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module over a monoid

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Idea

In a monoidal category, there is a notion of modules over monoid objects which generalizes the classical notion of modules over rings. This is a special case of module over a monad where the monad is taken to be AA \otimes -, with AA some monoid object.

Definition

Let (𝒱,,I)(\mathcal{V}, \otimes, I) be a monoidal category and AA a monoid object in 𝒱\mathcal{V}, hence an object A𝒱A \in \mathcal{V} equipped with a multiplication morphism

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

and a unit element

e:IA e : I \to A

satisfying the associativity law and the unit law.

Definition

A (left) module over AA in (𝒱,,I)(\mathcal{V}, \otimes, I) is

  • an object N𝒱N \in \mathcal{V}

  • equipped with a morphism

    λ:ANN \lambda : A \otimes N \to N

    in 𝒱\mathcal{V}

such that this satisfies the axioms of an action, in that the following are commuting diagrams in 𝒱\mathcal{V}:

AAN id Aλ AN id n λ AN λ N \array{ A \otimes A \otimes N &\stackrel{id_A \otimes \lambda}{\to}& A \otimes N \\ \downarrow^{\mathrlap{\cdot \otimes id_n}} && \downarrow^{\mathrlap{\lambda}} \\ A \otimes N &\stackrel{\lambda}{\to}& N }

and

IN eid N AN λ N. \array{ I \otimes N &&\stackrel{e \otimes id_N}{\to}&& A \otimes N \\ & \searrow && \swarrow_{\mathrlap{\lambda}} \\ && N } \,.

Examples

Modules over monoids in abelian groups

Recall that a ring, in the classical sense, is a monoid object in the category Ab of abelian groups with monoidal structure given by the tensor product of abelian groups \otimes. Accordingly a module over RR is a module in (Ab,)(Ab,\otimes) according to def. 1.

We unwind what this means in terms of abelian groups regarded as sets with extra structure:

Definition

A module NN over a ring RR is

  1. an object NN \in Ab, hence an abelian group;

  2. equipped with a morphism

    α:RNN \alpha : R \otimes N \to N

    in Ab; hence a function of the underlying sets that sends elements

    (r,n)rnα(r,n) (r,n) \mapsto r n \coloneqq \alpha(r,n)

    and which is a bilinear function in that it satisfies

    (r,n 1+n 2)rn 1+rn 2 (r, n_1 + n_2) \mapsto r n_1 + r n_2

    and

    (r 1+r 2,n)r 1n+r 2n (r_1 + r_2, n) \mapsto r_1 n + r_2 n

    for all r,r 1,r 2Rr, r_1, r_2 \in R and n,n 1,n 2Nn,n_1, n_2 \in N;

  3. such that the diagram

    RRN RId N RN Id Rα α RN N \array{ R \otimes R \otimes N &\stackrel{\cdot_R \otimes Id_N}{\to}& R \otimes N \\ {}^{\mathllap{Id_R \otimes \alpha}}\downarrow && \downarrow^{\mathrlap{\alpha}} \\ R \otimes N &\to& N }

    commutes in Ab, which means that for all elements as before we have

    (r 1r 2)n=r 1(r 2n). (r_1 \cdot r_2) n = r_1 (r_2 n) \,.
  4. such that the diagram

    1N 1id N RN α N \array{ 1 \otimes N &&\stackrel{1 \otimes id_N}{\to}&& R \otimes N \\ & \searrow && \swarrow_{\mathrlap{\alpha}} \\ && N }

    commutes, which means that on elements as above

    1n=n. 1 \cdot n = n \,.
Remark

The category of all modules over all commutative rings is Mod. It is a bifibration

ModCRing Mod \to CRing

over CRing.

This fibration may be characterized intrinsically, which gives yet another way of defining RR-modules. This we turn to below.

GG-sets

Simpler than the traditionally default notion of a module in (Ab,)(Ab,\otimes), as above is that of a module in Set, equipped with its cartesian monoidal structure. (These days one may want to think of this as a notion of modules over F1.)

A monoid object in (Set,×)(Set,\times) is just a monoid, for instance a discrete group GG. A GG-module in (Set,×)(Set,\times) is simpy an action, say a group action.

Definition

For SS \in Set and GG a discrete group, a GG-action of GG on SS is a function

λ:G×SS \lambda \colon G \times S \to S

such that

  1. the neutral element acts trivially

    *×S S (e,id S) λ G×S \array{ * \times S &&\stackrel{\simeq}{\to}&& S \\ & {}_{(e,id_S)}\searrow && \nearrow_{\mathrlap{\lambda}} \\ && G \times S }
  2. the action property holds: for all g 1,g 2Gg_1, g_2 \in G and sSs \in S we have λ(g 1,λ(g 2,s))=λ(g 1g 2,s)\lambda(g_1,\lambda(g_2, s)) = \lambda(g_1 \cdot g_2, s).

Abelian groups with GG-action as modules over the group ring

If a discrete group acts, as in def. 3, on the set underlying an abelian group and acts by linear maps (abelian group homomorphisms), then this action is equivalently a module over the group ring [G]\mathbb{Z}[G] as in def. 2.

Definition

For GG a discrete group, write [G]\mathbb{Z}[G] \in Ring for the ring

  1. whose underlying abelian group is the free abelian group on the set underlying GG;

  2. whose multiplication is given on basis elements by the group operation in GG.

Remark

For GG a finite group an element rrr of [G]\mathbb{Z}[G] is for the form

r= gGr gg r = \sum_{g \in G} r_g g

with r gr_g \in \mathbb{Z}. Addition is given by addition of the coefficients r gr_g and multiplication is given by the formula

rr˜ = gG g˜G(r gr˜ g˜)gg˜ = qG( gg˜=qr gr˜ g˜)q. \begin{aligned} r \cdot \tilde r & = \sum_{g \in G} \sum_{\tilde g \in G} (r_g \tilde r_{\tilde g}) g \cdot \tilde g \\ & = \sum_{q \in G} \left( \sum_{g \tilde g = q} r_g \tilde r_{\tilde g} \right) q \end{aligned} \,.
Proposition

For AA \in Ab an abelian group with underlying set U(A)U(A), GG-actions λ:G×U(A)U(A)\lambda \colon G \times U(A) \to U(A) such that for each element gGg \in G the function λ(g,):U(A)U(A)\lambda(g,-) \colon U(A) \to U(A) is an abelian group homomorphism are equivalently [G]\mathbb{Z}[G]-module structures on AA.

Proof

Since the underlying abelian group of [G]\mathbb{Z}[G] is a free by definition, a bilinear map [G]×AA\mathbb{Z}[G] \times A \to A is equivalently for each basis element gGg \in G a linear map AAA \to A. Similarly the module property is determined on basis elements, where it reduces manifestly to the action property of GG on U(A)U(A).

Remark

This reformulation of linear GG-actions in terms of modules allows to treat GG-actions in terms of homological algebra. See at Ext – Relation to group cohomology.

more examples

References

The basic properties of categories of modules over monoid objects in symmetric monoidal categories are spelled out in sections 1.2 and 1.3 of

  • Florian Marty, Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Thesis, 2009, web

A summary is in section 4.1 of

See also MO/180673, and the references at modules over a monad.

For the classical case of the symmetric monoidal category Ab, a standard textbook is

  • F.W. Anderson, K.R. Fuller, Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, Springer-Verlag, New York, (1992).

Revised on August 27, 2017 16:27:49 by Bartosz Milewski? (70.89.112.166)