Rigidification of algebras over multi-sorted theories

Bergner, Julia E¹

¹Kansas State University, 138 Cardwell Hall, Manhattan, KS 66506, USA

Algebraic and Geometric Topology, Tome 6 (2006) no. 4, pp. 1925-1955

Résumé

We define the notion of a multi-sorted algebraic theory, which is a generalization of an algebraic theory in which the objects are of different “sorts.” We prove a rigidification result for simplicial algebras over these theories, showing that there is a Quillen equivalence between a model category structure on the category of strict algebras over a multi-sorted theory and an appropriate model category structure on the category of functors from a multi-sorted theory to the category of simplicial sets. In the latter model structure, the fibrant objects are homotopy algebras over that theory. Our two main examples of strict algebras are operads in the category of simplicial sets and simplicial categories with a given set of objects.

Détail

DOI : 10.2140/agt.2006.6.1925

Keywords: algebraic theories, model categories, operads, simplicial categories

Bibliographie

[1] J Adámek, J Rosický, Locally presentable and accessible categories, London Mathematical Society Lecture Note Series 189, Cambridge University Press (1994)

[2] B Badzioch, Algebraic theories in homotopy theory, Ann. of Math. $(2)$ 155 (2002) 895

[3] J E Bergner, Simplicial monoids and Segal categories, to appear in Proc. Conf. Categories in Alg. Geom. and Math. Phys.

[4] J E Bergner, Three models for the homotopy theory of homotopy theories, PhD thesis

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

[6] F Borceux, Handbook of categorical algebra. 2, Encyclopedia of Mathematics and its Applications 51, Cambridge University Press (1994)

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

[8] P G Goerss, J F Jardine, Simplicial homotopy theory, Progress in Mathematics 174, Birkhäuser Verlag (1999)

[9] P S Hirschhorn, Model categories and their localizations, Mathematical Surveys and Monographs 99, American Mathematical Society (2003)

[10] M Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathematical Society (1999)

[11] M Johnson, R F C Walters, Algebra objects and algebra families for finite limit theories, J. Pure Appl. Algebra 83 (1992) 283

[12] F W Lawvere, Functorial semantics of algebraic theories, Proc. Nat. Acad. Sci. USA 50 (1963) 869

[13] S Mac Lane, Categories for the working mathematician, Graduate Texts in Mathematics 5, Springer (1998)

[14] M Markl, S Shnider, J Stasheff, Operads in algebra, topology and physics, Mathematical Surveys and Monographs 96, American Mathematical Society (2002)

[15] D G Quillen, Homotopical algebra, Lecture Notes in Mathematics 43, Springer (1967)

[16] C Rezk, Spaces of algebra structures and cohomology of operads, PhD thesis, MIT (1996)

[17] J Rosický, On homotopy varieties

[18] S Schwede, Stable homotopy of algebraic theories, Topology 40 (2001) 1