nLab
category of monoids

Definition

Definition

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

• objects are monoids in $C$;

• morphisms are morphisms in $C$ of the underlying objects that respect the monoid structure.

Remarks

• If Assoc denotes the associative operad in $C$, then $\mathrm{Mon}(C)={\mathrm{Alg}}_C\mathrm{Assoc}$ is the category of algebras over an operad for $\mathrm{Assoc}$.

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

Properties

The properties of the category of monoids $\mathrm{Mon}(C)$, 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 $\mathrm{Mon}(C)$, 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. $U:\mathrm{Mon}(C)\to C$ is a finitary monadic functor.

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

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 $F_C:C\to \mathrm{Mon}(C)$. On an object $X\in C$ the underlying object of $F_{C}X$ is

$$U_C{F}_{C}X=\coprod _{n\in \mathbb{N}}{X}^{\otimes n}=I_C\coprod X\coprod (X\otimes X)\coprod \cdots$$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 $\mathrm{Mon}(C)$ 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 $F_{C}X$

$$\begin{array}{ccc}{X}^{\times k}\otimes X^{\otimes (n-k)}& \stackrel{f_k\otimes f_{n-k}}{\to }& A\otimes A\\ {}^{\simeq }\downarrow & & {\downarrow }^{{\mu }_A}\\ X^{\otimes n}& \stackrel{f_n}{\to }& A\end{array}$$\array{ X^{\times 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

$$f_n:X^{\otimes n}\stackrel{f_1^{\otimes n}}{\to }{A}^{\otimes n}\stackrel{{\mu }_A}{\to }A.$$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 $\mathrm{Mon}(C)$ 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

Of noncommutative monoids

Proposition

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

$$\begin{array}{ccc}F(K)& \stackrel{F(f)}{\to }& F(L)\\ {}^p\downarrow & & \downarrow \\ X& \to & P\end{array}$$\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 $F:C\to \mathrm{Mon}(C)$ the free monoid functor from above.

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

$$P\simeq \underrightarrow{\lim }(X:=P_0\to P_1\to P_2\to \cdots )$$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 $\mathrm{Sub}(n)$ for the poset of subsets of the $n$-element set $n$ (this is the poset of paths along the edges of an $n$-dimensional cube). Define a diagram

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

by setting on subsets $S\subset n$

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

where

$$V_i:=\{\begin{array}{cc}K& \mathrm{if}\mathrm{not}i\in S\\ L& \mathrm{if}i\in S\end{array}$$V_i := \left\{ \array{ K & if not i \in 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 $S=n$.

Now take $P_n$ to be the pushout

$$\begin{array}{ccc}\underrightarrow{\lim }{K}^{-}& \to & Kn\\ \downarrow & & \downarrow \\ P_{n-1}& \to & P_n\end{array},$$\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 $S\subset n$ a proper subset by the tensor product morphisms of the form

$$(\cdots X\otimes K\otimes \cdots \otimes L\otimes \cdots )\stackrel{\cdots \otimes {\mu }_X\circ (\mathrm{Id}\otimes p)\otimes \cdots \otimes {\mathrm{Id}}_L\otimes \cdots }{\to }(X\otimes L{)}^{\mid S\mid }\otimes X\to P_{\mid S\mid }\to P_{n-1}.$$(\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

$$e_P:I_C\stackrel{e_X}{\to }X\to P$$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 $\mathrm{lk}+l$ as follows. we observe that we have a pushout diagram

$$\begin{array}{ccc}{Q}_k\otimes (X\otimes L{)}^{\otimes l}\coprod _{Q_k\otimes Q_l}(X\otimes L{)}^{\otimes l}\otimes X\otimes Q_l& \to & (X\otimes L{)}^{\otimes k}\otimes X\otimes (X\otimes L{)}^{\otimes l}\otimes X\\ \downarrow & & \downarrow \\ P_{k-1}\otimes P_l\coprod _{P_{k-1}\otimes P_{l-1}}{P}_k\otimes P_{l-1}& \to & P_k\otimes P_l\end{array},$$\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:

(…)

Of commutative monoids

Proposition

Suppose that $C$ is

• a symmetric monoidal category;

• with reflexive coequalizers

• that are preserved by the tensoring functors $A\otimes (-):C\to C$ for all objcts $A$ in $C$.

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

$$U(B\coprod _{A}C)\simeq B{\otimes }_{A}C.$$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.

Filtered colimits

Proposition

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

$$U:\mathrm{CMon}(C)\to C$$U : CMon(C) \to C

from commutative monoids to $C$ created 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

$$\mathrm{Mon}(F):\mathrm{Mon}(C)\to \mathrm{Mon}(D).$$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 $\mathrm{Mon}(C)$ may inherit itself the structure of a model category. See model structure on monoids in a monoidal model category.

References

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

• Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)

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

• Peter Johnstone, Sketches of an Elephant

category: category