symmetric monoidal (∞,1)-category of spectra
homotopy theory, (∞,1)-category theory, homotopy type theory
flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…
models: topological, simplicial, localic, …
see also algebraic topology
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
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 $A \otimes -$, with $A$ some monoid object.
Let $(\mathcal{V}, \otimes, I)$ be a monoidal category and $A$ a monoid object in $\mathcal{V}$, hence an object $A \in \mathcal{V}$ equipped with a multiplication morphism
and a unit element
satisfying the associativity law and the unit law.
A (left) module over $A$ in $(\mathcal{V}, \otimes, I)$ is
an object $N \in \mathcal{V}$
equipped with a morphism
in $\mathcal{V}$
such that this satisfies the axioms of an action, in that the following are commuting diagrams in $\mathcal{V}$:
and
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 $R$ is a module in $(Ab,\otimes)$ according to def. 1.
We unwind what this means in terms of abelian groups regarded as sets with extra structure:
A module $N$ over a ring $R$ is
an object $N \in$ Ab, hence an abelian group;
equipped with a morphism
in Ab; hence a function of the underlying sets that sends elements
and which is a bilinear function in that it satisfies
and
for all $r, r_1, r_2 \in R$ and $n,n_1, n_2 \in N$;
such that the diagram
commutes in Ab, which means that for all elements as before we have
such that the diagram
commutes, which means that on elements as above
The category of all modules over all commutative rings is Mod. It is a bifibration
over CRing.
This fibration may be characterized intrinsically, which gives yet another way of defining $R$-modules. This we turn to below.
Simpler than the traditionally default notion of a module in $(Ab,\otimes)$, 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 $(Set,\times)$ is just a monoid, for instance a discrete group $G$. A $G$-module in $(Set,\times)$ is simpy an action, say a group action.
For $S \in$ Set and $G$ a discrete group, a $G$-action of $G$ on $S$ is a function
such that
the neutral element acts trivially
the action property holds: for all $g_1, g_2 \in G$ and $s \in S$ we have $\lambda(g_1,\lambda(g_2, s)) = \lambda(g_1 \cdot g_2, s)$.
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 $\mathbb{Z}[G]$ as in def. 2.
For $G$ a discrete group, write $\mathbb{Z}[G] \in$ Ring for the ring
whose underlying abelian group is the free abelian group on the set underlying $G$;
whose multiplication is given on basis elements by the group operation in $G$.
For $G$ a finite group an element $r$r of $\mathbb{Z}[G]$ is for the form
with $r_g \in \mathbb{Z}$. Addition is given by addition of the coefficients $r_g$ and multiplication is given by the formula
For $A \in$ Ab an abelian group with underlying set $U(A)$, $G$-actions $\lambda \colon G \times U(A) \to U(A)$ such that for each element $g \in G$ the function $\lambda(g,-) \colon U(A) \to U(A)$ is an abelian group homomorphism are equivalently $\mathbb{Z}[G]$-module structures on $A$.
Since the underlying abelian group of $\mathbb{Z}[G]$ is a free by definition, a bilinear map $\mathbb{Z}[G] \times A \to A$ is equivalently for each basis element $g \in G$ a linear map $A \to A$. Similarly the module property is determined on basis elements, where it reduces manifestly to the action property of $G$ on $U(A)$.
This reformulation of linear $G$-actions in terms of modules allows to treat $G$-actions in terms of homological algebra. See at Ext – Relation to group cohomology.
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
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