Algebraic theories in homotopy theory

By Bernard Badzioch. 2001

77 citations recorded by the source corpus.

It is well known in homotopy theory that given a loop space X one can always find a simplicial group G weakly equivalent to X, such that the weak equivalence can be realized by maps preserving multiplication. It is also known that loop spaces are not the only class of spaces for which a statement of this kind holds; for example, any Aoo-space is weakly equivalent to a simplicial monoid and every Eilenberg-Mac Lane space K(G, n) with G an abelian group is equivalent to a simplicial abelian monoid. Results like this suggest that there might be some general principle comparing homotopy structures on a space to algebraic structures. Our aim in this paper is to show that there is, in fact, such a principle. To make this precise we need a few definitions.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/3085e9a8524206b5ba35cc78106aefb948f823f2

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Citations in the corpus

Homotopy Theory of Higher Categories: THE MODEL STRUCTURE (2010, methodology): “The homotopical analogue, whose origins go back to the various theories of H-spaces and loop spaces, was considered by Badzioch in [5], and his treatment was used by Bergner for Segal categories in [34] [39], and also generalized to models in simplicial categories in [37].”
Every homotopy theory of simplicial algebras admits a proper model (2000, methodology): “l category structure on the category of simplicial algebras over any I-sorted theory Tbased on simplicial sets. The model category structure we construct coincides with those constructed in [Sch] and [Bad00]. 7.1. Closed model category structure. Let Tbe an I-sorted theory over sS. Recall that there”
THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES (2005, methodology): “Consider the category I [1] which consists of two objects x and y and exactly two non-identity maps which are inverse to one another, x ! y and y ! x.”
RESOLUTION OF COLOURED OPERADS AND RECTIFICATION OF HOMOTOPY ALGEBRAS (2005, unknown): “Notes Math.”