group cohomology, nonabelian group cohomology, Lie group cohomology
cohomology with constant coefficients / with a local system of coefficients
differential cohomology
To a permutative category $C$ is naturally associated a Gamma-space, hence a symmetric spectrum. The generalized (Eilenberg-Steenrod) cohomology theory represented by this is called the (algebraic) K-theory of (or represented by) $C$.
If the category is even a bipermutative category then the corresponding K-theory of a bipermutative category in addition has E-infinity ring structure, hence is a multiplicative cohomology theory.
For $\mathcal{C}$ a category, write ${\vert \mathcal{C}\vert} \in$ sSet $\simeq_{Qu}$ Top for its nerve/geometric realization. (Beware that this is often denoted $B C$, instead, but that notation clashes with that for delooping, which we also need in the following.)
For $\mathcal{C}$ a permutative category its nerve/geometric realization $\vert \mathcal{C} \vert$ is naturally a topological monoid (Quillen 70 see e.g. May 13, theorem 4.10), hence admits a bar construction/classifying space $B {\vert \mathcal{C}\vert}$. The loop space of that
being an ∞-group, may be regarded as the homotopy theoretic group completion of the topological monoid ${\vert \mathcal{C}\vert}$.
This is the degree-0 space in the algebraic K-theory spectrum $\mathbb{K}\mathcal{C}$ of the permutative category $\mathcal{C}$ (see e.g. May 13, def 4.11).
By (Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7) the operation $\Omega B (-)$ is the derived functor of group completion, so that this construction ought to be a model for the K-theory of a symmetric monoidal (∞,1)-category.
In particular, under decategorification, its group of connected components is the actual Grothendieck group $K(-)$ of the isomorphism classes of objects in $\mathcal{C}$:
(recalled e.g. in Bohmann-Osorno 14, p. 14).
In particular for $R$ a topological ring one considers $C$ a skeleton of the groupoid of (finitely generated) projective modules over $R$. Then the K-theory of $C$ is the algebraic K-theory of $R$ (e.g. May 13, p. 25)
Write $FinSet^{*/}$ for the category of pointed finite sets.
For $C$ a permutative category, there is naturally a functor
such that (…).
(Elmendorf-Mandell, theorem 4.2)
Accordingly, postcomposition with the nerve $N : Cat \to sSet$ produces from $C$ a Gamma-space $N \widebar C$. To this corresponds a spectrum
This is the K-theory spectrum of $C$.
The construction of K-theory spectra of permutative categories constitutes a multifunctor
between multicategories of permutative categories (under Deligne tensor product) and spectra (under smash product of spectra).
Hence for a multilinear functor of permutative categories
there is a compatibly induced morphism of K-spectra out of the smash product
This implies that the construction further extends to a 2-functor from the 2-category $PermCat Cat$ of enriched categories over the multicategory of permutative categories to that of enriched categories of spectra:
which applies $\mathbb{K}$ to each hom-object.
(May 80, theorem 1.6, Theorem 2.1, Elmendorf-Mandell 04, theorem 1.1, Guillou 10, Theorem 1.1
(ordinary algebraic K-theory)
For $R$ a commutative ring, let $\mathcal{C} = R Mod_{pr}$ its category of finitely generated projective modules regarded as a permutative category. Then
is the classical algebraic K-theory spectrum of the ring $R$.
(e.g. Elmendorf-Mandell 04, p. 10).
(stable cohomotopy is K-theory of FinSet)
Let $\mathcal{C} =$ FinSet be a skeleton of the category of finite sets, regarded as a permutative category. Then
is the sphere spectrum, hence represents the cohomology theory called stable cohomotopy.
(due to Segal 74, Prop. 3.5, see also Priddy 73)
(stable cohomotopy as algebraic K-theory over the field with one element)
Since (pointed) finite sets may be regarded as the modules over the “field with one element” $\mathbb{F}_1$ (see there),
one may read example in view of example as saying that stable cohomotopy is the algebraic K-theory of the field with one element:
This perspective is highlighted for instance in (Deitmar 06, p. 2, Guillot 06).
Daniel Quillen, Cohomology of groups, Proceedings of the international congress of mathematics 1970
Daniel Quillen, On the group completion of a simplicial monoid
Stewart Priddy, Transfer, symmetric groups, and stable homotopy theory, in Higher K-Theories, Springer, Berlin, Heidelberg, 1973. 244-255 (pdf)
Graeme Segal, Catgeories and cohomology theories, Topology vol 13 (1974) (pdf)
Peter May, The spectra associated to permutative categories, Topology 17 (1978) (pdf)
Peter May, $E_\infty$-Spaces, group completions, and permutative categories, in Graeme Segal (ed.) New Developments in Topology, Cambirdge University Press 2013 (pdf, doi:10.1017/CBO9780511662607.008)
Peter May, Pairings of categories and spectra, J. Pure Appl. Algebra, 19:299–346, 1980
William Dwyer, Daniel Kan, Simplicial localization of categories, Journal of pure and applied algebra 17 (1980) 267-284
Anthony Elmendorf, Michael Mandell, Permutative categories as a model of connective stable homotopy, in: Birgit Richter (ed.) Structured Ring spectra, Cambridge University Press (2004)
Anthony Elmendorf, Michael Mandell, Rings, modules and algebras in infinite loop space theory, Adv. in Math. 205 (2006), no. 1, 163-228 (arXiv:math/0403403)
Anthony Elmendorf, Michael Mandell, Permutative categories, multicategories, and algebraic K-theory, Algebraic & Geometric Topology 9 (2009) 2391-2441 (arXiv:0710.0082, euclid:1513797088)
Bertrand Guillou, Strictification of categories weakly enriched in symmetric monoidal categories, Theory Appl. Categ., 24:No. 20, 564–579, 2010.
The interpretation of stable cohomotopy as the algebraic K-theory over the field with one element is adopted in
Anton Deitmar, Remarks on zeta functions and K-theory over $\mathbb{F}_1$ (arXiv:math/0605429)
Pierre Guillot, Adams operations in cohomotopy (arXiv:0612327)
The generalization of K-theory of permutative categories to Mackey functors is discussed in
Generalization to equivariant stable homotopy theory and G-spectra is discussed in
Last revised on June 24, 2023 at 15:22:07. See the history of this page for a list of all contributions to it.