category of monoids in nLab

Context

Monoidal categories

Algebra

Monoid theory

• Definition
• Properties
• Local presentability
• Free and relative free monoids
• Pushouts
• Of commutative monoids
• Of noncommutative monoids
• Filtered colimits
• Structure induced from monoidal functors
• Model structure
• Enrichment over CMonCMon
• Related categories
• References

Definition

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:A→Bf \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.

Proposition

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:C→Mon(C)F_C : C \to Mon(C). On an object X∈CX \in C the underlying object of F CXF_C X is

U CF CX=∐ n∈ℕX ⊗n=I C∐X∐(X⊗X)∐⋯ 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.

Proof

A morphism f:F CX→Af : F_C X \to A in Mon(C)Mon(C) with components f k:X ⊗k→U CAf_k : X^{\otimes k} \to U_C A is entirely fixed by its component f˜=f 1:X→U 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 ⊗k⊗X ⊗(n−k) →f k⊗f n−k A⊗A ≃↓ ↓ 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 ⊗n→f 1 ⊗nA ⊗n→μ 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.

Examples

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.

Proposition

Suppose that 𝒞\mathcal{C} is

• a symmetric monoidal category;

• with reflexive coequalizers

• that are preserved by the tensor product functors A⊗(−):𝒞→𝒞A \otimes (-) \colon \mathcal{C} \to \mathcal{C} for all objects AA in 𝒞\mathcal{C}.

Then for f:A→Bf \colon A \to B and g:A→Cg \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.

Proposition

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:K→Lf : K \to L a morphism in CC and F:C→Mon(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

P≃lim →(X:=P 0→P 1→P 2→⋯) P \simeq \lim_{\to}( X := P_0 \to P_1 \to P_2 \to \cdots )

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

Assume P n−1P_{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:Subn→C K : Sub \mathbf{n} \to C

by setting on subsets S⊂nS \subset \mathbf{n}

K S:=X⊗V 1⊗X⊗V 2⊗⋯⊗V n⊗X K_S := X \otimes V_1 \otimes X \otimes V_2 \otimes \cdots \otimes V_n \otimes X

where

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

and by assigning to a morphism S 1⊂S 2S_1 \subset S_2 the morphism which is the tensor product of identities on XX, identities on LL and the given morphism f:K→Lf : 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 n−1 → 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 S⊂nS \subset \mathbf{n} a proper subset by the tensor product morphisms of the form

(⋯X⊗K⊗⋯⊗L⊗⋯)→⋯⊗μ X∘(Id⊗p)⊗⋯⊗Id L⊗⋯(X⊗L) |S|⊗X→P |S|→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 PP. Take the monoid structure on it as follows. The unit of PP is the composite

e P:I C→e XX→P 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 k⊗P k→P 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⊗(X⊗L) ⊗l∐ Q k⊗Q l(X⊗L) ⊗l⊗X⊗Q l → (X⊗L) ⊗k⊗X⊗(X⊗L) ⊗l⊗X ↓ ↓ P k−1⊗P l∐ P k−1⊗P l−1P k⊗P l−1 → P k⊗P 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 (X⊗L) ⊗k+l⊗X→P k+l(X \otimes L)^{\otimes k+l} \otimes X \to P_{k+l}. These are compatible and hence give the desired morphism P k⊗P k→P k+lP_k \otimes P_k \to P_{k+l}.

Proof

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

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That X→PX \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:

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