nLab category of monoids

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, i.e., $(A,\nabla_{A},\eta_{A}), (B,\nabla_{B},\eta_{B})$ 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

• For commutative monoids

has a simple description. The case

• For non-commutative monoids

is more involved.

Of commutative monoids

Proposition

Suppose that $\mathcal{C}$ is

• a symmetric monoidal category;

• with reflexive coequalizers

• 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.

Related categories

• Grp, Ab

• Ring, CRing

• Rng

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

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