nLab Grothendieck group

Idea

The term Grothendieck group has two closely related meanings:

• Grothendieck group of commutative monoids

• Grothendieck group of stable infinity-categories

In its restricted sense the Grothendieck 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 such that the class of $(n_+, n_-)$ corresponds to the difference $n_+ - n_-$.

A vaguely similar procedure applied to isomorphism classes of objects in a Quillen exact category computes a group that is called the algebraic K-theory of that category. 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

• abelian category

and a

• Quillen exact category

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

• The Grothendieck group of the Quillen exact category of vector bundles on a space $X$ is called the topological K-theory of $X$.

  Notice that vector bundles do not form an abelian category.

• The Grothendieck group of the category of finite-dimensional complex-linear representations of a group is called its representation ring.

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.

Related concepts

• group completion

• K-theory

• Eilenberg swindle

References

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

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

• Mikhail Khovanov (notes by You Qi), §4 in: Introduction to categorification, lecture notes, Columbia University (2010, 2020) [web, web, full:pdf]

See also

• The Grothendieck Construction (UCSB ITP Seminar)

• Wikipedia, Grothendieck group