K-theory of a category (original) (raw)

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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[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

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

The additivity theorem

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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)} {\displaystyle {\mathcal {E}}(C)} has as objects the sequences A ↣ B ↠ A ′ {\displaystyle A\rightarrowtail B\twoheadrightarrow A'} {\displaystyle A\rightarrowtail B\twoheadrightarrow A'} in C, where the first map is a cofibration, and B ↠ A ′ {\displaystyle B\twoheadrightarrow 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')} {\displaystyle [A\rightarrowtail B\twoheadrightarrow A']\mapsto (A,A')} from E ( C ) {\displaystyle {\mathcal {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)} {\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 } {\displaystyle C\cong \langle C_{1},C_{2}\rangle }. Then the map of K-theory spectra K(C) → K(_C_1) ⊕ K(_C_2) 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

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Consider the category of pointed finite sets. This category has an object k + = { 0 , 1 , … , k } {\textstyle k_{+}=\{0,1,\ldots ,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_{+}} {\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]

Group completion method

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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

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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

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

  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](/wiki/Template:Cite%5Fjournal "Template: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.

For the recent ∞-category approach, see