Group Operads and Homotopy Theory

Abstract

We introduce the classical theory of the interplay between group theory and topology into the context of operads and explore some applications to homotopy theory. We first propose a notion of a group operad and then develop a theory of group operads, extending the classical theories of groups, spaces with actions of groups, covering spaces and classifying spaces of groups. In particular, the fundamental groups of a topological operad is naturally a group operad and its higher homotopy groups are naturally operads with actions of its fundamental groups operad, and a topological K(π, 1) operad is characterized by and can be reconstructed from its fundamental groups operad. Two most important examples of group operads are the symmetric groups operad and the braid groups operad which provide group models for Ω ∞ Σ ∞ X (due to Barratt and Eccles) and Ω 2 Σ 2 X (due to Fiedorowicz) respectively. We combine the two models together to produce a free group model for the canonical stabilization Ω 2 Σ 2 X ֒→ Ω ∞ Σ ∞ X, in particular a free group model for its homotopy fibre.

Operads with general groups of equivariance, and some 2-categorical aspects of operads in Cat

“…Reading symmetric group elements as permutations from top to bottom, below is a pictorial representation of the final axiom for the map µ : • Our definition of an action operad is the same as that appearing in Wahl's thesis [22], but without the condition that each π n is surjective. It is also the same as that appearing in work of Zhang [23], although we prove later (see Lemma 1.21) that the condition e 1 = id in Zhang's definition follows from the rest of the axioms.…”

“…In Wahl's thesis [22], the essential definitions appear but not in complete generality as she requires a surjectivity condition. Zhang [23] also studies these notions 1 , once again in the context of homotopy theory, but requires the superfluous condition that e 1 = id (see Lemma 1.21).…”

A general method to construct cube-like categories and applications to homotopy theory

“…We know the sequence Σ = (Σ n ) n≥0 of symmetric groups form a nonsymmetric operad. The notion of group operads is a generalization of it, which appears in [37] and an axiomatic definition is given in [39]. A group operad is a (Set-valued) non-symmetric operad G = (G n ) n≥0 together with an operad map π :…”

“…Moreover, we can also considered the symmetrized version of them by group operads. Group operad G, which is discussed in [37] and [39], is a non-symmetric operad such that each G(n) has a group structure with certain compatibility with the operad operations. We can define G the cubical category with symmetry of G as follows: For a goup operad G, we can naturally regard it as a crossed ∆-group defined in [3].…”

Categories of operators and actions of group operads

“…The original definition of categories of operators, however, only covers the symmetric operads (or multicategories more generally). On the other hand, Zhang introduced group operads [22] as generalizations of the operad S of symmetric groups. As pointed out by Gurski [8], a group operad G may have an action on multicategories; namely, for a non-symmetric multicategory M, the action of G on M is defined to be a right action σ∈Sn M(a σ (1) .…”