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 ${\displaystyle Ar(C)}$  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]

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.

• Volodin space
• Cotriple homology

• Lurie, J. "Higher Algebra" (PDF). last updated August 2017
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• 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.

• 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.