K-theory of a category

Short description: 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 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 and small stable ∞-categories.

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. 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 [math]\displaystyle{ Ar(C) }[/math] 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 [math]\displaystyle{ [n] = \{ 0 \lt 1 \lt 2 \lt \cdots \lt n \} }[/math] be viewed as a category in the usual way.

Let C be a category with cofibrations and let [math]\displaystyle{ S_n C }[/math] be a category whose objects are functors [math]\displaystyle{ f: Ar[n] \to C }[/math] such that, for [math]\displaystyle{ i \le j \le k }[/math], [math]\displaystyle{ f(i = i) = * }[/math], [math]\displaystyle{ f(i \le j) \to f(i \le k) }[/math] is a cofibration, and [math]\displaystyle{ f(j \le k) }[/math] is the pushout of [math]\displaystyle{ f(i \le j) \to f(i \le k) }[/math] and [math]\displaystyle{ f(i \le j) \to f(j = j) = * }[/math]. The category [math]\displaystyle{ S_n C }[/math] defined in this manner is itself a category with cofibrations. One can therefore iterate the construction, forming the sequence[math]\displaystyle{ S^{(m)}C = S \cdots SC }[/math]. 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. 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 [math]\displaystyle{ \mathcal{E}(C) }[/math] has as objects the sequences [math]\displaystyle{ A \rightarrowtail B \twoheadrightarrow A' }[/math] in C, where the first map is a cofibration, and [math]\displaystyle{ B \twoheadrightarrow A' }[/math] 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 [math]\displaystyle{ [ A \rightarrowtail B \twoheadrightarrow A' ] \mapsto (A, A') }[/math] from [math]\displaystyle{ \mathcal{E}(C) }[/math] to C × C respects it. The additivity theorem says that the induced map on K-theory spaces [math]\displaystyle{ K(\mathcal{E}(C)) \to K(C) \times K(C) }[/math] is a homotopy equivalence.

For dg-categories the statement is similar. Let C be a small pretriangulated dg-category with a semiorthogonal decomposition [math]\displaystyle{ C \cong \langle C_1, C_2 \rangle }[/math]. Then the map of K-theory spectra K(C) → K(C₁) ⊕ K(C₂) is a homotopy equivalence. In fact, K-theory is a universal functor satisfying this additivity property and Morita invariance.

Category of finite sets

Consider the category of pointed finite sets. This category has an object [math]\displaystyle{ k_+ = \{0, 1, \ldots, k\} }[/math] for every natural number k, and the morphisms in this category are the functions [math]\displaystyle{ f : m_+ \to n_+ }[/math] 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.

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.

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.

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

• J. Lurie, Higher Algebra, last updated August 2017
• Toën, B.; Vezzosi, G. (2004). "A remark on K-theory and S-categories". Topology 43 (4): 765–791. doi:10.1016/j.top.2003.10.008. 
• Carlsson, Gunnar (2005). "Deloopings in Algebraic K-Theory". in Friedlander, Eric M. (in en). Handbook of K-Theory. Springer Berlin Heidelberg. pp. 3–37. doi:10.1007/978-3-540-27855-9_1. ISBN 9783540230199. http://math.stanford.edu/~gunnar/handbook.two.pdf. 
• 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, 341, Berlin, New York: Springer-Verlag, pp. 85–147, doi:10.1007/BFb0067053, ISBN 978-3-540-06434-3 
• Waldhausen, Friedhelm (1985). "Algebraic K-theory of spaces". Algebraic and Geometric Topology. Lecture Notes in Mathematics. 1126. pp. 318–419. doi:10.1007/BFb0074449. ISBN 978-3-540-15235-4. https://pub.uni-bielefeld.de/record/1782197. 
• Thomason, Robert W. (1979). "First quadrant spectral sequences in algebraic K-theory". Algebraic Topology Aarhus 1978. Springer. pp. 332–355. http://www.maths.ed.ac.uk/~aar/papers/thomason1.pdf. 
• Blumberg, Andrew J; Gepner, David; Tabuada, Gonçalo (2013-04-18). "A universal characterization of higher algebraic K-theory". Geometry & Topology 17 (2): 733–838. doi:10.2140/gt.2013.17.733. ISSN 1364-0380. 

Further reading

• Geisser, Thomas (2005). "The cyclotomic trace map and values of zeta functions" (in en). Algebra and Number Theory. Hindustan Book Agency, Gurgaon. pp. 211–225. 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. 2244. doi:10.1007/978-3-030-27124-4. ISBN 978-3-030-27122-0.