K-theory of a category

Concept in algebra

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 Ki(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 Ki(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 Ki(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[edit]

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[edit]

The arrow category A r ( C ) {\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  [ n ] = { 0 < 1 < 2 < ⋯ < n } {\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  S n C {\displaystyle S_{n}C} be a category whose objects are functors  f : A r [ n ] -> C {\displaystyle f:Ar[n]\to C} such that, for  i ≤ j ≤ k {\displaystyle i\leq j\leq k} ,  f ( i = i ) = ∗ {\displaystyle f(i=i)=*} ,  f ( i ≤ j ) -> f ( i ≤ k ) {\displaystyle f(i\leq j)\to f(i\leq k)} is a cofibration, and  f ( j ≤ k ) {\displaystyle f(j\leq k)} is the pushout of  f ( i ≤ j ) -> f ( i ≤ k ) {\displaystyle f(i\leq j)\to f(i\leq k)} and  f ( i ≤ j ) -> f ( j = j ) = ∗ {\displaystyle f(i\leq j)\to f(j=j)=*} . The category  S n C {\displaystyle 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 ⋯ S C {\displaystyle S^{(m)}C=S\cdots SC} . This sequence is a spectrum called the K-theory spectrum of C.

## The additivity theorem[edit]

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  E ( C ) {\displaystyle {\mathcal {E}}(C)} has as objects the sequences  A ↣ B ↠ A ′ {\displaystyle A\rightarrowtail B\twoheadrightarrow A'} in C, where the first map is a cofibration, and  B ↠ A ′ {\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 [ A ↣ B ↠ A ′ ] ↦ ( A , A ′ ) {\displaystyle [A\rightarrowtail B\twoheadrightarrow A']\mapsto (A,A')} from  E ( C ) {\displaystyle {\mathcal {E}}(C)} to C × C respects it. The additivity theorem says that the induced map on K-theory spaces  K ( E ( C ) ) -> K ( C ) × K ( C ) {\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 C ≅ ⟨ C 1 , C 2 ⟩ {\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]

## Category of finite sets[edit]

Consider the category of pointed finite sets. This category has an object  k + = { 0 , 1 , … , k } {\textstyle k_{+}=\\{0,1,\ldots ,k\\}} for every natural number k, and the morphisms in this category are the functions  f : m + -> n + {\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[edit]

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[edit]

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[edit]

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[edit]

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

## See also[edit]

* Volodin space
* Cotriple homology

## Notes[edit]

1. ^ a b 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.
2. ^ *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.
3. ^ Boyarchenko, Mitya (4 November 2007). "K-theory of a Waldhausen category as a symmetric spectrum" (PDF).
4. ^ a b Dundas, Bjørn Ian; Goodwillie, Thomas G.; McCarthy, Randy (2012-09-06). The Local Structure of Algebraic K-Theory. Springer Science & Business Media. ISBN 9781447143932.
5. ^ Staffeldt, Ross (1989). "On fundamental theorems of algebraic K-theory". K-theory. 2 (4): 511–532. doi:10.1007/bf00533280.
6. ^ 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.
7. ^ 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)
8. ^ "K-theory in nLab". ncatlab.org. Retrieved 22 August 2017.
9. ^ 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.

## References[edit]

* 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.

## Further reading[edit]

* 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. Lecture Notes in Mathematics. Vol. 2244\. arXiv:1212.3563. doi:10.1007/978-3-030-27124-4. ISBN 978-3-030-27122-0. S2CID 117874919.