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

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.


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
Full Text: DOI arXiv Link