K-theory of a category

Last updated June 24, 2024

In algebraic K-theory, the K-theory of a category C (usually equipped with some kind of additional data) is a sequence of abelian groups K_i(C) associated to it. If C is an abelian category, there is no need for extra data, but in general it only makes sense to speak of K-theory after specifying on C a structure of an exact category, or of a Waldhausen category, or of a dg-category, or possibly some other variants. Thus, there are several constructions of those groups, corresponding to various kinds of structures put on C. Traditionally, the K-theory of C is defined to be the result of a suitable construction, but in some contexts there are more conceptual definitions. For instance, the K-theory is a 'universal additive invariant' of dg-categories^[1] and small stable ∞-categories.^[2]

The motivation for this notion comes from algebraic K-theory of rings. For a ring R Daniel Quillen in Quillen (1973) introduced two equivalent ways to find the higher K-theory. The plus construction expresses K_i(R) in terms of R directly, but it's hard to prove properties of the result, including basic ones like functoriality. The other way is to consider the exact category of projective modules over R and to set K_i(R) to be the K-theory of that category, defined using the Q-construction. This approach proved to be more useful, and could be applied to other exact categories as well. Later Friedhelm Waldhausen in Waldhausen (1985) extended the notion of K-theory even further, to very different kinds of categories, including the category of topological spaces.

K-theory of Waldhausen categories

In algebra, the S-construction is a construction in algebraic K-theory that produces a model that can be used to define higher K-groups. It is due to Friedhelm Waldhausen and concerns a category with cofibrations and weak equivalences; such a category is called a Waldhausen category and generalizes Quillen's exact category. A cofibration can be thought of as analogous to a monomorphism, and a category with cofibrations is one in which, roughly speaking, monomorphisms are stable under pushouts.^[3] According to Waldhausen, the "S" was chosen to stand for Graeme B. Segal.^[4]

Unlike the Q-construction, which produces a topological space, the S-construction produces a simplicial set.

Details

The arrow category $Ar(C)$ of a category C is a category whose objects are morphisms in C and whose morphisms are squares in C. Let a finite ordered set $[n]=\{0<1<2<\cdots <n\}$ be viewed as a category in the usual way.

Let C be a category with cofibrations and let $S_{n}C$ be a category whose objects are functors $f:Ar[n]\to C$ such that, for $i\leq j\leq k$ , $f(i=i)=*$ , $f(i\leq j)\to f(i\leq k)$ is a cofibration, and $f(j\leq k)$ is the pushout of $f(i\leq j)\to f(i\leq k)$ and $f(i\leq j)\to f(j=j)=*$ . The category $S_{n}C$ defined in this manner is itself a category with cofibrations. One can therefore iterate the construction, forming the sequence $S^{(m)}C=S\cdots SC$ . This sequence is a spectrum called the K-theory spectrum of C.

The additivity theorem

Most basic properties of algebraic K-theory of categories are consequences of the following important theorem.^[5] There are versions of it in all available settings. Here's a statement for Waldhausen categories. Notably, it's used to show that the sequence of spaces obtained by the iterated S-construction is an Ω-spectrum.

Let C be a Waldhausen category. The category of extensions ${\mathcal {E}}(C)$ has as objects the sequences $A\rightarrowtail B\twoheadrightarrow A'$ in C, where the first map is a cofibration, and $B\twoheadrightarrow A'$ is a quotient map, i.e. a pushout of the first one along the zero map A → 0. This category has a natural Waldhausen structure, and the forgetful functor $[A\rightarrowtail B\twoheadrightarrow A']\mapsto (A,A')$ from ${\mathcal {E}}(C)$ to C × C respects it. The additivity theorem says that the induced map on K-theory spaces $K({\mathcal {E}}(C))\to K(C)\times K(C)$ is a homotopy equivalence.^[6]

For dg-categories the statement is similar. Let C be a small pretriangulated dg-category with a semiorthogonal decomposition $C\cong \langle C_{1},C_{2}\rangle$ . Then the map of K-theory spectra K(C) → K(C₁) ⊕ K(C₂) is a homotopy equivalence.^[7] In fact, K-theory is a universal functor satisfying this additivity property and Morita invariance.^[1]

Category of finite sets

Consider the category of pointed finite sets. This category has an object ${\textstyle k_{+}=\{0,1,\ldots ,k\}}$ for every natural number k, and the morphisms in this category are the functions ${\textstyle f:m_{+}\to n_{+}}$ which preserve the zero element. A theorem of Barratt, Priddy and Quillen says that the algebraic K-theory of this category is a sphere spectrum.^[4]

Miscellaneous

More generally in abstract category theory, the K-theory of a category is a type of decategorification in which a set is created from an equivalence class of objects in a stable (∞,1)-category, where the elements of the set inherit an Abelian group structure from the exact sequences in the category.^[8]

Group completion method

The Grothendieck group construction is a functor from the category of rings to the category of abelian groups. The higher K-theory should then be a functor from the category of rings but to the category of higher objects such as simplicial abelian groups.

Topological Hochschild homology

Waldhausen introduced the idea of a trace map from the algebraic K-theory of a ring to its Hochschild homology; by way of this map, information can be obtained about the K-theory from the Hochschild homology. Bökstedt factorized this trace map, leading to the idea of a functor known as the topological Hochschild homology of the ring's Eilenberg–MacLane spectrum.^[9]

K-theory of a simplicial ring

If R is a constant simplicial ring, then this is the same thing as K-theory of a ring.

Notes

1 2 Tabuada, Goncalo (2008). "Higher K-theory via universal invariants". Duke Mathematical Journal. 145 (1): 121–206. arXiv: 0706.2420 . doi:10.1215/00127094-2008-049. S2CID 8886393.
↑
- Blumberg, Andrew J; Gepner, David; Tabuada, Gonçalo (2013-04-18). "A universal characterization of higher algebraic K-theory". Geometry & Topology. 17 (2): 733–838. arXiv: 1001.2282 . doi:10.2140/gt.2013.17.733. ISSN 1364-0380. S2CID 115177650.
↑ Boyarchenko, Mitya (4 November 2007). "K-theory of a Waldhausen category as a symmetric spectrum" (PDF).
1 2 Dundas, Bjørn Ian; Goodwillie, Thomas G.; McCarthy, Randy (2012-09-06). The Local Structure of Algebraic K-Theory. Springer Science & Business Media. ISBN 9781447143932.
↑ Staffeldt, Ross (1989). "On fundamental theorems of algebraic K-theory". K-theory. 2 (4): 511–532. doi:10.1007/bf00533280.
↑ Weibel, Charles (2013). "Chapter V: The Fundamental Theorems of higher K-theory". The K-book: an introduction to algebraic K-theory. Graduate Studies in Mathematics. Vol. 145. AMS.
↑ Tabuada, Gonçalo (2005). "Invariants additifs de dg-catégories". International Mathematics Research Notices. 2005 (53): 3309–3339. arXiv: math/0507227 . Bibcode:2005math......7227T. doi:10.1155/IMRN.2005.3309. S2CID 119162782.{{cite journal}}: CS1 maint: unflagged free DOI (link)
↑ "K-theory in nLab". ncatlab.org. Retrieved 22 August 2017.
↑ Schwänzl, R.; Vogt, R. M.; Waldhausen, F. (October 2000). "Topological Hochschild Homology". Journal of the London Mathematical Society. 62 (2): 345–356. CiteSeerX 10.1.1.1020.4419 . doi:10.1112/s0024610700008929. ISSN 1469-7750. S2CID 122754654.

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References

Lurie, J. "Higher Algebra" (PDF). last updated August 2017
Toën, B.; Vezzosi, G. (2004). "A remark on K-theory and S-categories". Topology. 43 (4): 765–791. arXiv: math/0210125 . doi:10.1016/j.top.2003.10.008. S2CID 14744110.
Carlsson, Gunnar (2005). "Deloopings in Algebraic K-Theory" (PDF). In Friedlander, Eric M.; Grayson, Daniel R. (eds.). Handbook of K-Theory. Springer Berlin Heidelberg. pp. 3–37. doi:10.1007/978-3-540-27855-9_1. ISBN 9783540230199. S2CID 16536324.
Quillen, Daniel (1973), "Higher algebraic K-theory. I", Algebraic K-theory, I: Higher K-theories (Proc. Conf., Battelle Memorial Inst., Seattle, Wash., 1972), Lecture Notes in Math, vol. 341, Berlin, New York: Springer-Verlag, pp. 85–147, doi:10.1007/BFb0067053, ISBN 978-3-540-06434-3, MR 0338129
Waldhausen, Friedhelm (1985). "Algebraic K-theory of spaces". Algebraic and Geometric Topology. Lecture Notes in Mathematics. Vol. 1126. pp. 318–419. doi:10.1007/BFb0074449. ISBN 978-3-540-15235-4.
Thomason, Robert W. (1979). "First quadrant spectral sequences in algebraic K-theory" (PDF). Algebraic Topology Aarhus 1978. Springer. pp. 332–355.
Blumberg, Andrew J; Gepner, David; Tabuada, Gonçalo (2013-04-18). "A universal characterization of higher algebraic K-theory". Geometry & Topology. 17 (2): 733–838. arXiv: 1001.2282 . doi:10.2140/gt.2013.17.733. ISSN 1364-0380. S2CID 115177650.