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cohomology

# Contents

The way I first visualized a K-group was as a group of “classes of objects” of an abelian (or more generally, additive) category, such as coherent sheaves on an algebraic variety, or vector bundles, etc. I would presumably have called this group $C(X)$ ($X$ being a variety or any other kind of “space”), $C$ the initial letter of ‘class’, but my past in functional analysis may have prevented this, as $C(X)$ designates also the space of continuous functions on $X$ (when $X$ is a topological space). Thus, I reverted to $K$ instead of $C$, since my mother tongue is German, Class = Klasse (in German), and the sounds corresponding to $C$ and $K$ are the same.

## Idea

Given a stable (∞,1)-category $C$, its naive decategorification (its set of objects modulo the relation of equivalence) naturally inherits the structure of an abelian monoid from the biproducts in $C$. That is, we define $[a]+[b] = [a\oplus b]$. We can then further pass to the Grothendieck group, adding formal additive inverses to get an abelian group.

Note that such a biproduct sits in a split (co)fibration sequence $a\to a\oplus b \to b$. Often we would like to obtain an analogous additivity result for arbitrary (co)fibration sequences (a.k.a. distinguished triangles). This requires imposing the relation $[c] = [a] + [b]$ for any fibration sequence $a\to c\to b$, thereby passing to a quotient abelian group called $K_0(C)$, the K-group or Grothendieck group of $C$; see the latter entry for more details.

The “K” is chosen by Grothendieck for the German word Klasse for “class”. The K-group of $C$ is the group of equivalence classes of $C$: it is a group due to the existence of a notion of exact sequences in $C$.

K-theory starts with the study of these K-groups and their higher analogues $K_n(C)$, collectively denoted $K(C)$. Sometimes the K-groups themselves are called “K-theory”. One would say for instance: “$K(C)$ is the K-theory of $C$.”

More generally, there is a symmetric groupal ∞-groupoid $\mathbf{K}(C)$ – i.e. a connective spectrum – in between the decategorification from $C$ to $K(C)$, for which $K_0(C)$ is the set of connected components

$C \mapsto \mathbf{K}(C) \to \pi_0 \mathbf{K}(C) = K_0(C) \,.$

and more generally $K_n(C) = \pi_n \mathbf{K}(C)$.

In nice cases this is the degree 0 part of a non-connective spectrum which is then the K-theory spectrum of $C$. This is also called the Waldhausen K-theory of $C$.

## Special cases and models

Much of the literature on K-theory discusses constructions that model the above abstract setup in terms of model categories, or just their homotopy categories, often of the derived catgeories type and then often expressed in terms of the abelian category or more generally Quillen exact category from which the derived category is derived.

Only a subset of the structure on a model category is necessary in order to conveniently extract the K-groups of the presented stable (∞,1)-category. For that reason the axioms of a Waldhausen category have been devised to provide just the necessary convenient prerequisites to compute the K-groups of the (∞,1)-category presented by the underlying homotopical category.

• In particular, the K-group associated to the stable (∞,1)-category $Ch^b(A)$ of bounded chain complexes in an abelian category or exact category $A$ is often called the K-group of $A$ itself and just denoted

$K(A) := K(Ch^b(A)) \,.$

Most explicit constructions of K-theory spectra start with the data of an exact category, such as notably Quillen’s Q-construction and the Waldhausen S-construction.

• In particular if the exact category $A$ is that of vector bundles on a topological space $X$

$A = VectBund(X)$

the corresponding K-group is degree 0 topological K-theory. This was the original of the notion and the term K-theory.

## Definition

Recall that given a (∞,1)-category $C$, we may regard it as a complete Segal space $C_{\bullet,\bullet}$, a bisimplicial set. For instance if $C$ is originally given as a quasicategory then

$C_{\bullet,\bullet} : [n],[m] \mapsto Core(Func(\Delta^n,C))_{m} \,,$

where $Core(Func(\Delta^n,C))$ denotes the maximal Kan complex inside the (∞,1)-category of (∞,1)-functors from $\Delta^n$ to $C$.

###### Definition

For $\mathcal{C}$ an (∞,1)-category and $n \in \mathbb{N}$, write $Gap(\Delta^n, \mathcal{C})$ for the full sub-$\infty$-category on $Func(Arr(\Delta^n),\mathcal{C} )$ on those objects $F$ for which

• the diagonal $F(n,n)$ is inhabited by zero objects, for all $n$;

• all diagrams of the form

$\array{ F(i,j) &\to& F(i,k) \\ \downarrow && \downarrow \\ F(j,j) &\to& F(j,k) }$

is an (∞,1)-pushout.

###### Definition

Let $C$ be a stable (∞,1)-category. Then its Waldhausen K-theory

$\mathbf{K}(C) := \underset{\rightarrow}{\lim}_n Core(Gap(C^{\Delta^n}))$

is the geometric realization of/homotopy colimit of the degreewise core of the $Gap$, def. , of the corresponding complete Segal space (as a simplicial diagram of $\infty$-groupoids).

This construction is also conjectured in the last section of Toen-Vezzosi’s A remark on K-theory .

###### Remark

In the case that $C$ is the simplicial localization of a Waldhausen category $\bar C$ the explicit way to obtain this is the Waldhausen S-construction.

###### Remark

It should be true that with this definition we have an isomorphism of groups

$K(C) \simeq \pi_0 \mathbf{K}(C) \,.$
###### Remark

This Waldhausen/hocolim-construction gives the connective K-theory, taking values in connective spectra. The universal completion to functor that sends homotopy cofibers of stable (infinity,1)-categories to homotopy cofibers of spectra is the corresponding unconnective $\mathbb{K}$-functor.

There is a universal characterization of the construction of the $\mathbb{K}$-theory spectrum of a stable $(\infty,1)$-category $A$:

there is an $(\infty,1)$-functor

$U : (\infty,1)StabCat \to N$

to a stable $(\infty,1)$-category which is universal with the property that it respects colimits and exact sequences in a suitable way. Given any stable $(\infty,1)$-category $A$, its (connective or non-connective, depending on details) algebraic K-theory spectrum is the hom-object

$K(A) \simeq Hom(U(Sp), U(A)) \,,$

where $Sp$ denotes the stable $(\infty,1)$-category of compact spectra. (BGT)

It was in

• Bertrand Toen, Gabriele Vezzosi, A remark on K-theory and $S$-categories (arXiv).

that it was proven that the the Waldhausen S-construction of the K-theory spectrum depends precisely on the simplicial localization of the Waldhausen category, i.e. of the (∞,1)-category that it presents.

In view of this remark 11.4 in

interprets the construction of the K-theory spectrum as a natural operation of stable (∞,1)-categories, as described above.

The universal property of the $(\infty,1)$-categorical definition is studied in

The standard constructions of K-theory spectra from Quillen exact categories are discussed in detail in chapter 1 of

A useful introduction to the definition and computation of K-groups (with a little on K-spectra) is

• Charles Weibel, The K-book: An introduction to algebraic K-theory (web)