Rigidification of algebras over multi-sorted theories

By Julia E. Bergner. 2005

69 citations recorded by the source corpus.

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.

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

View Paper: https://www.semanticscholar.org/paper/0312829572cf94c1dc4b0cb55eb6d06680a5e03d

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Algebra and Number Theory

1. Carlos Simpson
2. Ieke Moerdijk
3. C. Berger
4. Julia E. Bergner
5. Denis-Charles Cisinski
6. Emily Riehl
7. Clark Barwick
8. Michael Shulman

In Geometry and Topology

1. Carlos Simpson
2. Ieke Moerdijk
3. C. Berger
4. Julia E. Bergner
5. Denis-Charles Cisinski
6. Emily Riehl
7. Michael Shulman
8. Clark Barwick

In Mathematical Physics

1. Carlos Simpson
2. Ieke Moerdijk
3. C. Berger
4. Denis-Charles Cisinski
5. Julia E. Bergner
6. Emily Riehl
7. Clark Barwick
8. Joachim Kock

Citations in the corpus

• RESOLUTION OF COLOURED OPERADS AND RECTIFICATION OF HOMOTOPY ALGEBRAS (2005, result): “For such an algebraic theory T, this gives rise to a somewhat dierent notion of homotopyT-algebra (not based on a resolution ofT), for which a rectification result has been proved in the context of simplicial sets by Badzioch [1] and Bergner [ 6 ].”
• 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].”
• THREE MODELS FOR THE HOMOTOPY THEORY OF HOMOTOPY THEORIES (2005, methodology): “This generalization uses the notion of multi-sorted algebraic theory [4].”