K-theory of a category

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.

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.

The arrow category ${\displaystyle Ar(C)}$ 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 ${\displaystyle [n]=\{0<1<2<\cdots <n\}}$ be viewed as a category in the usual way.

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

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 ${\displaystyle {\mathcal {E}}(C)}$ has as objects the sequences ${\displaystyle A\rightarrowtail B\twoheadrightarrow A'}$ in C, where the first map is a cofibration, and ${\displaystyle B\twoheadrightarrow A'}$ 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 ${\displaystyle [A\rightarrowtail B\twoheadrightarrow A']\mapsto (A,A')}$ from ${\displaystyle {\mathcal {E}}(C)}$ to C × C respects it. The additivity theorem says that the induced map on K-theory spaces ${\displaystyle K({\mathcal {E}}(C))\to K(C)\times K(C)}$ is a homotopy equivalence.^[6]

For dg-categories the statement is similar. Let C be a small pretriangulated dg-category with a semiorthogonal decomposition ${\displaystyle C\cong \langle C_{1},C_{2}\rangle }$. 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]

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]

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]

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.

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]

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

• Volodin space
• Cotriple homology

• 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. Berlin, Heidelberg: Springer. 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. Lecture Notes in Mathematics. Vol. 763. Springer. pp. 332–355. doi:10.1007/BFb0088093. ISBN 978-3-540-09721-1.
• 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.

• Geisser, Thomas (2005). "The cyclotomic trace map and values of zeta functions". Algebra and Number Theory. Hindustan Book Agency, Gurgaon. pp. 211–225. arXiv:math/0406547. doi:10.1007/978-93-86279-23-1_14. ISBN 978-81-85931-57-9.

For the recent ∞-category approach, see

• Dyckerhoff, Tobias; Kapranov, Mikhail (2019). Higher Segal spaces I. Lecture Notes in Mathematics. Vol. 2244. Cham: Springer. arXiv:1212.3563. doi:10.1007/978-3-030-27124-4. ISBN 978-3-030-27122-0. S2CID 117874919.