Global algebraic K‐theory

Abstract

We introduce a global equivariant refinement of algebraic K‐theory; here ‘global equivariant’ refers to simultaneous and compatible actions of all finite groups. Our construction turns a specific kind of categorical input data into a global Ω$\Omega$‐spectrum that keeps track of genuine G$G$‐equivariant infinite loop spaces, for all finite groups G$G$. The resulting global algebraic K‐theory spectrum is a rigid way of packaging the representation K‐theory, or ‘Swan K‐theory’ into one highly structured object.

Genuine versus naïve symmetric monoidal $G$-categories

“…As mentioned without proof in [39,Remark 4.20], parsummable categories can be identified with tame «-algebras, also see [36,Theorem A.13] for the corresponding Set-level statement. Using Theorem 5.15 we can now easily give a full proof of this as well as of its simplicial counterpart, for which we begin with the following observation.…”

“…In fact, we develop the whole theory both for algebras in simplicial sets as well as algebras in categories. As an upshot, this then allows us to give a new model for the category theory of genuine symmetric monoidal G-categories-which unlike the respective homotopy theories differs from the one for naïve symmetric monoidal G-categories-in terms of the G-parsummable categories studied in [23,39]. These represent a rather different approach to "coherent commutativity," similar to the "ultra-commutative" philosophy of [35,38]; somewhat loosely speaking, we can think of them as G-categories equipped with a strictly equivariant, unital, associative, and commutative, but only partially defined operation.…”

“…This is in turn immediate from universality of K (also see [20,Example 1.2.35]). Now we are ready to define parsummable categories as first introduced in [39] as well as their simplicial counterpart, the parsummable simplicial sets [23]. One can show that C D is again tame and that defines a subfunctor of the Cartesian product.…”

“…One can show that C D is again tame and that defines a subfunctor of the Cartesian product. The usual coherence isomorphisms of the Cartesian symmetric monoidal structure then restrict to make the monoidal product of a preferred symmetric monoidal structure on EM-Cat , see [39,Proposition 2.35]. Taking diagonal G-actions this then more generally provides a symmetric monoidal structure on EM-G -Cat for every (discrete) group G. Definition 5.9.…”

“…For G D 1 and O D E † the above in particular yields an equivalence between permutative categories and parsummable categories, both viewed with respect to the underlying equivalences of categories. On the other hand, we previously proved in [21] that a specific functor ˆconstructed by Schwede [39,Construction 11.1] from permutative to parsummable categories descends to an equivalence of associated quasi-categories. This induced functor is in fact inverse to the above equivalence for abstract reasons: namely, both ˆand the above equivalence preserve underlying categories in the sense that they come with an equivalence forget ı ˆ' forget and similarly for the above composition.…”