# nLab Grothendieck group

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

# Contents

## Idea

The term Grothendieck group has two closely related meanings:

In its restricted sense the Grothedieck group of a commutative monoid (i.e. of a commutative semi-group with unit) $A$ is a specific presentation of its group completion, given as a certain group structure on a quotient of the Cartesian product $A \times A$. This is such that applied to the additive monoid of natural numbers $\mathbb{N}$ it produces the additive group of integers $\mathbb{Z}$ represented by pairs of natural numbers $(n_+, n_-)$ subject to an equivalence relation which identifies them with their difference $n_+ - n_-$.

A vaguely similar procedure applied to isomorphism classes of certain Quillen exact categories happens to compute a group that is called the algebraic K-theory of these categories. See K-theory for some general abstract nonsense behind this.

Notably, the Grothendieck group completion of the decategorification of the category of topological vector bundles on some (compact Hausdorff) topological space $X$ produces the group known as the topological K-theory of $X$.

Given this one speaks more generally of the (algebraic) K-theory group of a suitable category (one presenting a stable (∞,1)-category in some way) as its Grothendieck group .

In that sense, the Grothendieck group of a ($\infty$-)category $C$ with a notion of cofibration sequences is the decategorification set $K(C)$ equipped with a notion of addition that is encoded in these homotopy exact sequences.

We now first state the definition of “Grothendieck group completion” – which is really just the free group completion of an abelian monoid – and then the definition of Grothendieck group in the sense of algebraic K-theory. Notice that a priori both concepts are entirely independent constructions on different entities. But in various special case both can be applied to specific objects so as to produce the same result.

## Of commutative monoids

Every abelian group is in particular a commutative monoid. The forgetful functor $U \colon Ab \to CMon$ from the category Ab to the category CMon has a left adjoint $F$ (the corresponding free functor), the group completion functor

$Ab \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} CMon$

The Grothendieck group construction on commutative monoids (def. below) is an explict presentation of this group completion functor.

For more details see at Grothendieck group of a commutative monoid.

###### Remark

The idea of the free group on an abelian monoid is a very simple algebraic idea that, at least for a cancellative monoid (so that the unit is monic and one can reasonably use the term ‘completion’) certainly predates Grothendieck. That the integers $\mathbb{Z}$ is the group completion of the natural numbers $\mathbb{N}$ goes back at least to Kronecker.

###### Definition

(Grothendieck group of a commutative monoid)

Let $(A,+)$ be a commutative monoid (i.e. a commutative semi-group).

On the Cartesian product of underlying sets $A \times A$ (the set of ordered pairs of elements in $A$), consider the equivalence relation

$\big( (a_+, a_-) \sim_1 (b_+, b_-) \big) \;\Leftrightarrow\; \left( \underset{k \in A}{\exists} \left( a_+ + b_- + k = b_+ + a_- + k \right) \right)$

or equivalently the equivalence relation

$\big( (a_+, a_-) \sim_2 (b_+, b_-) \big) \;\Leftrightarrow\; \left( \underset{k_1, k_2 \in A}{\exists} \left( (a_+ + k_1, a_- + k_1) = (b_+ + k_2, b_- + k_2) \right) \right) \,.$

Write

$G(A) \coloneqq (A \times A)/\sim$

for the set of equivalence classes under this equivalence relation. This inherits a binary operation

$+ \;\colon\; G(A) \times G(A) \longrightarrow G(A)$

by applying the addition in $A$ on representatives:

$[a_+,a_-] + [b_+,b_-] \coloneqq [ a_+ + b_+ , a_- + b_- ] \,.$

This defines the structure of an abelian group

$(G(A),+)$

and this is the Grothendieck group of $A$.

This comes with a canonical homomorphism of semigroups

$\array{ A &\overset{\phantom{A} \eta_A \phantom{A} }{\longrightarrow}& G(A) \\ a &\overset{\phantom{AAA}}{\mapsto}& [a,0] } \,.$

## Of stable $\infty$-categories

In this sense, a Grothendieck group is fundamentally something assigned to a stable (∞,1)-category. We start with the naive decategorification of $C$, i.e. the set of equivalence classes of objects, which inherits the structure of an abelian monoid. Then in addition to group-completing it as above, we add additional relations by the rule that for every fibration sequence

$A \to X \to B$

in $C$, the equivalence classes $[A]$, $[B]$ and $[X]$ must satisfy

$[X] = [A] + [B] \,.$

The result is also called the K-theory of $C$.

In particular, the additive inverse $-[A]$ of an element $[A]$ is the class of its loop space object $\Omega A$, or equivalently of its delooping $\mathbf{B} A$ called the suspension $\Sigma A$, since by definition the sequences

$\Omega A \to 0 \to A$

and

$A \to 0 \to \Sigma A$

are fibration sequence, so that

$[A] + [\Omega A] = 0$

and

$[A] + [\Sigma A] = 0 \,.$

But there are many ways to model a stable (∞,1)-category by an ordinary category. Essentially for each of these ways there is a seperate prescription for how to model the above general construction in terms of concrete 1-categorical constructions.

In particular from an

and a

one obtains the corresponding categories of chain complexes. These are stable (∞,1)-categories. Below we list presciptions for how to compute the Grothendieck/K-theory groups of these in terms of the underlying 1-categories.

Apart from the case of abelian categories, this requires some handle on the fibration sequences. A tool developed to handle exactly this for the purpose of computing Grothendieck/K-theory groups is the notion of a Waldhausen category. That provides the sufficient extra information to get a hand on the homotopy exact sequences.

### of an abelian category

Let $C$ be an abelian category. The Grothendieck group or algebraic K-theory group of $C$, denoted $K(C)$, is the abelian group generated by isomorphism classes of objects of $C$, with relations of the form

$[X] = [A] + [B]$

whenever there is a short exact sequence

$0 \to A \to X \to B \to 0$

### of a Quillen exact category

An exact category $C$ in the present sense is a full subcategory of an abelian category $\hat C$ such that the collection of all sequences $0 \to A \to X \to B \to 0$ in $C$ that are exact sequences in $\hat C$ has the property that for every exact sequence $A \to X \to B$ in $\hat C$ with $A$ and $B \in C$ also their “sum” $X$ is in $C$.

Given an exact category $C$ with the inherited notion of exact sequences this way, the definition of its Grothendieck group is as above.

### of a Waldhausen category

A Waldhausen category is a category with weak equivalences with an initial object – called $0$ – and equipped with the notion of auxiliary morphism called cofibrations. These satisfy some axioms which are such that the ordinary 1-categorical cokernel of a cofibration $A \hookrightarrow X$, i.e. the ordinary pushout

$\array{ A &\hookrightarrow& X \\ \downarrow && \downarrow \\ 0 &\to & B }$

computes the desired homotopy pushout. (This is exactly dual to the reasoning by which one computes homotopy pullbacks in a category of fibrant objects. See there for details.)

Therefore in a Waldhausen category a cofibration sequence is a pushout sequence

$A \hookrightarrow X \to B$

where the first morphism is a cofibration.

The Grothendieck/K-theory-group of the Waldhausen category $C$ is then, as before, on the decategorification $K(C)$ the abelian group structure given by

$[X] = [A] + [B]$

for all cofibration sequences as above.

### Examples

These two examples illustrate a general fact: the Grothendieck group of a monoidal abelian category inherits a ring structure from the tensor product in this category, and thus becomes a ring, called the Grothendieck ring. See also the general discussion at decategorification.

• Every Quillen exact category $C$ is canonically equipped with the structure of a Waldhausen category. The two different prescriptions for forming the Grothendieck group $K(C)$ of $C$ do coincide.

## References

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

section 2: The Grothendieck group $K_0$ (pdf)