group theory

categorification

# Contents

## Idea

The term Grothendieck group has a restricted and a more general meaning.

In its restricted sense the Grothendieck group of an abelian monoid $A$ is a specific presentation of its group completions, given as a certain group structure on a quotient of $A \times A$. This is such that applied to the additive monoid of natural numbers $\mathbb{N}$ it produces the additive group of integers $\mathbb{Z}$.

The same 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. There is a very simple very general nonsense behind this, that is described at K-theory.

Notably the Grothendieck group completion of the decategorification of the category of vector bundles on some 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.

## Definition

We 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.

### Group completion of a commutative monoid

Every abelian group is in particular a commutative monoid. There is a forgetful functor

$U \;\colon\; Ab \to CMon \,.$

To go the other way around we may ask for a left adjoint to this forgetful functor. This will take an commutative monoid $A$ and send it to an abelian group $G(A)$ with the property that for every abelian group $K$ with underlying abelian monoid $F(K)$ every morphism of monoids $A \to F(K)$ corresponds to a morphism of groups $G(A) \to K$.

This $G(A)$ is called the group completion or the Grothendieck group completion of $A$.

There are several ways to describe this. A popular one is to realize $G(A)$ as the quotient of the product abelian monoid $A \times A$ under the equivalence relation

$((a_1,b_1) \sim (a_2,b_2)) \Leftrightarrow \exists k : a_1 + b_2 + k = a_2 + b_1 + k$

###### 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 $\mathbb{Z}$ is the group completion of $\mathbb{N}$ goes back at least to Kronecker.

### $\infty$-Group completion

More generally, ∞-group completion

$K \;\colon\; CMon_\infty(\infty Grpd) \longrightarrow AbGrp_\infty(\infty Grpd)$

is the left adjoint (∞,1)-functor to the inclusion of abelian ∞-groups (connective spectra) into commutative ∞-monoids? in ∞Grpd.

Write

$\mathcal{K} \;\colon\; CMon_\infty(\infty Cat) \longrightarrow Spectra$

for the composite of this with the core functor from symmetric monoidal (∞,1)-categories and the inclusion of connective spectra into all spectra. This is the algebraic K-theory of symmetric monoidal (∞,1)-categories.

Fundamentally a Grothendieck group is something assigned to a stable (∞,1)-category. It is the group structure $+ : K(C)\times K(C) \to K(C)$ on the decategorification $K(C)$ of $C$ defined by the rule that for every fibration sequence

$A \to X \to B$

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

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

In particular the 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

$0 \to A \to X \to C \to 0$

### 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.

Urs Schreiber: the category of vector bundles is not abelian, but just Quillen exact. I added a clause to that effect above.

Generally, my feeling is that “Grothendieck group” refers to “Grothendieck group completion of a monoid”. The first definition at the Wikipedia entry on Gorthendieck group.

What is described here – the group defined by sequences – is the definition of the K-theory group in algebraic K-theory. This is described at the beginning of K-theory.

I’d think that the logic is rather that in some cases both concepts happen to coincide: the algebraic K-theory of the category of bounded complexes of vector bundles on a space happens to coincide with the Grothendieck group completion of the monoid of isomorphism classes of vector bundles.

My feeling is that the Wikipedia entry is a bit suboptimal in its second part, where it effectively calls algebraic K-theory the “Grothendieck group completion”. I’d think one shouldn’t do that in this generality.

Another thing in this context is the statement about the exact sequences. It is really the homotopy exact sequences namely the fibration sequences that count. Of course some of them can be computed by ordinary exact sequences if these are sufficiently cofibrant. That’s what the structure provided by a Waldhausen category structure provides: a realization of the homotopy exact sequences as pushouts of cofibration morphisms.

So in summary my feeling is: the entry in its current form actually tries to define algebraic K-theory and not the Grothendieck group concept, and that description with a few subtleties taken care of is currently at K-theory and should eventually be copied/moved to algebraic K-theory. The entry titled “Grothendieck group” should discuss the concept where from a monoid $A$ a group structure on $A \times A$ is defined by dividing out the equivalence relation $(a_1,b_1) \sim (a_2,b_2) \Leftrightarrow \exists k : a_1 + b_2 + k = a_2 + b_1 + k$.

But maybe I am mixed up. Let me know what you think.

Toby: Well, I'll say this: I always understood ‘Grothendieck group’ in the article text above, and I never understood ‘K-theory’. That may have been my fault, of course, and it may be telling that I was John's student.

Urs Schreiber: so Weibel in his book referenced below indeed says “Grothendieck group” synonomously for “algebraic K-theory group” and says just “group completion” (chapter to section 1) for what I suggested should properly be called “Grothendieck group completion”.

I added corresponding remarks to the beginning of the entry. And I created two subsections under “Definintion”. One for plain group completion. The other for algebraic K-theory groups.

Toby: 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 $\mathbb{Z}$ is the group completion of $\mathbb{N}$ goes back at least to Kronecker.

Urs Schreiber: I copied that last paragraph of yours into the section above. I still think that many people will say “Grothendieck group” for this group completion. But maybe I am wrong and we should branch off another page on “free group completion”.

Colin Tan: I suggest branching off another page on “free group completion” and including the noncommutative case in there.

## References

In

see

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