# nLab category of monoids

Contents

### Context

#### Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

#### Algebra

higher algebra

universal algebra

## Theorems

#### Monoid theory

monoid theory in algebra:

# Contents

## Definition

###### Definition

For $C$ a monoidal category, the category of monoids $Mon(C)$ in $C$ is the category whose

• objects are monoids in $C$;

• morphisms are morphisms in $C$ of the underlying objects that respect the monoid structure ie. $(A,\nabla_{A},\eta_{A}), (B,\nabla_{B},\eta_{B})$ being two objects in $Mon(C)$, a morphism $f \colon A \rightarrow B$ is a morphism in $Mon(C)$ if these two diagrams commute:

Similarly for the category of commutative monoids $CMon(C)$, if $C$ is symmetric monoidal.

###### Remark

If Assoc denotes the associative operad in $C$, then $Mon(C) = Alg_C Assoc$ is the category of algebras over an operad for $Assoc$.

###### Remark

Every category of monoids comes with a forgetful functor $U \colon Mon(C) \to C$ which is faithful and conservative. In many cases it is monadic.

## Properties

The properties of the category of monoids $Mon (C)$, especially with respect to colimits, are markedly different according to whether or not the tensor product of $C$ preserves colimits in each variable. (This is automatically the case if $C$ is closed.)

Most “algebraic” situations have this property, but others do not. For instance, the category of monads on a fixed category $A$ is $Mon (C)$, where $C= [A,A]$ is the category of endofunctors of $A$ with composition as its monoidal structure. This monoidal product preserves colimits in one variable (since colimits in $[A,A]$ are computed pointwise), but not in the other (since most endofunctors do not preserve colimits). So far, the material on this page focuses on the case where $\otimes$ does preserve colimits in both variables, although some of the references at the end discuss the more general case.

### Local presentability

###### Theorem

Let $C$ be a closed symmetric monoidal category with countable coproducts which is locally presentable.

Then

1. $U : Mon(C) \to C$ is a finitary monadic functor.

2. If $C$ is a $\lambda$-locally presentable category then so is $Mon(C)$.

This appears in (Porst, page 7).

### Free and relative free monoids

###### Proposition

Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the forgetful functor $U_C$ has a left adjoint $F_C : C \to Mon(C)$. On an object $X \in C$ the underlying object of $F_C X$ is

$U_C F_C X = \coprod_{n \in \mathbb{N}} X^{\otimes n} = I_C \coprod X \coprod (X \otimes X) \coprod \cdots$

in $C$, with the monoidal structure given by tensor product/juxtaposition.

###### Proof

A morphism $f : F_C X \to A$ in $Mon(C)$ with components $f_k : X^{\otimes k} \to U_C A$ is entirely fixed by its component $\tilde f = f_1 : X \to U_C A$ on $X$, because by the homomorphism property and the special free nature of the product in $F_C X$

$\array{ X^{\otimes k} \otimes X^{\otimes (n-k)} &\stackrel{f_k \otimes f_{n-k}}{\to}& A \otimes A \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_A}} \\ X^{\otimes n} &\stackrel{f_{n}}{\to}& A }$

it follows that

$f_n : X^{\otimes n} \stackrel{f_1^{\otimes n}}{\to} A^{\otimes n} \stackrel{\mu_A}{\to} A \,.$

Conversely, every choice for $f_1$ extends to a morphism $f$ in $Mon(C)$ this way.

###### Examples

Free algebras of the form $F(A)$ are called tensor algebras, at least for $C =$ Vect and similar.

The elements of the free algebra $F(A)$ are somtimes called lists, at least for $C =$ Set and similar.

### Pushouts

We discuss forming pushouts in a category of monoids. The case

has a simple description. The case

is more involved.

#### Of commutative monoids

###### Proposition

Suppose that $\mathcal{C}$ is

• that are preserved by the tensor product functors $A \otimes (-) \colon \mathcal{C} \to \mathcal{C}$ for all objects $A$ in $\mathcal{C}$.

Then for $f \colon A \to B$ and $g \colon A \to C$ two morphisms in the category $CMon(\mathcal{C})$ of commutative monoids in $\mathcal{C}$, the underlying object in $\mathcal{C}$ of the pushout in $CMon(\mathcal{C})$ coincides with that of the pushout in the category $A$Mod of $A$-modules

$U(B \coprod_A C) \simeq B \otimes_A C \,.$

Here $B$ and $C$ are regarded as equipped with the canonical $A$-module structure induced by the morphisms $f$ and $g$, respectively.

This appears for instance as (Johnstone, page 478, cor. 1.1.9).

#### Of noncommutative monoids

###### Proposition

If $C$ is cocomplete and its tensor product preserves colimits on both sides, then the category $Mon(C)$ of monoids has all pushouts

$\array{ F(K) &\stackrel{F(f)}{\to}& F(L) \\ {}^{\mathllap{p}}\downarrow && \downarrow \\ X &\to& P }$

along morphisms $F(f) : F(K) \to F(L)$, for $f : K \to L$ a morphism in $C$ and $F : C \to Mon(C)$ the free monoid functor from above.

Moreover, these pushouts in $Mon(C)$ are computed in $C$ as the colimit over a sequence

$P \simeq \lim_{\to}( X := P_0 \to P_1 \to P_2 \to \cdots )$

of objects $(P_n)_{n \in \mathbb{N}}$, which are each given by pushouts in $C$ inductively as follows.

Assume $P_{n-1}$ has been defined. Write $Sub(\mathbf{n})$ for the poset of subsets of the $n$-element set $\mathbf{n}$ (this is the poset of paths along the edges of an $n$-dimensional cube). Define a diagram

$K : Sub \mathbf{n} \to C$

by setting on subsets $S \subset \mathbf{n}$

$K_S := X \otimes V_1 \otimes X \otimes V_2 \otimes \cdots \otimes V_n \otimes X$

where

$V_i := \left\{ \array{ K & if \, i \notin S \\ L & if \, i \in S } \right.$

and by assigning to a morphism $S_1 \subset S_2$ the morphism which is the tensor product of identities on $X$, identities on $L$ and the given morphism $f : K \to L$.

Write $K^-$ for the same diagram minus the terminal object $S = \mathbf{n}$.

Now take $P_n$ to be the pushout

$\array{ \lim_{\to} K^- &\to& K \mathbf{n} \\ \downarrow && \downarrow \\ P_{n-1} &\to& P_n } \,,$

where the top morphism is the canonical one induced by the commutativity of the diagram $K$, and where the left morphism is defined in terms of components $K^-(S)$ of the colimit for $S \subset \mathbf{n}$ a proper subset by the tensor product morphisms of the form

$(\cdots X \otimes K \otimes \cdots \otimes L \otimes \cdots) \stackrel{\cdots \otimes \mu_X \circ (Id \otimes p) \otimes \cdots \otimes Id_L \otimes \cdots}{\to} (X \otimes L)^{|S|} \otimes X \to P_{|S|} \to P_{n-1} \,.$

This gives the underlying object of the monoid $P$. Take the monoid structure on it as follows. The unit of $P$ is the composite

$e_P : I_C \stackrel{e_X}{\to} X \to P$

with the unit of $X$. The product we take to be the image in the colimit of compatible morphisms $P_k \otimes P_k \to P_{k + l}$ defined by induction on $lk + l$ as follows. we observe that we have a pushout diagram

$\array{ Q_k \otimes (X \otimes L)^{\otimes l} \coprod_{Q_k \otimes Q_l} (X \otimes L)^{\otimes l} \otimes X \otimes Q_l &\to& (X \otimes L)^{\otimes k} \otimes X \otimes (X \otimes L)^{\otimes l} \otimes X \\ \downarrow && \downarrow \\ P_{k-1} \otimes P_l \coprod_{P_{k-1} \otimes P_{l-1}} P_k \otimes P_{l-1} &\to& P_k \otimes P_l } \,,$

where $Q_n := (\lim_{\to} K)_n$ is the colimit as in the above at stage $n$.

There is a morphism from the bottom left object to $P_{k+l}$ given by the induction assumption. Moreover we have a morphism from the top right object to $P_{k+1}$ obtained by first multiplying the two adjacent factors of $X$ and then applying the morphism $(X \otimes L)^{\otimes k+l} \otimes X \to P_{k+l}$. These are compatible and hence give the desired morphism $P_k \otimes P_k \to P_{k+l}$.

This construction is spelled out for instance in the proof of SchwedeShipley, lemma 6.2

###### Proof

First we need to discuss that this definition is actually consistent, in that the morphism $\lim_\to K^- \to P_{n-1}$ is well defined and the monoid structure on $P$ is well defined.

(…)

That $X \to P$ is a morphism of monoids follows then essentially by the definition of the monoid structure on $P$.

Finally we need to check the universal property of the cocone $P$ obtained this way:

(…)

### Filtered colimits

###### Proposition

For $C$ a closed symmetric monoidal category the forgetful functor

$U : CMon(C) \to C$

from commutative monoids to $C$ creates filtered colimits.

This appears for instance as (Johnstone, C1.1 lemma 1.1.8).

### Structure induced from monoidal functors

If $F : C\to D$ is a lax monoidal functor, then it induces canonically a functor between categories of monoids

$Mon(F) : Mon(C) \to Mon(D) \,.$

This is one good way to remember the difference between lax and colax monoidal functors.

### Model structure

If $C$ is a monoidal model category, then $Mon(C)$ may inherit itself the structure of a model category. See model structure on monoids in a monoidal model category.

### Enrichment over $CMon$

Some categories are implicitly enriched over commutative monoids, in particular semiadditive categories. Also Ab-enriched categories (and hence in particular abelian categories) of course have an underlying $CMon$-enrichment.

A general discussion of categories of monoids in symmetric monoidal categories is in

• Hans Porst, On Categories of Monoids, Comonoids and bimonoids (pdf)

Free monoid constructions are discussed in

• Eduardo Dubuc, Free monoids Algebra J. 29, 208–228 (1974)

• Max Kelly, A unified treatment of transfinite constructions for free algebras, free monoids,colimits, associated sheaves, and so on Bull. Austral. Math. Soc. 22(1), 1–83 (1980)

• Stephen Lack, Note on the construction of free monoids Appl Categor Struct (2010) 18:17–29

The detailed discussion of pushouts along free monoid morphisms is in the proof of lemma 6.2 of

Some remarks on commutative monoids are in section C1.1 of

Discussion of the closed monoidal category structure on a category of algebras of a commutative algebraic theory is in

• Peter Freyd, Algebra valued functors in general and tensor products in particular, Colloq. Math. 14 (1966), 89-106.
category: category