Monoid Objects

This post was machine-translated from the Korean original by Marvin (via Kimi). It may contain errors or awkward phrasing — the Korean original is the source of truth.

Monoid Objects

Now we can define monoid objects.

Definition 1 A monoid object in a monoidal category \((\mathcal{A},\otimes, I)\) consists of the following data:

• an object \(M\),
• multiplication \(\mu:M\otimes M \rightarrow M\),
• unit \(\eta:I \rightarrow M\).

These satisfy the following conditions.

• (Associativity)¹
• (Unit)

Any monoidal category \((\mathcal{A},\otimes, I)\) always has the monoid object \(I\). Also, if \(M\) is a monoid object in a symmetric monoidal category, then one can easily check that \(M\otimes M\) is also a monoid object.

Example 2 The following are all examples of monoid objects.

• In the cartesian monoidal category \(\Set\), a monoid object is a monoid in the usual sense.
• In \(\Top\), a monoid object is a topological monoid.
• For any commutative ring \(R\), a monoid object in \((\lMod{R},\otimes_R, R)\) is an associative unital \(R\)-algebra.
• For any commutative ring \(R\), a monoid object in \((\Ch(R),\otimes_R, R)\) is a differential graded \(R\)-algebra. Here the unit \(R\) is the chain complex with \(R\) in degree \(0\) and \(0\) in all other degrees.

We need to explain the above examples not from a categorical perspective, but in the algebraic language we already know.

For the first example, the fact that a monoid object \((M,\mu,\eta)\) in \(\Set\) can be thought of as an ordinary monoid means the following. The underlying set of the monoid \(M\) is \(M\), and an operation on \(M\) is defined via the multiplication \(\mu:M\times M \rightarrow M\). Meanwhile, since the terminal object in \(\Set\) is a singleton, the image of the unit \(\eta\) in \(M\) will be some element of \(M\), which can be regarded as the unit of the monoid. The second example can be explained similarly.

To examine the third example, it is best to first look at the symmetric monoidal category structure of \(\lMod{R}\). Unlike cartesian monoidal categories, the monoidal product in \(\lMod{R}\) is given by the tensor product rather than the categorical product, and thus the unit object is not a terminal object but \(R\). For the unitors, whenever an \(R\)-module \(M\) is given, \(\lambda_M\) is the isomorphism

\[\lambda_M: R\otimes M \rightarrow M;\quad r\otimes m\mapsto rm\]

determined by this formula, and similarly \(\rho_M\) is the uniquely determined \(R\)-linear map given by \(m\otimes r\mapsto rm\).

Every object of \(\lMod{R}\) already carries an additive structure. A monoid object \((M,\mu,\eta)\) in \(\lMod{R}\) can be understood as endowing \(M\), which already has an additive structure, with a multiplicative structure compatible with this addition, and in this way \(M\) becomes an \(R\)-algebra. Here, the compatibility of the additive and multiplicative structures—that is, the validity of distributivity and similar properties—follows from the existence of a one-to-one correspondence between arbitrary elements of \(M\otimes M \rightarrow M\) and \(R\)-bilinear maps from \(M\times M\) to \(M\).

The last thing remaining to give \(M\) a multiplicative structure is the identity element for this multiplication, and this information can be specified by \(\eta:R \rightarrow M\). Considering the left \(R\)-module structure defined on \(R\), the information encoded in \(\eta\) is exactly equivalent to \(\eta(1)\), and this element \(\eta(1)\in M\) serves as the identity element for the newly defined multiplication:

\[\mu(\eta(1)\otimes m)=\mu((\eta\otimes\id_M)(1\otimes m))=\lambda_M(1\otimes m)=m\]

and similarly, using the right unitor one can also show \(\mu(m\otimes\eta(1))=m\).

For any monoidal category \(\mathcal{A}\), we can also define morphisms between monoid objects defined over it, and thus we can consider the category of monoid objects. However, we will not pursue the definition of a monoid category in this direction.

Group Objects

As with the previous definition, we can define group objects. To do so, as was the case when defining monoid objects, we need to express each property of groups as a diagram. A group \((G, \mu, e,(-)^{-1})\) precisely satisfies the following conditions.

• \((G,\mu,e)\) is a monoid.

• \((-)^{-1}:G \rightarrow G\) satisfies, for all \(g\in G\), the equation

  \[\mu(g^{-1},g)=\mu(g,g^{-1})=e\]

However, there is a problem if we try to translate this into the language of monoidal categories. If we write the second condition as a diagram,

it should look as follows. Here \(e_G\) is the group homomorphism sending every element of \(G\) to the identity element of \(G\), and \((-1)^{-1}\times \id_G\) is the product of the two maps \((-)^{-1}:G \rightarrow G\) and \(\id_G:G \rightarrow G\). Of course one could add both pieces of data and call this a group object, but doing so would not be a good solution because, for example, the unit \(\eta:I \rightarrow G\) (as a monoid object) and the newly defined morphism \(e_G\) would be entirely unrelated.

But if the original category were not merely a monoidal category but a cartesian monoidal category, then all these problems would be neatly resolved. First, \(e_G\) is given by the composition

\[G\overset{\epsilon_G}{\longrightarrow}\{e\}\overset{\eta}{\longrightarrow}G\]

where \(\epsilon_G\) is the unique morphism from \(G\) to the terminal object \(\{e\}\), and \(\eta\) is the unit of \(G\) as a monoid object. Moreover, in a cartesian monoidal category the monoidal product is the categorical product, so the following diagram

shows that \((-1)^{-1}\times \id_G\) is well defined.

Definition 3 In a cartesian monoidal category \((\mathcal{A},\times, I)\), a group object consists of the following data:

• an object \(G\),
• multiplication \(\mu:G\otimes G \rightarrow G\),
• unit \(\eta:I \rightarrow G\),
• inverse \(\iota:G \rightarrow G\).

Let \(e_G\) denote the composition \(G\rightarrow I\overset{\eta}{\rightarrow}G\). These satisfy the following conditions.

• (Associativity) The following diagram commutes.
• (Unit element) The following diagram commutes.
• (Inverse element) The following diagram commutes.

Since Definition 3 started from a cartesian monoidal category, we were able to draw the diagrams as above using the universal property of the categorical product, omitting the associators and unitors. If we write them all out explicitly, the first two diagrams are exactly the conditions for a monoid object, and the last condition can be seen as the newly added one.

Example 4 The following are all group objects.

• In \(\Set\), a group object is a group.
• In \(\Top\), a group object is a topological group.
• In \(\Man^\infty\), a group object is a Lie group.
• In \(\Var\), a group object is an algebraic group.
• In \(\Sch\), a group object is a group scheme.
• In \(\Grp\), a group object is an abelian group.

Only the last example may look slightly less obvious, but this can be verified by checking that since the inverse \(\iota\) must be a group homomorphism, commutativity follows from this condition.

Hopf Monoid

Looking at what was needed when constructing Definition 3 above, what we needed was precisely the diagonal map \(\Delta: G \rightarrow G\otimes G\), the augmentation map \(G \rightarrow I\), and the inverse map \(I \rightarrow G\). Sorting out what is needed here, we can first define the following.

Definition 5 Let a monoidal category \((\mathcal{A},\otimes,I)\) be given. An object \(M\) of \(\mathcal{A}\) is called a comonoid if \(M\) is a monoid object in \(\mathcal{A}^\op\).

Unpacking this, the data constituting a comonoid are comultiplication \(\Delta: G \rightarrow G\otimes G\) and counit \(\epsilon:G \rightarrow I\), and these satisfy the dual versions of the two conditions in Definition 1.

Definition 6 Let a symmetric monoidal category \((\mathcal{A},\otimes,I)\) be given. Then \((M,\mu,\eta,\Delta,\epsilon)\) is called a bimonoid if the following hold.

• \((M,\mu,\eta)\) is a monoid.
• \((M,\Delta,\epsilon)\) is a comonoid.
• The comultiplication and counit are both monoid morphisms.

When a monoid object \(M\) is given, the role of the symmetry is important in order to give \(M\otimes M\) a monoid structure, so generally the notion of a bimonoid is also defined only in a symmetric monoidal category. We now define a Hopf monoid as follows.

Definition 7 In a symmetric monoidal category \((\mathcal{A},\otimes,I)\), \((H,\mu,\eta,\Delta,\epsilon,\iota)\) is called a Hopf monoid if \((H,\mu,\eta,\Delta,\epsilon)\) is a bimonoid and \(\iota\) satisfies the same condition as the last diagram in Definition 3.

To write the condition for \(\iota\) explicitly, one must translate all the diagrams given in Definition 3 into the information that a Hopf monoid carries; for example, one of the triangles is the following diagram

which can be written out explicitly, and similarly using \(\iota\otimes\id_H\) one obtains the other triangle.

Example 8 The following are all examples of Hopf monoids.

• Any monoid object in a cartesian monoidal category has a natural bimonoid structure, and thus every group object in a cartesian monoidal category is a Hopf monoid.
• In \(\Vect\), a Hopf monoid is a Hopf algebra.

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References

[nLab] nLab. Monoidal category. (Link)
[Rie] Emily Riehl. Category Theory in Context. Dover Publications, 2016.

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1. In the diagram for the associativity of a monoid that we examined for motivation in a previous post, \((M\times M)\times M\) and \(M\times(M\times M)\) were regarded as the same thing, so the diagram was a quadrilateral, but here \((M\otimes M)\otimes M\) and \(M\otimes(M\otimes M)\) are different objects, so it becomes a pentagon. ↩