Adding inverses to diagrams encoding algebraic structures. (English) Zbl 1154.55013

Bergner, Julia E.

Homology Homotopy Appl. 10, No. 2, 149-174 (2008); erratum ibid. 14, No. 1, 287-291 (2012).

The author showed in a previous work [J. E. Bergner, Simplicial monoids and Segal categories. Categories in algebra, geometry and mathematical physics. Conference and workshop in honor of Ross Street’s 60th birthday, Sydney and Canberra, Australia, July 11–16/July 18–21, 2005. Providence, RI: American Mathematical Society (AMS). Contemporary Mathematics 431, 59–83 (2007; Zbl 1134.18006)] that certain diagrams of spaces are simplicial monoids. In this paper, focusing on diagrams of spaces which give one of the spaces the structure of a monoid, the argument is modified to obtain a diagram encoding the structure of a simplicial group rather than a simplicial monoid. As the construction for simplicial monoids generalizes to the many object case of simplicial categories, models for simplicial groupoids can be obtained as well. Finally, Segal’s argument [G. Segal, Topology 13, 293–312 (1974; Zbl 0284.55016)] for a model for simplicial abelian monoids is modified to obtain a model for simplicial abelian groups.

Reviewer: J. Remedios (La Laguna)

MSC:

55U10	Simplicial sets and complexes in algebraic topology
18B40	Groupoids, semigroupoids, semigroups, groups (viewed as categories)
18C10	Theories (e.g., algebraic theories), structure, and semantics
55P35	Loop spaces

Keywords:

simplicial monoid; simplicial group; simplicial category

Citations:

Zbl 1134.18006; Zbl 0284.55016