Operations in A-theory

Segal operations in the algebraic K-theory of topological spaces

“…Definition 3.11 (Compare 2.1, p.268, [5] and Section 4, [2]). Let Z be a simplicial object in a category D. Define a concatenation operation con : ∆ k → ∆.…”

“…Under the assumption that the category C satisfy the extension property for cofibrations, which has been verified for R f (X) and R f (X, Σ n , {all}) in lemma 3.16 part 2, one may identify the iterated G • -construction G n • C with Exact(Γ(−) n , C) according to lemma 3.16, part 1. Using the adjointness relation, or diagonals, we have • : F → Cat ∆ op following the recipe in [5]. Thus, on objects x ∈ Ob(F + ) and A ∈ ∆, put G x A C := Exact(Γ(A) x , C).…”

“…17 (Proposition 4.5 of[5]). The box-tensor operations and the diamond operations fulfill properties (E1) through (E5).Proof.…”

“…Proof. Part 1 is[5, 4.4 Remark, p.274]. For part 2, because we are working inside R(X) with simplicial sets, cofibrations are the injective maps.…”

$$G $$-theory of $$\mathbb F_1$$-algebras I: the equivariant Nishida problem

“…Grayson enlisted five conditions ((E1), · · · , (E5)), which ⊠ and ✸ k must satisfy so that one can construct the desired operations [31] (see also pages 270-271 of [32]). The author is unable to provide a proof of the following lemma and this makes the rest of the section conjectural.…”