What Are Groupification and Perfection of a Commutative Monoid?

For a commutative monoid, groupification has a standard meaning: it is the universal abelian group receiving a monoid homomorphism from the monoid. Perfection is less unambiguous. In characteristic $p$, it commonly means adjoining compatible $p$-th roots by iterating the Frobenius map. If a text uses “perfection” without naming a prime or defining the term, check its convention before assuming which construction it means.

The two constructions do different things. Groupification adds inverses; $p$-perfection adds roots. Neither should be confused with taking a quotient to make a monoid cancellative or with saturating a monoid.

Groupification: adjoining inverses

Write the operation of a commutative monoid $M$ additively, with identity $0$. Its groupification, often written $M^{\mathrm{gp}}$, consists of formal differences $a-b$, where $a,b\in M$. The difference $a-b$ is intended to act like $a+(-b)$, even if $M$ itself has no negatives.

One construction starts with pairs $(a,b)\in M\times M$. Declare

$$(a,b)\sim(c,d)\quad\text{if there is some }e\in M\text{ such that }a+d+e=c+b+e.$$

The groupification is the set of equivalence classes, with addition $[(a,b)]+[(c,d)]=[(a+c,b+d)]$. The monoid map is $a\mapsto[(a,0)]$. The extra $e$ in the equivalence relation matters when $M$ is not cancellative: it avoids assuming that equal terms can be cancelled.

The defining property is universal. For any abelian group $G$ and monoid homomorphism $f:M\to G$, there is a unique group homomorphism $\widetilde f:M^{\mathrm{gp}}\to G$ such that $\widetilde f(a-b)=f(a)-f(b)$. Thus every monoid map from $M$ to a group factors uniquely through its groupification.

Example: the natural numbers

Take $M=\mathbb N=\{0,1,2,\ldots\}$ under addition. Formal differences $a-b$ produce every integer, so

$$\mathbb N^{\mathrm{gp}}\cong\mathbb Z.$$

For example, $[(0,3)]$ represents $-3$. The map $\mathbb N\to\mathbb Z$ is injective because addition on $\mathbb N$ is cancellative.

In general, the map $M\to M^{\mathrm{gp}}$ need not be injective. Two elements $a,b\in M$ have the same image exactly when there is some $c\in M$ with $a+c=b+c$. Cancellation, the condition that $a+c=b+c$ always implies $a=b$, therefore guarantees injectivity. If cancellation fails, groupification identifies elements that become equal after adding the same element.

$p$-perfection: adjoining roots

For a fixed prime $p$, the $p$-folding endomorphism on an additive commutative monoid is

$$F:M\longrightarrow M,\qquad F(m)=pm.$$

In this additive-monoid setting, this map is also called the $p$-power or Frobenius endomorphism. Its direct limit under repeated applications of $F$ is called the $p$-perfection in settings that use this terminology:

$$M^{\mathrm{perf}}=\varinjlim\bigl(M\xrightarrow{F}M\xrightarrow{F}M\xrightarrow{F}\cdots\bigr).$$

In this limit, Frobenius becomes invertible, so elements have compatible $p$-th roots. The analogous multiplicative notation uses $F(m)=m^p$. For a ring of characteristic $p$, the usual Frobenius is $x\mapsto x^p$, not multiplication by $p$ in its additive monoid; ring perfection is formed using that ring endomorphism. This construction is especially familiar for rings and schemes in characteristic $p$; a monoid version uses the same direct-limit idea.

For example, begin with $\mathbb N$ under addition and use $F(n)=pn$. Its $p$-perfection can be represented by nonnegative rational numbers whose denominators are powers of $p$:

$$\mathbb N^{\mathrm{perf}}\cong\mathbb Z[1/p]_{\ge 0}.$$

The element $1$ acquires a compatible $p$-th root represented by $1/p$, then a further root $1/p^2$, and so on. Groupifying this perfected monoid gives $\mathbb Z[1/p]$.

How the constructions relate

Groupification is a left adjoint: it turns a commutative monoid into a group in the universal way. Left adjoints preserve colimits, including the direct limit used for $p$-perfection. Consequently, there is a natural isomorphism

$$(M^{\mathrm{perf}})^{\mathrm{gp}}\cong(M^{\mathrm{gp}})^{\mathrm{perf}}.$$

So for this meaning of perfection, you can first adjoin $p$-th roots and then groupify, or groupify and then adjoin roots; the result is the same up to natural isomorphism.

Common terminology traps

• Cancellative quotient: Identifying $a$ and $b$ when $a+c=b+c$ for some $c$ produces a cancellative version of the monoid. This is sometimes called cancellativization; it is not the same as adding inverses.
• Saturation: For a monoid embedded in a group, saturation typically adds elements whose positive integer multiples already lie in the monoid. That is different from $p$-perfection, which repeatedly adjoins roots for one fixed prime.
• Unspecified perfection: Without a prime, a characteristic, or an explicit definition, “perfection” alone may not determine a unique construction. State the convention being used, especially when comparing results from different sources.