K-theory of a category

In algebraic K-theory, the K-theory of a category is a set of functors $K_{i}$ from the category to Abelian groups satisfying some naturalness conditions. For instance, given a commutative ring $R$, one may construct a symmetric monoidal category of projective modules using the tensor product of modules. Then the $K_{0}$-theory of the ring is the Grothendieck group of its associated category.

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.

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.

S-construction

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. According to Waldhausen, the "S" was chosen to stand for Graeme B. Segal.

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.

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.

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.

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.

See also

• Volodin space
• Cotriple homology

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.
• Carlsson, Gunnar (2005). Friedlander, Eric M.; Grayson, Daniel R. (eds.). Handbook of K-Theory (PDF). Springer Berlin Heidelberg. pp. 3–37. doi:10.1007/978-3-540-27855-9_1. ISBN 9783540230199.
• Thomason, Robert W. (1979). "First quadrant spectral sequences in algebraic K-theory". Algebraic Topology Aarhus 1978 (PDF). Springer. pp. 332–355.

Further reading

• 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.

For the recent ∞-category approach, see

• 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.
• Dyckerhoff, Tobias; Kapranov, Mikhail (2012-12-14). "Higher Segal spaces I". arXiv:1212.3563 [math].

External links

• http://ncatlab.org/nlab/show/Waldhausen+S-construction