Geodesic
Classifying spaces of algebras over a prop
Yalin, Sinan1
1Laboratoire Paul Painlevé, Université de Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France
Algebraic and Geometric Topology, Tome 14 (2014) no. 5, pp. 2561-2593
Cet article a éte moissonné depuis la source Mathematical Sciences Publishers

Voir la notice de l'article

We prove that a weak equivalence between cofibrant props induces a weak equivalence between the associated classifying spaces of algebras. This statement generalizes to the prop setting a homotopy invariance result which is well known in the case of algebras over operads. The absence of model category structure on algebras over a prop creates difficulties and we introduce new methods to overcome them. We also explain how our result can be extended to algebras over colored props in any symmetric monoidal model category tensored over chain complexes.

DOI : 10.2140/agt.2014.14.2561
Classification : 18G55, 18D10, 18D50
Keywords: props, classifying spaces, moduli spaces, bialgebras category, homotopical algebra, homotopy invariance

Yalin, Sinan  1

1 Laboratoire Paul Painlevé, Université de Lille 1, Cité Scientifique, 59655 Villeneuve d’Ascq Cedex, France 
@article{10_2140_agt_2014_14_2561,
     author = {Yalin, Sinan},
     title = {Classifying spaces of algebras over a prop},
     journal = {Algebraic and Geometric Topology},
     pages = {2561--2593},
     year = {2014},
     volume = {14},
     number = {5},
     doi = {10.2140/agt.2014.14.2561},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.2561/}
}
TY  - JOUR
AU  - Yalin, Sinan
TI  - Classifying spaces of algebras over a prop
JO  - Algebraic and Geometric Topology
PY  - 2014
SP  - 2561
EP  - 2593
VL  - 14
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.2561/
DO  - 10.2140/agt.2014.14.2561
ID  - 10_2140_agt_2014_14_2561
ER  - 
%0 Journal Article
%A Yalin, Sinan
%T Classifying spaces of algebras over a prop
%J Algebraic and Geometric Topology
%D 2014
%P 2561-2593
%V 14
%N 5
%U http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.2561/
%R 10.2140/agt.2014.14.2561
%F 10_2140_agt_2014_14_2561
Yalin, Sinan. Classifying spaces of algebras over a prop. Algebraic and Geometric Topology, Tome 14 (2014) no. 5, pp. 2561-2593. doi: 10.2140/agt.2014.14.2561

[1] J M Boardman, R M Vogt, Homotopy-everything $H\!$–spaces, Bull. Amer. Math. Soc. 74 (1968) 1117

[2] J M Boardman, R M Vogt, Homotopy invariant algebraic structures on topological spaces, Lecture Notes in Mathematics 347, Springer (1973)

[3] W G Dwyer, D M Kan, Function complexes in homotopical algebra, Topology 19 (1980) 427

[4] W G Dwyer, J Spaliński, Homotopy theories and model categories, from: "Handbook of algebraic topology", North-Holland (1995) 73

[5] B Enriquez, P Etingof, On the invertibility of quantization functors, J. Algebra 289 (2005) 321

[6] B Fresse, Modules over operads and functors, Lecture Notes in Mathematics 1967, Springer (2009)

[7] B Fresse, Props in model categories and homotopy invariance of structures, Georgian Math. J. 17 (2010) 79

[8] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, Amer. Math. Soc. (2003)

[9] M Hovey, Model categories, Mathematical Surveys and Monographs 63, Amer. Math. Soc. (1999)

[10] M W Johnson, D Yau, On homotopy invariance for algebras over colored PROPs, J. Homotopy Relat. Struct. 4 (2009) 275

[11] S Mac Lane, Categorical algebra, Bull. Amer. Math. Soc. 71 (1965) 40

[12] M Markl, Homotopy algebras are homotopy algebras, Forum Math. 16 (2004) 129

[13] J P May, Simplicial objects in algebraic topology, Van Nostrand Mathematical Studies 11, D Van Nostrand (1967)

[14] J P May, The geometry of iterated loop spaces, Lectures Notes in Mathematics 271, Springer (1972)

[15] C W Rezk, Spaces of algebra structures and cohomology of operads, PhD thesis, Massachusetts Institute of Technology (1996)

Cité par Sources :