nLab tensor product of commutative monoids




Monoidal categories

monoidal categories

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Closed structure

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Internal monoids



In higher category theory

Monoid theory



For AA and BB two commutative monoids, their tensor product ABA \otimes B is a new commutative monoid such that a monoid homomorphism ABCA \otimes B \to C is equivalently a bilinear map out of AA and BB.



Let CMon be the collection of commutative monoids, regarded as a multicategory whose multimorphisms are the multilinear maps A 1,,A nBA_1, \cdots, A_n \to B.

The tensor product A,BABA, B \mapsto A \otimes B in this multicategory is the tensor product of commutative monoids.

Equivalently this means explicitly:


For A,BA, B two commutative monoids, their tensor product of commutative monoids is the commutative monoid ABA \otimes B which is the quotient of the free commutative monoid on the product of their underlying sets A×BA \times B by the relations

  • (a 1,b)+(a 2,b)(a 1+a 2,b)(a_1,b)+(a_2,b)\sim (a_1+a_2,b)

  • (a,b 1)+(a,b 2)(a,b 1+b 2)(a,b_1)+(a,b_2)\sim (a,b_1+b_2)

  • (0,b)0(0,b)\sim 0

  • (a,0)0(a,0)\sim 0

for all a,a 1,a 2Aa, a_1, a_2 \in A and b,b 1,b 2Bb, b_1, b_2 \in B.


By definition of the free construction and the quotient there is a canonical function of the underlying sets

p A,B:U(A)×U(B)U(AB) p_{A,B} \;\colon\; U(A) \times U(B) \overset{}{\longrightarrow} U(A \otimes B)

(where U:CMonSetU \colon CMon \to Set is the forgetful functor).

On elements this sends (a,b)(a,b) to the equivalence class that it represents under the above equivalence relations.

The following relates the tensor product to bilinear functions. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of commutative monoids.


A function of underlying sets f:A×BCf : A \times B \to C is a bilinear function precisely if it factors by the morphism of through a monoid homomorphism ϕ:ABC\phi : A \otimes B \to C out of the tensor product:

f:A×BABϕC. f : A \times B \stackrel{\otimes}{\to} A \otimes B \stackrel{\phi}{\to} C \,.


Symmetric monoidal category structure


Equipped with the tensor product \otimes of def. and the exchange map σ A,B:ABBA\sigma_{A, B}: A\otimes B \to B \otimes A generated by σ A,B(a,b)=(b,a)\sigma_{A, B}(a, b) = (b, a), CMon becomes a symmetric monoidal category.

The unit object in (CMon,)(CMon, \otimes) is the additive monoid of natural numbers \mathbb{N}.


To see that \mathbb{N} is the unit object, consider for any commutative monoid AA the map

AA A \otimes \mathbb{N} \to A

which sends for nn \in \mathbb{N}

(a,n)naa+a++a nsummands. (a, n) \mapsto n \cdot a \coloneqq \underbrace{a + a + \cdots + a}_{n\;summands} \,.

Due to the quotient relation defining the tensor product, the element on the left is also equal to

(a,n)=(a,1+1+1 nsummands)=(a,1)+(a,1)++(a,1) nsummands. (a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands}) = \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands} \,.

This shows that AAA \otimes \mathbb{N} \to A is in fact an isomorphism.

Showing that σ A,B\sigma_{A, B} is natural in A,BA, B is trivial, so σ\sigma is a braiding. σ 2\sigma^2 is identity, so it gives CMon a symmetric monoidal structure.


The tensor product of commutative monoids distributes over the biproduct of commutative monoids

A sSB s sS(AB s). A \otimes \oplus_{s \in S} B_s \simeq \oplus_{s \in S} ( A \otimes B_s ) \,.



A monoid in (CMon,)(CMon, \otimes) is equivalently a rig.


Let (A,)(A, \cdot) be a monoid in (CMon,)(CMon, \otimes). The fact that the multiplication

:AAA \cdot : A \otimes A \to A

is bilinear means by the above that for all a 1,a 2,bAa_1, a_2, b \in A we have

(a 1+a 2)b=a 1b+a 2b (a_1 + a_2) \cdot b = a_1 \cdot b + a_2 \cdot b
b(a 1+a 2)=ba 1+ba 2. b \cdot (a_1 + a_2) = b \cdot a_1 + b \cdot a_2 \,.
0b=0. 0 \cdot b = 0 \,.
b0=0. b \cdot 0 = 0 \,.

This is precisely the distributivity law and absorption law? of the rig.

In (,1)(\infty,1)-category theory

The assignment 𝒞CMon(𝒞)\mathcal{C} \mapsto \mathrm{CMon}(\mathcal{C}) satisfies base change for presentable (∞,1)-categories, i.e.

𝒞CMon(𝒮)CMon(𝒮). \mathcal{C} \otimes \mathrm{CMon}(\mathcal{S}) \simeq \mathrm{CMon}(\mathcal{S}).

In particular, CMon(S)\mathrm{CMon}(S) is a mode; this implies that for 𝒞\mathcal{C} a presentably symmetric monoidal(∞,1)-category, CMon(𝒞)\mathrm{CMon}(\mathcal{C}) possesses a unique symmetric monoidal structure subject to the condition that the free functor

𝒞CMon(𝒞) \mathcal{C} \rightarrow \mathrm{CMon}(\mathcal{C})

is symmetric monoidal. This is due to Gepner-Groth-Nikolaus


Last revised on April 19, 2024 at 20:19:52. See the history of this page for a list of all contributions to it.