nLab category of monoids



Monoidal categories

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products



Internal monoids



In higher category theory


Monoid theory




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

  • objects are monoids in CC;

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

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


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


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


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

Most “algebraic” situations have this property, but others do not. For instance, the category of monads on a fixed category AA is Mon(C)Mon (C), where C=[A,A]C= [A,A] is the category of endofunctors of AA with composition as its monoidal structure. This monoidal product preserves colimits in one variable (since colimits in [A,A][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


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


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

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

This appears in (Porst, page 7).

Free and relative free monoids


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

U CF CX= nX n=I CX(XX) U_C F_C X = \coprod_{n \in \mathbb{N}} X^{\otimes n} = I_C \coprod X \coprod (X \otimes X) \coprod \cdots

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


A morphism f:F CXAf : F_C X \to A in Mon(C)Mon(C) with components f k:X kU CAf_k : X^{\otimes k} \to U_C A is entirely fixed by its component f˜=f 1:XU CA\tilde f = f_1 : X \to U_C A on XX, because by the homomorphism property and the special free nature of the product in F CXF_C X

X kX (nk) f kf nk AA μ A X n f n A \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 nf 1 nA nμ AA. f_n : X^{\otimes n} \stackrel{f_1^{\otimes n}}{\to} A^{\otimes n} \stackrel{\mu_A}{\to} A \,.

Conversely, every choice for f 1f_1 extends to a morphism ff in Mon(C)Mon(C) this way.


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

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


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

has a simple description. The case

is more involved.

Of commutative monoids


Suppose that 𝒞\mathcal{C} is

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

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

Here BB and CC are regarded as equipped with the canonical AA-module structure induced by the morphisms ff and gg, respectively.

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

Of noncommutative monoids


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

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

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

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

Plim (X:=P 0P 1P 2) P \simeq \lim_{\to}( X := P_0 \to P_1 \to P_2 \to \cdots )

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

Assume P n1P_{n-1} has been defined. Write Sub(n)Sub(\mathbf{n}) for the poset of subsets of the nn-element set n\mathbf{n} (this is the poset of paths along the edges of an nn-dimensional cube). Define a diagram

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

by setting on subsets SnS \subset \mathbf{n}

K S:=XV 1XV 2V nX K_S := X \otimes V_1 \otimes X \otimes V_2 \otimes \cdots \otimes V_n \otimes X


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

and by assigning to a morphism S 1S 2S_1 \subset S_2 the morphism which is the tensor product of identities on XX, identities on LL and the given morphism f:KLf : K \to L.

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

Now take P nP_n to be the pushout

lim K Kn P n1 P n, \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 KK, and where the left morphism is defined in terms of components K (S)K^-(S) of the colimit for SnS \subset \mathbf{n} a proper subset by the tensor product morphisms of the form

(XKL)μ X(Idp)Id L(XL) |S|XP |S|P n1. (\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 PP. Take the monoid structure on it as follows. The unit of PP is the composite

e P:I Ce XXP e_P : I_C \stackrel{e_X}{\to} X \to P

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

Q k(XL) l Q kQ l(XL) lXQ l (XL) kX(XL) lX P k1P l P k1P l1P kP l1 P kP l, \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 K) nQ_n := (\lim_{\to} K)_n is the colimit as in the above at stage nn.

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

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


First we need to discuss that this definition is actually consistent, in that the morphism lim K P n1\lim_\to K^- \to P_{n-1} is well defined and the monoid structure on PP is well defined.


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

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


Filtered colimits


For CC a closed symmetric monoidal category the forgetful functor

U:CMon(C)C U : CMon(C) \to C

from commutative monoids to CC creates filtered colimits.

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

Structure induced from monoidal functors

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

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

Enrichment over CMonCMon

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 CMonCMon-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

Last revised on March 11, 2024 at 06:07:56. See the history of this page for a list of all contributions to it.