) can be equipped with an appropriate model structure. If \(X\) is an \({\mathcal O}\)-algebra (respectively \({\mathcal O}\)-module) and the simplicial bar construction \(B({\mathcal O},{\mathcal O},X)\) is objectwise cofibrant in \({\mathcal A}lg_{\mathcal O}\), then there is a zigzag of weak equivalences \(Lf_*(X) \simeq |B({\mathcal O},\mathcal{O'},X)|\). For example, if \({\mathcal O}\) is a cofibrant operad and \(X\) is a cofibrant \({\mathcal O}\)-algebra, then \(B({\mathcal O},{\mathcal O},X)\) is object wise cofibrant.
For the main theorem on topological Quillen homology, consider an augmented operad \({\mathcal O}\) in symmetric spectra or unbounded chain complexes over a commutative ring. If \(X\) is an \({\mathcal O}\)-algebra and the simplicial bar construction \(B({\mathcal O},{\mathcal O},X)\) is objectwise cofibrant, then there is a zigzag of weak equivalences \(I \circ^{\text{L}}_{\mathcal O} X \simeq |B(I,{\mathcal O},X)|\). A similar statement holds for left \({\mathcal O}\)-modules.
An auxiliary result is that the forgetful functor from \({\mathcal O}\)-algebras to symmetric spectra or unbounded chain complexes preserves, up to weak equivalence, homotopy colimits indexed by \(\Delta^{\text{op}}\).
The author compares his main theorem with a similar result by M. Basterra on the total left derived “indecomposables” functor in the context of \(S\)-modules [“André-Quillen cohomology of commutative \(S\)-algebras”, J. Pure Appl. Algebra 144, No. 2, 111–143 (1999; Zbl 0937.55006)] and a result by B. Fresse on the total left derived “indecomposables” functor in the context of non-negative chain complexes [“Koszul duality of operads and homology of partition posets”, Contemp. Math. 346, 115–215 (2004; Zbl 1077.18007)].
Reviewer: Thomas Fiore (Dearborn)
MSC:
| 18D50 | Operads (MSC2010) |
| 55P43 | Spectra with additional structure (\(E_\infty\), \(A_\infty\), ring spectra, etc.) |
| 55P48 | Loop space machines and operads in algebraic topology |
| 18G55 | Nonabelian homotopical algebra (MSC2010) |
| 55U35 | Abstract and axiomatic homotopy theory in algebraic topology |
References:
| [1] | M Basterra, André-Quillen cohomology of commutative \(S\)-algebras, J. Pure Appl. Algebra 144 (1999) 111 · Zbl 0937.55006 · doi:10.1016/S0022-4049(98)00051-6 |
| [2] | M Basterra, M A Mandell, Homology and cohomology of \(E_{\infty}\) ring spectra, Math. Z. 249 (2005) 903 · Zbl 1071.55006 · doi:10.1007/s00209-004-0744-y |
| [3] | C Berger, I Moerdijk, Axiomatic homotopy theory for operads, Comment. Math. Helv. 78 (2003) 805 · Zbl 1041.18011 · doi:10.1007/s00014-003-0772-y |
| [4] | W Chachólski, J Scherer, Homotopy theory of diagrams, Mem. Amer. Math. Soc. 155 (2002) · Zbl 1006.18015 |
| [5] | D Dugger, D C Isaksen, Topological hypercovers and \(\mathbbA^1\)-realizations, Math. Z. 246 (2004) 667 · Zbl 1055.55016 · doi:10.1007/s00209-003-0607-y |
| [6] | W G Dwyer, H W Henn, Homotopy theoretic methods in group cohomology, Advanced Courses in Mathematics. CRM Barcelona, Birkhäuser Verlag (2001) · Zbl 1047.55001 |
| [7] | W G Dwyer, J Spaliński, Homotopy theories and model categories, North-Holland (1995) 73 · Zbl 0869.55018 |
| [8] | A D Elmendorf, I Kriz, M A Mandell, J P May, Rings, modules, and algebras in stable homotopy theory, Mathematical Surveys and Monographs 47, American Mathematical Society (1997) · Zbl 0894.55001 |
| [9] | A D Elmendorf, M A Mandell, Rings, modules, and algebras in infinite loop space theory, Adv. Math. 205 (2006) 163 · Zbl 1117.19001 · doi:10.1016/j.aim.2005.07.007 |
| [10] | B Fresse, Lie theory of formal groups over an operad, J. Algebra 202 (1998) 455 · Zbl 1041.18009 · doi:10.1006/jabr.1997.7280 |
| [11] | B Fresse, Koszul duality of operads and homology of partition posets, Contemp. Math. 346, Amer. Math. Soc. (2004) 115 · Zbl 1077.18007 |
| [12] | B Fresse, Modules over operads and functors, Lecture Notes in Mathematics 1967, Springer (2009) · Zbl 1178.18007 · doi:10.1007/978-3-540-89056-0 |
| [13] | P Gabriel, M Zisman, Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 35, Springer New York, New York (1967) · Zbl 0186.56802 |
| [14] | P G Goerss, On the André-Quillen cohomology of commutative \(\mathbbF_2\)-algebras, Astérisque (1990) 169 · Zbl 0742.13008 |
| [15] | P G Goerss, M J Hopkins, André-Quillen (co)-homology for simplicial algebras over simplicial operads, Contemp. Math. 265, Amer. Math. Soc. (2000) 41 · Zbl 0999.18009 |
| [16] | P G Goerss, M J Hopkins, Moduli spaces of commutative ring spectra, London Math. Soc. Lecture Note Ser. 315, Cambridge Univ. Press (2004) 151 · Zbl 1086.55006 |
| [17] | P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag (1999) · Zbl 0949.55001 |
| [18] | P Goerss, K Schemmerhorn, Model categories and simplicial methods, Contemp. Math. 436, Amer. Math. Soc. (2007) 3 · Zbl 1134.18007 |
| [19] | V K A M Gugenheim, J P May, On the theory and applications of differential torsion products, Memoirs of the American Mathematical Society 142, American Mathematical Society (1974) · Zbl 0292.55019 |
| [20] | J E Harper, Homotopy theory of modules over operads in symmetric spectra, Algebr. Geom. Topol. 9 (2009) 1637 · Zbl 1235.55004 · doi:10.2140/agt.2009.9.1637 |
| [21] | J E Harper, Homotopy theory of modules over operads and non-\(\Sigma\) operads in monoidal model categories, J. Pure Appl. Algebra (to appear) · Zbl 1231.55011 · doi:10.1016/j.jpaa.2009.11.006 |
| [22] | V Hinich, Homological algebra of homotopy algebras, Comm. Algebra 25 (1997) 3291 · Zbl 0894.18008 · doi:10.1080/00927879708826055 |
| [23] | P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003) · Zbl 1017.55001 |
| [24] | M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999) · Zbl 0909.55001 |
| [25] | M Hovey, B Shipley, J Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149 · Zbl 0931.55006 · doi:10.1090/S0894-0347-99-00320-3 |
| [26] | M Kapranov, Y Manin, Modules and Morita theorem for operads, Amer. J. Math. 123 (2001) 811 · Zbl 1001.18004 · doi:10.1353/ajm.2001.0033 |
| [27] | I K\vz, J P May, Operads, algebras, modules and motives, Astérisque 233 (1995) · Zbl 0840.18001 |
| [28] | S Mac Lane, Homology, Classics in Mathematics, Springer (1995) · Zbl 0818.18001 |
| [29] | S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1998) · Zbl 0906.18001 |
| [30] | M A Mandell, \(E_\infty\) algebras and \(p\)-adic homotopy theory, Topology 40 (2001) 43 · Zbl 0974.55004 · doi:10.1016/S0040-9383(99)00053-1 |
| [31] | M A Mandell, J P May, S Schwede, B Shipley, Model categories of diagram spectra, Proc. London Math. Soc. \((3)\) 82 (2001) 441 · Zbl 1017.55004 · doi:10.1112/S0024611501012692 |
| [32] | J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972) · Zbl 0244.55009 |
| [33] | J P May, Classifying spaces and fibrations, Mem. Amer. Math. Soc. 1 (1975) · Zbl 0321.55033 |
| [34] | J P May, Simplicial objects in algebraic topology, Chicago Lectures in Mathematics, University of Chicago Press (1992) · Zbl 0769.55001 |
| [35] | J E McClure, J H Smith, A solution of Deligne’s Hochschild cohomology conjecture, Contemp. Math. 293, Amer. Math. Soc. (2002) 153 · Zbl 1009.18009 |
| [36] | H Miller, Correction to: “The Sullivan conjecture on maps from classifying spaces” [Ann. of Math. (2) 120 (1984) 39-87], Ann. of Math. \((2)\) 121 (1985) 605 · Zbl 0575.55011 · doi:10.2307/1971212 |
| [37] | D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer (1967) · Zbl 0168.20903 · doi:10.1007/BFb0097438 |
| [38] | D Quillen, Rational homotopy theory, Ann. of Math. \((2)\) 90 (1969) 205 · Zbl 0191.53702 · doi:10.2307/1970725 |
| [39] | D Quillen, On the (co-) homology of commutative rings, Amer. Math. Soc. (1970) 65 · Zbl 0234.18010 |
| [40] | C Rezk, Spaces of Algebra Structures and Cohomology of Operads, PhD thesis, MIT (1996) |
| [41] | C Rezk, Notes on the Hopkins-Miller theorem, Contemp. Math. 220, Amer. Math. Soc. (1998) 313 · Zbl 0910.55004 |
| [42] | S Schwede, Spectra in model categories and applications to the algebraic cotangent complex, J. Pure Appl. Algebra 120 (1997) 77 · Zbl 0888.55010 · doi:10.1016/S0022-4049(96)00058-8 |
| [43] | S Schwede, \(S\)-modules and symmetric spectra, Math. Ann. 319 (2001) 517 · Zbl 0972.55005 · doi:10.1007/PL00004446 |
| [44] | S Schwede, Stable homotopy of algebraic theories, Topology 40 (2001) 1 · Zbl 0964.55017 · doi:10.1016/S0040-9383(99)00046-4 |
| [45] | S Schwede, An untitled book project about symmetric spectra (2007) |
| [46] | S Schwede, B E Shipley, Algebras and modules in monoidal model categories, Proc. London Math. Soc. \((3)\) 80 (2000) 491 · Zbl 1026.18004 · doi:10.1112/S002461150001220X |
| [47] | B Shipley, A convenient model category for commutative ring spectra, Contemp. Math. 346, Amer. Math. Soc. (2004) 473 · Zbl 1063.55006 |
| [48] | M Spitzweck, Operads, algebras and modules in general model categories (2001) · Zbl 1103.18300 |
| [49] | C A Weibel, An introduction to homological algebra, Cambridge Studies in Advanced Mathematics 38, Cambridge University Press (1994) · Zbl 0797.18001 |
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