module

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

### Basic idea

The basic idea is that a _module_ $V$ is an object equipped with an [[action]] by a [[monoid]] $A$. This is closely related to the concept of a [[representation]] of a [[group]].

A familiar example of a module is a [[vector space]] $V$ over a [[field]] $k$: this is a _module_ over $k$ in the category [[Ab]] of abelian groups: every element in $k$ acts on the vector space by multiplication of vectors, and this action respects the addition of vectors, but nothing in the definition of [[vector space]] really depends on the fact that $k$ here is a [[field]]: more generally it could be any [[commutative ring]] (or even a general [[rig]]) $R$. The analog of a vector space for fields replaced by rings is that of a _module_ over the ring $R$.

An $R$-module in [[Ab]] can be thought of as a generalization of an abelian group, where the operation taking integer multiples of an element (seen as iterated addition) is extended to taking arbitrary multiples with coefficients in $R$. In the trivial case a $\mathbb{Z}$-module is simply an abelian group.

This is the traditional and maybe most common notion of modules. But the basic notion is easily much more general.

### Motivation for and role of modules: generalized vector bundles
 {#RelationToVectorBundlesInIntroduction}

The theory of monoids or rings and their modules, its "meaning" and usage, is naturally understood via the [[Isbell duality|duality]] between [[algebra]] and [[geometry]]:

1. a [[ring]] $R$ is to be thought of as the ring of [[functions]] on some [[space]],

1. an $R$-[[module]] is to be thought of as the space of [[sections]] of a [[vector bundle]] on that space.

A classical situation where this correspondence holds precisely is [[topology]], where

1. the [[Gelfand duality]] theorem says that sending a [[compact topological space]] $X$ to its [[C-star algebra]] $C(X,\mathbb{C})$ of [[continuous functions]] with values in the [[complex numbers]] constitutes an [[equivalence of categories]] between compact topological spaces and the [[opposite category]] of commutative $C^\ast$-algebras;

1. the [[Serre-Swan theorem]] says that sending a [[Hausdorff topological space|Hausdorff]] topological complex [[vector bundle]] $E \to X$ over a compact topological space to the $C(X,\mathbb{C})$-module of its continuous [[sections]] establishes an equivalence of categories between that of topological complex vector bundles over $X$ and that of [[finitely generated module|finitely generated]] [[projective modules]] over $C(X,\mathbb{C})$.

In fact, as this example already shows, modules faithfully subsume vector bundles, but are in fact more general. In many contexts one regard modules as the canonical generalization of the notion of vector bundles, with better formal properties.

This identification of vector bundles with $R$-modules being the spaces of sections of a vector bundle on the space whose ring of functions is $R$ can be taken then as the very _definition_: notably in [[algebraic geometry]] Gelfand duality is taken to "hold by definition" in that an [[algebraic variety]] is essentially by definition the formal dual of a given ring, and the Serre-Swan theorem similarly becomes the statement that the space of sections of a vector bundle over a variety is equivalently given by a module over that ring. (See also at [[quasicoherent module]] for more on this.)

This [[Isbell duality|duality]] between geometry and algebra allows us to re-interpret many statement about modules in terms of vector bundles. For instance

* the [[direct sum]] of modules corresponds to fiberwise direct sum of vector bundles;

* the [[extension of scalars]] of a module along a ring homomorphism corresponds to [[pullback]] of vector bundles along the dual map of spaces;

* etc.

Using this dictionary for instance the notion of [[descent]] of vector bundles can be expressed in terms of [[monadic descent]], see at _[[Sweedler coring]]_ for discussion of this point.

### More general perspectives

The notion of [[monoid]] in a [[monoidal category]] generalizes
directly to that of a monoid in a [[2-category]], where it is called a [[monad]]. Accordingly the notion of module generalizes to this more general case, where however it is called an _algebra over a monad_ . For more on this see [Modules for monoids in 2-categories: algebras over monads](#MonadAlgs) below.

Apart from this direct generalization, there are two distinct and separately important perspectives on the notion of module from the [[nPOV]]:

* modules may usefully be thought of in the context of [[enriched category theory]] (and the enrichment may be over a [[2-category]]);

* modules may usefully be thought of in terms of [[abelian category|abelianization]]/[[stabilization]] of [[overcategory|overcategories]].

#### Modules for monoids in 2-categories: modules over monads {#MonadAlgs}

The notion of [[monoid]] generalizes straightforwardly from
monoids in a [[monoidal category]] to monoids in a [[2-category]]:
for the [[2-category]] [[Cat]], and more generally for arbitrary
2-categories, these are called [[monad]]s.

A [[module over a monad]] (see there for more details) is defined essentially exactly as that
of module over a monoid. For historical reasons, a module over
a monad in [[Cat]] is called an _algebra over a monad_, because the algebras in the sense of [[universal algebra]] can be obtained as algebras/modules over a [[finitary monad]] in $Set$: the modules for a free algebra monad (for certain kind of algebras) on [[Set]], which are the composition of the free algebra functor and its [[right adjoint]] forgetful functor are exactly algebras of that type. Modules over a fixed monad (in $Cat$) are the objects of the [[Eilenberg-Moore category]] of the monad; in an arbitrary bicategory, this category generalized to Eilenberg-Moore objects which may or may not exist.

#### Enriched presheaves

See [[module over an enriched category]].

#### Stabilized overcategories

A module $N$ over a (commutative, unital) ring $R$ may be encoded in another ring: the one that as an abelian group is the [[direct sum]] $R \oplus N$ and whose product is defined by the formulas

$$
  (r_1, n_1) \cdot (r_2,n_2) := (r_1 r_2, r_2 n_1 + r_1 n_2)
  \,.
$$

This is a **square-0 extension** of $R$. It is canonically equipped with a ring homomorphism $R \oplus N \to R$ which is the identity on $R$ and sends all elements of $N$ to 0. As such, $R \oplus N \to R$ is an object in the [[overcategory]] $CRing/R$. But a special such object: it is in fact canonically an abelian [[group object]] in $CRing/R$, where the group operation (over $R$!) is given by addition of elements in $N$.

From this perspective, it makes sense for general [[category|categories]] $C$ to think of the [[abelian category|abelianization]] of their [[overcategory|overcategories]] $C/A$ as categories of modules over the object $A$.

Taken all together, this makes the fiberwise [[abelian category|abelianization]] of their [[codomain fibration]] $cod : [I,C] \to C$ the category of all possible modules over all objects of $C$.

This general perspective has a nice [[vertical categorification]] to the context of [[(∞,1)-category|(∞,1)-categories]]: abelianization becomes [[stabilization]] in this context, and the fiberwise stabilization of the [[codomain fibration]] of any [[(∞,1)-category]] $C$ is the [[tangent (∞,1)-category]] $T_C \to C$.

For instance for $sAlg_k$ the [[(∞,1)-category]] of [[simplicial ring|simplicial algebras]] over a ground field $k$ of characteristic 0, we have that the [[stabilization]] $Stab(sAlgk/A)$ of the [[over quasi-category|over (∞,1)-category]] over $A$ is equivalent to the $(\infty,1)$-category $A Mod$ of
$A$-modules.

## Definition
 {#Definitions}

We spell out the definition of _module_ for

* [modules over a monoid in a monoidal category](#ModuleOverMonoidObject)

Then we give more general definitions

* [modules as presheaves in enriched category theory](#InEnrichedCategory)

* [modules as objects in stabilized overcategories](#DefWithOCat).

### Modules over a monoid in  a monoidal category
 {#ModuleOverMonoidObject}

See [[module over a monoid]].

### Presheaves in enriched category theory
 {#InEnrichedCategory}

See [[module over an enriched category]].

### In terms of stabilized overcategories
 {#DefWithOCat}

There is a general definition of modules in terms of
stabilized slice-categories of the category of monoids: _[[Beck modules]]_,
_[[tangent (infinity,1)-categories]]_.

#### Modules over a ring
 {#ModulesOverARingInTermsOfStabilizedSlices}

The ordinary case of modules over rings is phrased
in terms of stabilized overcategories by the following
observation, which goes back at least to ([Beck 67](#Beck67)), and is found in the important paper of  ([Quillen 70](#Quillen70)); both listed below. For more see at _[[Beck module]]_.

+-- {: .num_prop #Square0ExtensionsOfRingsAreAbelianSliceObjectsAreModules}
###### Proposition

Let $R \in CRing$ be a commutative [[ring]]. Then there is a canonical [[equivalence of categories|equivalence]] between the [[category]] $R Mod$ of $R$-modules and the category $Ab(CRing/R)$ of abelian [[group object]]s in the [[overcategory]] of $CRing$ over $R$

$$
  R Mod \simeq Ab(CRing/R)
  \,.
$$

=--

+-- {: .proof}
###### Proof

We first unwind what the structure of an abelian group object
$(p:  K \to R)$ in the overcategory $CRing/R$ is explicitly

The unit of the abelian group object in $CRing/R$ is a diagram

$$
  \array{
     R &&\to&& K
     \\
     & {}_{\mathllap{Id}}\searrow && \swarrow_{\mathrlap{p}}
     \\
     && R
  }
  \,.
$$

This diagram identifies $K$ with a ring whose underlying [[abelian group]] is the [[direct sum]] $R \oplus ker(p)$ of some ring $R$ with the [[kernel]] of $p$ such that for $r \in R$ and $n \in ker(p)$ we have $r\cdot n \in ker(p)$.

The [[product]] of $R \oplus N \to R$ with itself in the [[overcategory]] is the [[fiber product]] over $R$ in the original category, hence is $R \oplus N \oplus N$.

The addition operation on the abelian group object is therefore a morphism

$$
  \array{
    R \oplus N \oplus N &&\to&& R \oplus N
    \\
    & \searrow && \swarrow
    \\
    && R
  }
  \,.
$$

With the above unit, the unit axiom on this operation together with the fact that the top morphism is a ring homomorphism says that this morphism is

$$
    R \oplus N \oplus N \stackrel{Id \oplus (Id + Id)}{\to}
    R \oplus N
  \,.
$$

Since the ring product in the direct product ring $R \oplus N \oplus N$ between two elements in the two copies of $N$ vanishes, it therefore has to vanish between two elements in the same copy, too.

This says that $R \oplus N$ is a square-0 extension of $R$. Conversely, for every square-0-extension we obtain an abelian group object this way.

=--

For instance the square-0-extension of a ring $R$ corresponding to the canonical $R$-module structure on $R$ itself is the [[ring of dual numbers]] for $R$.

#### Modules over a group
 {#ModulesOverAGroupInTermsOfStabilizedOvercategories}

Let $G$ be a [[group]]. Taking together the above desriptions

1. of [Modules over a group as modules over the group ring](#AbelianGroupsWithGAction)

1. of [Modules over a ring as stabilized overcategories](#ModulesOverARingInTermsOfStabilizedSlices)

one finds:

+-- {: .num_prop }
###### Proposition

The [[category]] of $G$-modules is [[equivalence of categories|equivalent]] to the category of [[abelian group objects]] in the [[slice]] of [[Ring]] over the [[group ring]]

$$
  G Mod \simeq Ab(Ring_{/\mathbb{Z}[G]})
  \,.
$$

=--

But there is also a more direct characterization along these lines, not involving the auxiliary construction of group rings.

+-- {: .num_prop }
###### Proposition

The [[category]] of $G$-modules is equivalent to the category of [[abelian group objects]] in the [[slice category]] of [[groups]] over $G$

$$
  G Mod \simeq Ab(Grp_{/G})
  \,.
$$

=--

+-- {: .proof}
###### Proof

The proof is analogous to that of prop. \ref{Square0ExtensionsOfRingsAreAbelianSliceObjectsAreModules}.
One checks that a group homomorphism $\hat G \to G$ with the structure of an abelian group object over $G$ is a [[central extension]] of $G$ by some [[abelian group]] $A$ which more over is a split extension (the is the neutral element of the abelian group object) and hence is a [[semidirect product group]] $\hat G \simeq G \ltimes A$. By the discussion there these are equivalently given by [[actions]] of $G$ on $A$ by group [[automorphisms]]. This is precisely what it means for $A$ to carry a $G$-module structure.

=--

This construction generalizes to [[∞-groups]]. See at _[[∞-action]]_ the section _[∞-action -- G-modules](infinity-action#GModules)_.

#### Modules over a simplicial ring
 {#SimpRings}

Let $sAlg_k$ (or $sAlg$ for short) be the [[(∞,1)-category]] of commutative [[simplicial ring|simplicial algebras]] over a base field $k$.

For $A \in sAlg_k$ there is generally a functor

$$
  A Mod \to Stab(sAlg_k/A)
$$

from the [[stable (∞,1)-category]] of $A$-modules to the [[stabilization]] of the [[overcategory]] of $sAlg$. But in general this functor is neither [[essentially surjective functor|essentially surjective]] nor [[full functor|full]]. If however $k$ has characteristic 0, then this is an equivalence.

### Modules over an algebra over an operad
 {#OverHigherAndGeneralizedAlgebras}

There is a notion of [[algebra over an operad]]. The corresponding notion of modules is described at _[[module over an algebra over an operad]]_.

### Multiplicatively cancellative module over a rig

A module $M$ over a [[rig]] $S$ is called multiplicatively cancellative (in [Nazari & Ghalandarzadeh (2019), Sec. 3](#NazariGhalandarzadeh19)) if for any $s,s' \in S$ and $0 \neq m \in M$, $sm = s'm$ implies $s = s'$.

## Examples

### Of modules over a ring
 {#ExamplesOfModulesOverARing}

Let $R$ be a [[commutative ring]].

+-- {: .num_example #RingAsModuleOverItself}
###### Example

The ring $R$ is naturally a module over itself, by regarding its multiplication map $R \otimes R \to R$ as a module action $R \otimes N \to N$ with $N \coloneqq R$.

=--

+-- {: .num_example}
###### Example

More generally, for $n \in \mathbb{N}$ the $n$-fold [[direct sum]] of the [[abelian group]] underlying $R$ is naturally a module over $R$

$$
  R^n \coloneqq R^{\oplus_n} \coloneqq \underbrace{R \oplus R \oplus \cdots \oplus R}_{n\;summands}
  \,.
$$

The module action is componentwise:

$$
  r \cdot (r_1, r_2, \cdots, r_n) = (r \cdot r_1, r\cdot r_2, \cdot r \cdot r_n)
  \,.
$$

=--

+-- {: .num_example}
###### Example

Even more generally, for $I \in $ [[Set]] any [[set]], the direct sum
$\oplus_{i \in I} R$ is an $R$-module.

This is the **[[free module]]** (over $R$) on the set $S$.

The set $I$ serves as the [[basis of a free module]]: a general element $v \in \oplus_i R$ is a [[formal linear combination]] of elements of $I$ with [[coefficients]] in $R$.

=--

For special cases of the ring $R$, the notion of $R$-module is equivalent to other notions:

+-- {: .num_example}
###### Example

For $R = \mathbb{Z}$ the [[integers]], an $R$-module is equivalently just an [[abelian group]].

=--

+-- {: .num_example}
###### Example

A $\mathbb{Z}$-module, hence an abelian group, is not a free module if it has a non-trivial [[torsion subgroup]].

=--

+-- {: .num_example #VectorSpacesArekModules}
###### Example

For $R = k$ a [[field]], an $R$-module is equivalently a [[vector space]] over $k$.

Every [[finitely generated module|finitely generated]] [[free module|free]] $k$-module is a [[free module]], hence every finite dimensional vector space has a [[basis of a free module|basis]]. For infinite dimensions this is true if the [[axiom of choice]] holds.

=--

+-- {: .num_example #RestrictionAndExtensionOfScalars}
###### Example

For $f : S \to R$ a [[homomorphism]] of [[rings]],
[[restriction of scalars]] produces $R$-modules $f_* N$ from $S$-modules $N$ and [[extension of scalars]] produces $S$-modules $f_! N$ from $R$-modules $N$.

=--

+-- {: .num_example #SubmodulesInExamples}
###### Example

For $N$ a module and $\{n_i\}_{i \in I}$ a set of elements, the
[[linear span]]

$$
  \langle n_i\rangle_{i \in I} \hookrightarrow N
  \,,
$$

(hence the completion of this set under addition in $N$ and multiplication by $R$) is a [[submodule]] of $N$.

=--

+-- {: .num_example}
###### Example

Consider example \ref{SubmodulesInExamples} for the case that the module is $N = R$, the ring itself, as in example \ref{RingAsModuleOverItself}. Then a [[submodule]] is equivalently (called) an _[[ideal]]_ of $R$.

=--

+-- {: .num_example}
###### Example

Let $X$ be a [[topological space]] and let

$$
  R \coloneqq  C(X,\mathbb{C})
$$

be the ring of [[continuous functions]] on $X$ with values in the
[[complex numbers]].

Given a complex [[vector bundle]] $E \to X$ on $X$, write $\Gamma(E)$
for its set of continuous sections. Since for each point $x \in X$ the [[fiber]] $E_x$ of $E$ over $x$ is a $\mathbb{C}$-module (by example \ref{VectorSpacesArekModules}), $\Gamma(X)$ is a $C(X,\mathbb{C})$-module.

By the [[Serre-Swan theorem]]
if $X$ is [[Hausdorff topological space|Hausdorff]] and [[compact topological space|compact]], then $\Gamma(X)$ is a [[projective module|projective]] $C(X,\mathbb{C})$-module and indeed there is an equivalence between projective $C(X,\mathbb{C})$-modules and complex vector bundles over $X$.

More on this below in
_[Vector bundle and modules](#VectorBundlesAndModules)_.

=--

### Vector bundles and modules
 {#VectorBundlesAndModules}

A [[vector space]] is a [[vector bundle]] over the [[point]]. For every [[vector bundle]] $E \to X$ over a [[space]] $X$, its collection $\Gamma(E)$ of [[section]]s is a module over the monoid/ring of functions on $X$. When $X$ is a [[ringed space]], $\Gamma(X)$ is usefully thought of as a [[sheaf]] of modules over the [[structure sheaf]] of $X$:

For describing vector bundles and their generalization it turns out that this perspective of encoding them in terms of their modules of sections is useful. For instance the category of vector bundles on a space typically fails to be an [[abelian category]]. But if instead of looking just as sheaves of modules on $X$ that arise as sections of vector bundles one generalizes to [[coherent sheaf|coherent sheaves of modules]] then one obtains an abelian category, something like the completion of $Vect(X)$ to an abelian category. If one further demands that the category be closed under push-forward operations, such as to obtain a [[bifibration]] of generalized vector bundles over spaces, one arrives at the notion of [[quasicoherent sheaf|quasicoherent sheaves of modules]] over the [[structure sheaf]].

But it turns out that the category of [[quasicoherent sheaf|quasicoherent sheaves]] over a test space (see there for details) is equivalent simply to the category of _all_ modules over the (functions on) this test space. This means that quasicoherent sheaves of modules have a nice description in terms of the general-abstract-nonsense characterization of modules discussed above:

For $C$ our [[(∞,1)-category]] of of test [[space]]s (hence the [[opposite category]] $C^{op}$ our [[(∞,1)-category]] of "functions rings" on test spaces), by the above the assignment of all modules over a test space is given by

$$
  Mod : C^{op} \to (\infty,1)Cat
$$

$$
  Mod : U \mapsto Stab( C/U )
  \,.
$$

Then for $X$ any [[space]] regarded as an [[∞-stack]] on $C$, a "quasicoherent $\infty$-stack of modules" on $X$ is a morphism

$$
  X \to Mod
  \,.
$$

## Related concepts

* [[action]], [[∞-action]]

* **module**, [[(∞,1)-module]], [[2-module]], [[module category]]

  * [[module object]]

  * [[operad for modules over an algebra]]

  * [[linear independence]]

  * [[injective module]], [[projective module]], [[free module]]

  * [[Noetherian module]]

  * [[completion of a module]]

  * [[quasicoherent module]]

    [[module over a derived stack]]

  * [[Fredholm module]]

  * [[Beck module]]

* [[representation]], [[∞-representation]]

* [[linear combination]]

* [[submodule]], [[quotient module]]

* [[finitely generated module]], [[presentable module]], [[finitely presented module]]

* [[projective module]], [[injective module]], [[free module]], [[flat module]]

* [[super module]], [[N-graded module]]

* [[Banach module]]

* [[module over a monoidal functor]]

* [[module over an algebra over an operad]]

## References

### General

Textbook accounts:

* [[Frank W. Anderson]], [[Kent R. Fuller]], _Rings and Categories of Modules_, Graduate Texts in Mathematics, **13** Springer (1992) [[doi:10.1007/978-1-4612-4418-9](https://doi.org/10.1007/978-1-4612-4418-9)]

* [[Paul Edwin Bland]], *Rings and Their Modules*, De Gruyter (2011) [[doi:10.1515/9783110250237](https://doi.org/10.1515/9783110250237), [pdf](http://site.iugaza.edu.ps/mashker/files/%D9%85%D8%AD%D8%A7%D8%B6%D8%B1%D8%A7%D8%AA-%D8%AC%D8%A8%D8%B1-%D8%AF%D9%83%D8%AA%D9%88%D8%B1%D8%A7%D8%A9.pdf)]

Lecture notes:

* {#Lawvere92} [[William Lawvere]], pp. 27 of: *Introduction to Linear Categories and Applications*, course lecture notes (1992) [[pdf](https://github.com/mattearnshaw/lawvere/blob/192dac273e8bf352f307f87b9ec4fe8ef7dc85b9/pdfs/1992-introduction-to-linear-categories-and-applications.pdf)]

  > (where $R$-modules are called $R$-*linear spaces*)

Lectures notes on [[sheaves]] of modules / modules over a [[ringed space]]:

* [[Daniel Murfet]], _Modules over a ringed space_ ([pdf](http://therisingsea.org/notes/RingedSpaceModules.pdf))

Exposition of basics of [[monoidal categories]] and [[categorical algebra]]:

* _[[geometry of physics -- categories and toposes]]_, Section 2: _[Basic notions of categorical algebra](geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra)_

Formalization in [[cubical type theory|cubical]] [[homotopy type theory]]:

* [[1lab]]: *[Algebra.Ring.Module](https://1lab.dev/Algebra.Ring.Module.html)*

### On modules as enriched presheaves

See also the references at [[enriched category theory]] and at [[profunctor]].

### On modules as stabilized overcategories

The observation that the category of modules over a ring $R$ is equivalent to the category of abelian group objects in the overcategory $CRing/R$ ([[Beck module]]) is due to

* {#Beck67} [[Jon Beck]], _Triples, algebras and cohomology_, Ph.D. thesis, Columbia University, 1967, Reprints in Theory and Applications of Categories, No. 2 (2003) pp 1-59 ([TAC](http://www.tac.mta.ca/tac/reprints/articles/2/tr2abs.html))

* {#Quillen70} [[Daniel G. Quillen]], _On the (co-)homology of commutative rings_, in Proc. Symp. on Categorical Algebra, 65 – 87, American Math. Soc.,  1970.

The fully abstract higher categorical concept in terms of [[stabilization|stabilized]] [[overcategory|overcategories]] and the [[tangent (∞,1)-category]] appears in

* [[Jacob Lurie]], _[[Deformation Theory]]_

[[(∞,1)-modules]] over [[A-∞ algebras]] are discussed in section 4.2 of

* [[Jacob Lurie]], _[[Higher Algebra]]_

Multiplicatively cancellative modules over a rig appear in

* {#NazariGhalandarzadeh19} Rafieh Razavi Nazari, Shaban Ghalandarzadeh, _Multiplication semimodules_, 2019 ([arXiv:0704.2106](https://arxiv.org/abs/1904.11729))

[[!redirects module]]
[[!redirects modules]]

[[!redirects left module]]
[[!redirects left modules]]

[[!redirects right module]]
[[!redirects right modules]]

[[!redirects module over a ring]]
[[!redirects modules over a ring]]
[[!redirects module over rings]]

[[!redirects module over a commutative ring]]
[[!redirects modules over a commutative ring]]
[[!redirects modules over commutative rings]]

[[!redirects multiplicatively cancellative module]]
[[!redirects multiplicatively cancellative modules]]


module 2-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _module 2-category_ is a [[2-category]] equipped with the [[action]] of a [[monoidal 2-category]].

The collection of all module 2-categories over a monoidal 2-category forms a _[[3-category]] of module 2-categories_.

## Related concepts

* [[module]]

* [[module category]]

[[!redirects module 2-categories]]
[[!redirects 3-category of module 2-categories]]
[[!redirects 3-categories of module 2-categories]]


module algebra

A __left module $k$-algebra__ is a $k$-algebra $A$ equipped with a left [[Hopf action]] (also sometimes said module algebra action) $\triangleright: B\otimes A\to A$ of a $k$-bialgebra $B$. Sometimes one talks about module algebras meaning a monoid with Hopf actions of a bimonoid in a more general symmetric monoidal category; a module monoid would be a better term if the category is not $k$-linear.

Related entries include [[smash product algebra]], [[comodule algebra]], [[gebra]], [[bigebra]]

* S. Majid, _Foundations of quantum group theory_, Cambridge University Press 1995, 2000.
* Susan Montgomery, _Hopf algebras and their actions on rings_, CBMS Lecture Notes 82, AMS 1993, 240p.
* Matthew Tucker-Simmons, _$\ast$-structures on module-algebras_, Ph. D. thesis, [arxiv/1211.6652](http://arxiv.org/abs/1211.6652)

[[!redirects module algebras]]


module bundle

#Contents#
* table of contents
{:toc}

## Idea

A [[fiber bundle]] of [[modules]] over some [[ring]].

If $R$ is a [[field]] then this is a [[vector bundle]].

## Properties

### Relation to algebraic K-theory

For $R$ a [[commutative ring]], then $R$-module bundles may serve to
represent [[cohomology]] classes in [[algebraic K-theory]].

## Related concepts

* [[(infinity,1)-module bundle]]

[[!redirects module bundles]]


module category

> *This page is about actions of tensor categories.
> For [[categories of modules]] or [[actegory|actions of monoidal categories]], see there.*

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea
A **module category** is a [[linear category]] equipped with an action of a [[tensor linear category]].
That is, a module category is an [[actegory]] [[enriched category|enriched]] in $K\mathbf{Mod}$, or a [[pseudoaction]] in $K\mathbf{Mod}$-$\mathbf{Cat}$.

## Related concepts

* [[actegory]]

* [[2-module]], [[n-module]]

* [[bimodule category]]

* [[module]]

* [[module 2-category]]

* [[module over a monoidal functor]]

## References

* Victor Ostrik, _Module categories, weak Hopf algebras and modular invariants_, ([doi](https://doi.org/10.1007/s00031-003-0515-6)), Transformation Groups, p.177-206, vol. 8, 2003, ([arXiv:0111139](https://arxiv.org/abs/math/0111139))

* {#DSPS13} [[Chris Douglas]], [[Chris Schommer-Pries]], [[Noah Snyder]], _Dualizable tensor categories_ ([arXiv:1312.7188](http://arxiv.org/abs/1312.7188))

[[!redirects module categories]]

[[!redirects 2-category of module categories]]
[[!redirects 2-categories of module categories]]


module in higher geometry

## Idea

This entry is about the notion of [[module]] or [[quasi-coherent sheaf|quasi-coherent]] [[chain complex|complexes]] in the setting of [[higher geometry]] and more specifically, [[E-infinity geometry]] and [[derived algebraic geometry]].

In this setting, modules are already derived, in the sense that modules over an ordinary [[scheme]] or [[stack]], viewed as a discrete [[derived scheme]] or [[derived stack]], are [[complexes]] of [[quasi-coherent sheaves]].

## Definition

Let $Sch^{aff}$ and $Stk$ denote the [[(infinity,1)-categories]] of affine [[derived schemes]] and [[derived stacks]], respectively.
Consider the [[(infinity,1)-prestack]] of [[stable (infinity,1)-categories]]
  $$ Mod : (Sch^{aff})^{op} \to Cat^{stab}_\infty $$
which associates to a [[commutative ring spectrum]] $A$ the [[stable (infinity,1)-category]] $Mod(A)$.

By taking the [[right Kan extension]] of this prestack along the ([[opposite (infinity,1)-category|opposite]] of the) [[Yoneda embedding]]
  $$ (Sch^{aff})^{op} \hookrightarrow Stk^{op}, $$
one gets an [[(infinity,1)-prestack]]
  $$ Mod : Stk^{op} \to Cat^{stab}_\infty. $$
In other words, for a derived stack $X$, $Mod(X)$ is given by the [[limit in an (infinity,1)-category|limit]]
  $$ Mod(X) = lim_{Spec(A) \to X} Mod(A). $$

## In ordinary algebraic geometry

In the case of ordinary [[affine schemes]], [[modules]] in this sense, i.e. [[modules]] over [[Eilenberg-Mac Lane spectra]], correspond by the [[stable Dold-Kan correspondence]] to [[chain complexes]].  The corresponding notion of module over an ordinary [[scheme]] or [[stack]] is then a _quasi-coherent complex_.
That is, for a [[commutative ring]] $A$,
  $$ Mod(H A) = D(Mod(A)) $$
(the [[derived category]] of [[chain complexes]] of $A$-[[modules]]), and for a classical [[scheme]] $X$,
  $$ Mod(X) = D(QCoh(A)) $$
(the [[derived category]] of [[chain complexes]] of [[quasi-coherent sheaves]]).

## Properties

The [[(infinity,1)-prestack]] $Mod$ satisfies [[Zariski descent]] and even [[Nisnevich descent]]; this is due to [[Jacob Lurie]] and [[Vladimir Drinfeld]].

$Mod$ lifts to a prestack of [[symmetric monoidal (infinity,1)-categories]].  The [[dualizable objects]] are precisely the [[perfect modules]].
In good cases, the [[stable (infinity,1)-category]] $Mod(X)$ is [[compactly generated (infinity,1)-category|compactly generated]] and the [[compact objects]] are precisely the [[perfect modules]].

## Related concepts

* [[perfect module]]
* [[triangulated categories of sheaves]]

## References

* [[Dennis Gaitsgory]], _[[Notes on geometric Langlands]]: quasi-coherent sheaves_.

* [[Bertrand Toën]], [[Gabriele Vezzosi]], _Homotopical algebraic geometry II: geometric stacks and applications_, 2004, [arXiv:math/0404373](http://arxiv.org/abs/math/0404373).

* [[Jacob Lurie]], [[Derived Algebraic Geometry]], thesis, [pdf](http://www.math.harvard.edu/~lurie/papers/DAG.pdf).

* {#BFN08} [[David Ben-Zvi]], [[John Francis]], [[David Nadler]], section 3.1 of _Integral Transforms and Drinfeld Centers in Derived Algebraic Geometry_, J. Amer. Math. Soc. 23 (2010), no. 4, 909-966 ([arXiv:0805.0157](http://arxiv.org/abs/0805.0157))

* [[B. Toen]], _Derived Azumaya algebras and generators for twisted derived categories_, [arXiv:1002.2599](http://arxiv.org/abs/1002.2599).

[[!redirects module over a derived stack]]
[[!redirects modules over a derived stack]]
[[!redirects modules over derived stacks]]
[[!redirects module over a derived scheme]]
[[!redirects modules over a derived scheme]]
[[!redirects modules over derived schemes]]

[[!redirects module over a stack]]
[[!redirects modules over a stack]]
[[!redirects modules over stacks]]
[[!redirects module over a scheme]]
[[!redirects modules over a scheme]]
[[!redirects modules over schemes]]

[[!redirects quasi-coherent complex over a stack]]
[[!redirects quasi-coherent complexes over a stack]]
[[!redirects quasi-coherent complexes over stacks]]
[[!redirects quasi-coherent complex over a scheme]]
[[!redirects quasi-coherent complexes over a scheme]]
[[!redirects quasi-coherent complexes over schemes]]


module object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _action object_ or _module object_ is the [[internalization]] of the notion of [[action]]/[[module]] of [[monoids]] (such as [[groups]] or [[rings]]) on [[sets]] (such as [[group representations]] or [[modules]]), into any [[monoidal category]] $\mathcal{C}$ to yield a notion of actions of [[monoid objects]] (such as [[group objects]] or [[ring objects]]) on the [[objects]] of that [[category]] $\mathcal{C}$.

## Definition
 {#Definition}

+-- {: .num_defn #MonoidsInMonoidalCategory}
###### Definition

Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, and given $(A,\mu,e)$ a [[monoid in a monoidal category|monoid in]] $(\mathcal{C}, \otimes, 1)$, then a **left module object** in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is

1. an [[object]] $N \in \mathcal{C}$;

1. a [[morphism]] $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the _[[action]]_);

such that

1. {#Unitality} ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       1 \otimes N
         &\overset{e \otimes id}{\longrightarrow}&
       A \otimes N
       \\
       & {}_{\mathllap{\ell}}\searrow
       & \downarrow^{\mathrlap{\rho}}
       \\
       && N
     }
     \,,
   $$

   where $\ell$ is the left unitor isomorphism of $\mathcal{C}$.

1. {#ActionProperty} (action property) the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (A\otimes A) \otimes N
         &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}&
       A \otimes (A \otimes N)
         &\overset{A \otimes \rho}{\longrightarrow}&
       A \otimes N
       \\
       {}^{\mathllap{\mu \otimes N}}\downarrow
         && &&
       \downarrow^{\mathrlap{\rho}}
       \\
       A \otimes N
         &\longrightarrow&
         &\overset{\rho}{\longrightarrow}&
       N
     }
     \,,
   $$

=--

## Generalisation
A module and the monoid it lies over do not necessarily belong to the same category, a fact suggested by the [[microcosm principle]]:

+-- {: .num_defn #MonoidsInActegory}
###### Definition
Given a [[monoidal category]] $(\mathcal{M}, \odot, 1)$ and an $\mathcal{M}$-module (also called $\mathcal{M}$-[[actegory]]) $\mathcal{C}$ (supported by the monoidal action $\bullet : \mathcal{M} \times \mathcal{C} \to \mathcal{C}$), and given $(A,\mu,e)$ a [[monoid in a monoidal category|monoid in]] $(\mathcal{M}, \odot, 1)$, then a **left module object** in $\mathcal{C}$ over $(A,\mu,e)$ is

1. an [[object]] $N \in \mathcal{C}$;

1. a [[morphism]] $\rho \;\colon\; A \bullet N \longrightarrow N$ (called the _[[action]]_);

such that

1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]:

   $$
     \array{
       1 \bullet N
         &\overset{e \bullet id}{\longrightarrow}&
       A \bullet N
       \\
       & {}_{\mathllap{\lambda}}\searrow
       & \downarrow^{\mathrlap{\rho}}
       \\
       && N
     }
     \,,
   $$

   where $\lambda$ is the [[actegory#Actegory|unitor]] of $\bullet$.

1. (action property) the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (A \odot A) \bullet N
         &\underoverset{\simeq}{\alpha_{A,A,N}}{\longrightarrow}&
       A \bullet (A \bullet N)
         &\overset{A \bullet \rho}{\longrightarrow}&
       A \bullet N
       \\
       {}^{\mathllap{\mu \bullet N}}\downarrow
         && &&
       \downarrow^{\mathrlap{\rho}}
       \\
       A \bullet N
         &\longrightarrow&
         &\overset{\rho}{\longrightarrow}&
       N
     }
     \,,
   $$

   where $\alpha$ is the [[actegory#Actegory|actor]] of $\bullet$.
=--

## Examples

\begin{example}\label{GeometricActions}
**(geometric actions)**

A [[group object]]-action

* internal to [[Sets]] is a [[G-set]]: a [[set]] equipped with a [[group action]] by a [[discrete group]] -- this is the plain notion of a group action;

* internal to [[TopologicalSpaces]] is a [[topological G-space]]: a [[topological group]] equipped with an [[action]] on a [[topological space]] by [[continuous functions]];

* internal to [[SmoothManifolds]] is a [[G-manifold]]: a [[Lie group]] equipped with an [[action]] on a [[smooth manifold]] by [[smooth functions]];

* internal to [[SimplicialSets]] is a [[simplicial group action]],

\end{example}

\begin{example}\label{EquivariantPrincipalBundle}
**([[equivariant principal bundles]])**

A $G$-[[equivariant principal bundle]] is an internal action of a [[group object]] internal to a [[category]] of internal $G$-[[actions]] as in Example \ref{GeometricActions}, such as [[G-sets]]/[[G-spaces]]/[[G-manifolds]] (an "[[equivariant group]]") which satisfies, internally, [[principal bundle|principality]] and [[locally trivial bundle|local triviality]]-condition.

\end{example}

\begin{example}\label{ActionObjectsInCat}
**(2-actions)**
\linebreak
The notion of [[coherence|coherent]] action object in the [[2-category]] [[Cat]] (of [[categories]] with [[functors]] and [[natural transformations]]) is a [[categorification|categorified]] notion of "action" (namely of [[monoidal categories]]), known as *[[module categories]]* (also: "*[[actegories]]*"), see also *[[2-module]]* and *[[n-module]]*).
\end{example}

\begin{example}
**([[module spectrum]])** \linebreak

A module object in a [[symmetric monoidal category of spectra]] is a [[module spectrum]] over a [[ring spectrum]].

Internal to just the [[stable homotopy category]] it is a _[[homotopy module spectrum]]_.

\end{example}

## Related concepts

* [[module over a monoidal functor]]

* [[(infinity,1)-module]], [[(infinity,n)-module]]

\linebreak

[[categorical algebra -- contents]]

**[[internalization]] and [[categorical algebra]]**

* [[monoid object]]

* [[group object]]

* [[ring object]]

* [[algebra object]] (associative, [[Lie algebra object|Lie]], ...)

* [[module object]]/[[action object]]

* [[internal locale]]

* [[internal category]]

  * [[internal groupoid]]

  * [[internal site]]

  * [[internal diagram]]

**[[universal algebra]]**

* [[Lawvere theory|algebras over]]$\,$ [[algebraic theories]],

* [[algebra over a monad|algebras over]]$\,$ [[monads]],

* [[algebra over an operad|algebras over]]$\,$ [[operads]].

**[[categorical semantics]]**

* [[internal logic]], [[internal language]]

* [[relation between category theory and type theory]]

## References

The general definition of internal actions seems to have first been formulated in:

* {#Grothendieck61} [[Alexander Grothendieck]], p. 106 (9 of 21) of: [[FGA]] _Techniques de construction et théorèmes d'existence en géométrie algébrique III: préschémas quotients_, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212,   ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf), [English transl. pdf](https://labs.thosgood.com/builds/fga-3-III.pdf))

following the general principle of [[internalization]] formulated in:

* {#Grothendieck60} [[Alexander Grothendieck]], p. 370 (3 of 23) in: [[FGA]] *Technique de descente et théorèmes d'existence en géométrie algébriques. II: Le théorème d'existence en théorie formelle des modules*, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 ([numdam:SB_1958-1960__5__369_0](http://www.numdam.org/item/SB_1958-1960__5__369_0), [pdf](http://www.numdam.org/item/SB_1958-1960__5__369_0.pdf), [English transl. pdf](https://labs.thosgood.com/builds/fga-3-II.pdf))

Review:

* [[Barbara Fantechi]], [[Lothar Göttsche]], [[Luc Illusie]], [[Steven L. Kleiman]], [[Nitin Nitsure]], [[Angelo Vistoli]], Section 2.2 of: _Fundamental algebraic geometry. Grothendieck's [[FGA explained]]_, Mathematical Surveys and Monographs __123__, Amer. Math. Soc. (2005) [[MR2007f:14001](http://www.ams.org/mathscinet-getitem?mr=2007f:14001), [ISBN:978-0-8218-4245-4](https://bookstore.ams.org/surv-123-s), [lecture notes](http://indico.ictp.it/event/a0255/other-view?view=ictptimetable)]

* [[Saunders MacLane]], Sections VII.4 in: *[[Categories for the Working Mathematician]]*, Springer  (1979, 2nd ed.) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

* [[John Michael Boardman]], *Algebraic objects in categories*, Chapter 7 of: _Stable Operations in Generalized Cohomology_ [[pdf](https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf), [[Boardman-StableOperations.pdf:file]]] in: [[Ioan Mackenzie James]] (ed.) _[[Handbook of Algebraic Topology]]_ Oxford (1995) [[doi:10.1016/B978-0-444-81779-2.X5000-7](https://doi.org/10.1016/B978-0-444-81779-2.X5000-7)]

* [[Magnus Forrester-Barker]], *Group Objects and Internal Categories* [[arXiv:math/0212065](https://arxiv.org/abs/math/0212065)]

See also:

* [[Francis Borceux]], [[George Janelidze]], [[Gregory Maxwell Kelly]], p. 8 of: _Internal object actions_, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 ([dml:249553](https://eudml.org/doc/249553))

* [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], pp. 15 in: *Symmetric spectra*, J. Amer. Math. Soc. **13** (2000) 149-208 [[arXiv:math/9801077](http://arxiv.org/abs/math/9801077), [doi:10.1090/S0894-0347-99-00320-3](https://doi.org/10.1090/S0894-0347-99-00320-3)]

Discussion that the [[Mod|category of module objects]] over a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] a [[bicomplete category|bicomplete]] [[closed symmetric monoidal category]] is itself [[bicomplete category|bicomplete]] [[closed symmetric monoidal category|closed symmetric monoidal]]:

* [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], Lemma 2.2.2 & 2.2.8 in: *Symmetric spectra*, J. Amer. Math. Soc. **13** (2000) 149-208 [[arXiv:math/9801077](http://arxiv.org/abs/math/9801077), [doi:10.1090/S0894-0347-99-00320-3](https://doi.org/10.1090/S0894-0347-99-00320-3)]

* [[Florian Marty]], Prop. 1.2.14, 1.2.16, 1.2.17 in: _Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative_, Ph.D. Toulouse (2009) [[theses:2009TOU30071](https://www.theses.fr/2009TOU30071), [[Marty-DesOuverts.pdf:file]]]

* [[Martin Brandenburg]], Prop. 4.1.10: *Tensor categorical foundations of algebraic geometry* [[arXiv:1410.1716](http://arxiv.org/abs/1410.1716)]

See also:

* [MO:180673](http://mathoverflow.net/questions/180673/category-of-modules-over-commutative-monoid-in-symmetric-monoidal-category).

Lecture notes:

* [[Urs Schreiber]]: *[Algebras and Modules](Introduction+to+Stable+homotopy+theory+--+1-2#AlgebrasAndModules)*, in *[[Introduction to Stable Homotopy Theory]]* 1.2: *[[Introduction to Stable homotopy theory -- 1-2|Structured Spectra]]*

[[!redirects module objects]]

[[!redirects module in a monoidal category]]
[[!redirects modules in a monoidal category]]

[[!redirects action object]]
[[!redirects action objects]]

[[!redirects internal action]]
[[!redirects internal actions]]

[[!redirects group action object]]
[[!redirects group action objects]]


module over a groupoid

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

A _module over a groupoid_ is a collection of [[abelian group]]s equipped with a linear  [[action]] by a [[groupoid]].

## Definition

+-- {: .un_def}
###### Definition
**(module over a groupoid)**

Let $\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G})$ be a [[groupoid]]. A **module over the groupoid** $\mathcal{G}$ is a collection $\{N_x\}_{x \in \mathcal{G}_0}$ of [[abelian group]]s equipped with a collection of maps

$$
  N_x \times \mathcal{G}(x,y) \to N_y
$$

that are linear and respect the groupoid composition in the obvious way.

=--

## References

* [[Ronnie Brown]], [[Philip Higgins]], [[Rafael Sivera]], _[[Nonabelian Algebraic Topology]]_

[[!redirects modules over a groupoid]]
[[!redirects modules over groupoids]]


module over a monad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Just as the notion of a [[monad]] in a [[bicategory]] $K$ generalizes that of a [[monoid in a monoidal category]], [[module over a monoid|modules over monoids]] generalize easily to modules over monads.

Beware that modules over monads in [[Cat]] are often called [[algebra over a monad|*algebras* for the monad]] (see there for more), since they literally are algebras in the sense of [[universal algebra]], see [below](#AlgebrasForMonadsInCat). By extension, one might speak of modules over monads in any [[2-category]] as "algebras for the monad".

The [[formal dual|formally dual]] concept is that of _[[coalgebra over a comonad]]_.

## Definition

### Modules

Let $K$ be a [[bicategory]] and $t \colon a \to a$ a [[monad]] in $K$ with structure 2-cells $\mu \colon t t \Rightarrow t$ and $\eta \colon 1_a \Rightarrow t$.  Then a **left $t$-module** (or **$t$-algebra**) is given by a 1-cell $x \colon b \to a$ and a 2-cell $\lambda \colon t x \Rightarrow x$, where
$$
\array{
  t t x & \overset{\mu x}{\to} & t x \\
  t\lambda\downarrow & & \downarrow \lambda \\
  t x & \underset{\lambda}{\to} & x
}
\qquad \qquad
\array{
  x & \overset{\eta x}{\to} & t x \\
  & 1\searrow & \downarrow \lambda \\
  & & x
}
$$
commute.  Similarly, a **right $t$-module** (or **$t$-opalgebra**) is given by a 1-cell $y \colon a \to c$ and a 2-cell $\rho \colon y t \Rightarrow y$, with commuting diagrams as above with $y$ on the left instead of $x$ on the right.

Clearly, a right $t$-module in $K$ is the same thing as a left $t$-module in $K^{\mathrm{op}}$.  A **left $t$-comodule** or **coalgebra** is then a left $t$-module in $K^{\mathrm{co}}$, and a **right $t$-comodule** is a left $t$-module in $K^{\mathrm{coop}}$.

A $t$-module of any of these sorts is _a fortiori_ an [[algebra over an endomorphism|algebra over the underlying endomorphism]] $t$.

### Bimodules

Given monads $s$ on $b$ and $t$ on $a$, an **$s,t$-bimodule** is given by a 1-cell $x\colon b \to a$, together with the structures of a right $s$-module $\rho \colon x s \Rightarrow x$ and a left $t$-module $\lambda \colon t x \Rightarrow x$ that are compatible in the sense that the diagram
$$
\array{
  t x s & \overset{t\rho}{\to} & t x \\
  \lambda s \downarrow & & \downarrow \lambda \\
  x s & \underset{\rho}{\to} & x
}
$$
commutes.  Such a bimodule may be written as $x \colon s ⇸ t$.

A **morphism** of left $t$-modules $(x,\lambda) \to (x',\lambda')$ is given by a 2-cell $\alpha \colon x \Rightarrow x'$ such that $\lambda' \circ t\alpha = \alpha \circ \lambda$.  Similarly, a morphism of right $t$-modules $(y,\rho) \to (y',\rho')$ is $\beta \colon y \Rightarrow y'$ such that $\rho' \circ \alpha s = \alpha \circ \rho$.  A morphism of bimodules $(x,\lambda,\rho) \to (x',\lambda',\rho')$ is given by $\alpha \colon x \Rightarrow x'$ that is a morphism of both left and right modules.

More abstractly, the monads $s$ and $t$ in $K$ give rise to ordinary monads $s^*$ and $t_*$ on the hom-category $K(b,a)$, by pre- and post-composition.  The associativity isomorphism of $K$ then gives rise to an invertible [[distributive law]] between these, so that the composite $s^* t_* \cong t_* s^* \colon x \mapsto t x s$ is again a monad.  Then the category $Mod_K(s,t)$ of bimodules from $s$ to $t$ is the ordinary [[Eilenberg--Moore category]] $K(b,a)^{s^* t_*}$.

### Algebras for monads in Cat
 {#AlgebrasForMonadsInCat}

If $K =$ [[Cat]] and $(T,\eta,\mu)$ is a monad on a [[category]] $C$, then a left $T$-module $A \colon 1 \to C$, where $1$ is the [[terminal category]], is usually called an **[[algebra over a monad|algebra over $T$]]** or **$T$-algebra** (see there): it is given by an object $A \in C$ together with a morphism $\alpha \colon T A \to A$, such that
$$ \array {
  T(T(A))              & \stackrel{\mu_A}\rightarrow  & T(A) \\
  T(\alpha) \downarrow &                              & \downarrow \alpha \\
  T(A)                 & \stackrel{\alpha}\rightarrow & A
}
$$
and
$$ \array {
  A & \stackrel{\eta_A}\rightarrow  & T(A) \\
    & id_A \searrow & \downarrow \alpha \\
    &               & A
}
$$
commute.

In particular, every algebra over a monad $(T,\eta,\mu)$ in $Cat$ has the structure of an [[algebra over an endofunctor|algebra over the underlying endofunctor]] $T$.

$T$-algebras can also be defined as left [[modules]] over $T$ _qua_ monoid in $End(C)$.  There the object $A$ is represented by the constant endofunctor at $A$.

The [[Eilenberg-Moore category]] of $T$ is the category of these algebras.  It has a universal property that allows the notion of
[[Eilenberg-Moore object]] to be defined in any bicategory.

### In a virtual double category
It'sthe notion of (bi)module makes sense in [[virtual double categories]], generalizing the previous definition.

A *monad* in a virtual double category is a [[loose morphism|loose]] endomorphism $t : a \nrightarrow a$ together with a binary map (i.e.~a square in the virtual double category) $\mu:t,t \to t$ giving the multiplication and a nullary map $\eta:() \to t$ giving the unit:
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	\arrow[""{name=1, anchor=center, inner sep=0}, draw=none, from=1-4, to=1-5]
	\arrow[""{name=2, anchor=center, inner sep=0}, "t"', "\shortmid"{marking}, from=2-4, to=2-5]
	\arrow["\mu"', shorten >=3pt, Rightarrow, from=1-2, to=0]
	\arrow[shorten <=4pt, Rightarrow, no head, from=1, to=2-4]
	\arrow[shorten <=4pt, Rightarrow, no head, from=1, to=2-5]
	\arrow["\eta"'{pos=0.6}, shorten <=9pt, shorten >=4pt, Rightarrow, from=1, to=2]
\end{tikzcd}
This data satisfies strict unitality and associativity equations.

A *left module* $m$ over a *monad* $t:a \nrightarrow a$ is another loose cell $m:a \nrightarrow b$ together with a binary map $\rhd : t,m \to m$:
<!--
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-->
\begin{tikzcd}[row sep=scriptsize]
	a & a & b \\
	a && b
	\arrow[""{name=0, anchor=center, inner sep=0}, "m"', "\shortmid"{marking}, from=2-1, to=2-3]
	\arrow["t", "\shortmid"{marking}, from=1-1, to=1-2]
	\arrow["m", "\shortmid"{marking}, from=1-2, to=1-3]
	\arrow[Rightarrow, no head, from=1-1, to=2-1]
	\arrow[Rightarrow, no head, from=1-3, to=2-3]
	\arrow["\rhd"'{pos=0.4}, shorten >=3pt, Rightarrow, from=1-2, to=0]
\end{tikzcd}
This again satisfies standard equations stating compatibility with the monad structure on $t$.

A *right module* is equipped with a binary map $\lhd : m, t \to m$ instead, and a bimodule has a ternary map $\rhd - \lhd : t,m,t \to m$.

## Properties

### Colimits

see at _[[colimits in categories of algebras]]_

### Tensor product

Given bimodules $x' \colon r ⇸ s$ and $x \colon s ⇸ t$, where $r,s,t$ are monads on $c,b,a$ respectively, we may be able to form the **tensor product** $x \otimes_s x' \colon r ⇸ t$ just as in the case of bimodules over rings.  If the hom-categories of the bicategory $K$ have [[reflexive coequalizer]]s that are preserved by composition on both sides, then the tensor product is given by the reflexive coequalizer in $K(c,a)$
$$
\array{
  x s x' & \overset{\to}{\to} & x x' & \to x \otimes_s x'
}
$$
where the parallel arrows are the two induced actions $\rho x'$ and $x \lambda$ on $s$.  Indeed, under the hypothesis on $K$ the forgetful functor $Mod_K(r,t) = K(c,a)^{r^* t_*} \to K(c,a)$ [[colimits in categories of algebras|reflects reflexive coequalizers]], because the monad $r^* t_*$ preserves them, and so $x \otimes_s x'$ is an $r,t$-bimodule.

If $K$ satisfies the above conditions then there is a bicategory $Mod(K)$ consisting of monads, bimodules and bimodule morphisms in $K$.  The identity module on a monad $t$ is $t$ itself, and the unit and associativity conditions follow from the [[universal property]] of the above coequalizer.  There is a [[lax functor|lax]] forgetful functor $Mod(K) \to K$, with comparison morphisms $1_a \to t$ the unit of $t$, and $x x' \to x \otimes_s x'$ the coequalizer map.

## Examples

If $K = Span(Set)$, the bicategory of [[span]]s of sets, then a monad in $K$ is precisely a small category.  Then $Mod(K) = Prof$, the category of small categories, [[profunctor]]s and natural transformations.

More generally, $Mod(Span(C))$, for $C$ any category with coequalizers and pullbacks that preserve them, consists of [[internal category|internal categories]] in $C$, together with [[internal profunctor|internal profunctors]] between them and transformations between those.

## Related concepts

* **algebra over a monad**, [[algebra over an endofunctor]], [[coalgebra over an endofunctor]], [[algebra over a profunctor]]

  [[∞-algebra over an (∞,1)-monad]]

  * [[model structure on algebras over a monad]]

* [[algebra over an algebraic theory]]

  [[∞-algebra over an (∞,1)-algebraic theory]]

  * [[homotopy T-algebra]] / [[model structure on simplicial T-algebras]]

* [[algebra over an operad]]

   [[∞-algebra over an (∞,1)-operad]]

   * [[model structure on algebras over an operad]]

* [[Eilenberg-Moore category]], [[Kleisli category]], [[Eilenberg-Moore object]], [[Kleisli object]]

## References

* [[John Isbell]], _Generic algebras_ Transactions of the AMS, vol 275, number 2 ([pdf](http://www.ams.org/journals/tran/1983-275-02/S0002-9947-1983-0682715-8/S0002-9947-1983-0682715-8.pdf))

* H. Lindner, _Commutative monads_ in _Deuxiéme colloque sur l'algébre des catégories_. Amiens-1975. Résumés des conférences, pages 283-288. Cahiers de topologie et géométrie différentielle catégoriques, tome 16, nr. 3, 1975.

* R. Guitart, _Tenseurs et machines_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 21(1):5-62, 1980.

* A. Kock. _Closed categories generated by commutative monads_, Journal of the Australian Mathematical Society, 12(04):405-424, 1971.

* G. J. Seal. _Tensors, monads and actions_, Theory Appl. Categ., 28:No. 15, 403-433, 2013.

Discussion of [[model category]] structures on categories of [[coalgebras]] over [[comonads]] is in

* [[Kathryn Hess]], [[Brooke Shipley]], _The homotopy theory of coalgebras over a comonad_ ([arXiv:1205.3979](http://arxiv.org/abs/1205.3979))

[[!redirects module of a monad]]
[[!redirects module for a monad]]
[[!redirects modules of a monad]]
[[!redirects modules for a monad]]
[[!redirects modules over a monad]]
[[!redirects modules for monads]]
[[!redirects modules over monads]]


module over a monoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## 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 $A \otimes -$, with $A$ some [[monoid object]].

## Definition

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

$$
  \cdot : A \otimes A \to A
$$

and a unit element

$$
  e : I \to A
$$

satisfying the [[associativity law]] and the [[unit law]].

+-- {: .num_defn #ModuleInMonoidalCategory}
###### Definition

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

* an [[object]] $N \in \mathcal{V}$

* equipped with a morphism

  $$
    \lambda : A \otimes N \to N
  $$

  in $\mathcal{V}$

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

$$
  \array{
     A \otimes A \otimes N &\stackrel{id_A \otimes \lambda}{\to}& A \otimes N
     \\
     \downarrow^{\mathrlap{\cdot \otimes id_n}} && \downarrow^{\mathrlap{\lambda}}
     \\
     A \otimes N &\stackrel{\lambda}{\to}& N
  }
$$

and

$$
  \array{
     I \otimes N &&\stackrel{e \otimes id_N}{\to}&& A \otimes N
     \\
     & \searrow && \swarrow_{\mathrlap{\lambda}}
     \\
     && N
  }
  \,.
$$

=--

\begin{definition} Similarly a **right module** over $A$ in $(\mathcal{V}, \otimes, I)$ is

* an [[object]] $N \in \mathcal{V}$

* equipped with a morphism

  $$
    \rho : N \otimes A \to N
  $$

  in $\mathcal{V}$

such that this satisfies the axioms of a (right) [[action]], in that the following are [[commuting diagrams]] in $\mathcal{V}$:

$$
  \array{
     N \otimes A \otimes A &\stackrel{\rho \otimes id_A}{\to}& N \otimes A
     \\
     \downarrow^{\mathrlap{\cdot \otimes id_n}} && \downarrow^{\mathrlap{\rho}}
     \\
     N \otimes A &\stackrel{\rho}{\to}& N
  }
$$

and

$$
  \array{
     N \otimes I &&\stackrel{id_N \otimes e}{\to}&& N \otimes A
     \\
     & \searrow && \swarrow_{\mathrlap{\rho}}
     \\
     && N
  }
  \,.
$$
\end{definition}

## 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 $R$_ is a module in $(Ab,\otimes)$ according to def. \ref{ModuleInMonoidalCategory}.

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

+-- {: .num_defn #ModuleOverARing}
###### Definition

A **module** $N$ over a ring $R$ is

1. an [[object]] $N \in $ [[Ab]], hence an [[abelian group]];

1. equipped with a [[morphism]]

   $$
     \alpha : R \otimes N \to N
   $$

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

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

   and which is a [[bilinear function]] in that it satisfies

   $$
    (r, n_1 + n_2) \mapsto r n_1 + r n_2
   $$

   and

   $$
    (r_1 + r_2, n) \mapsto r_1 n + r_2 n
   $$

   for all $r, r_1, r_2 \in R$ and $n,n_1, n_2 \in N$;

1. such that the [[diagram]]

   $$
     \array{
        R \otimes R \otimes N
        &\stackrel{\cdot_R \otimes Id_N}{\to}& R \otimes N
        \\
        {}^{\mathllap{Id_R \otimes \alpha}}\downarrow
        &&
        \downarrow^{\mathrlap{\alpha}}
        \\
        R \otimes N &\to& N
     }
   $$

   [[commuting diagram|commutes]] in [[Ab]],  which means that for all elements as before we have

   $$
     (r_1 \cdot r_2) n = r_1 (r_2 n)
      \,.
   $$

1. such that the diagram

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

   commutes, which means that on elements as above

   $$
     1 \cdot n = n
     \,.
   $$

=--

+-- {: .num_remark }
###### Remark

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

$$
  Mod \to CRing
$$

over [[CRing]].

This fibration may be characterized intrinsically, which gives
yet another way of defining $R$-modules. This we turn to
[below](#ModulesOverARingInTermsOfStabilizedSlices).

=--

### $G$-sets
 {#GSets}

Simpler than the traditionally default notion of a module in
$(Ab,\otimes)$, as [above](#Rings) is that of a module in
[[Set]], equipped with its [[cartesian monoidal category|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]].

+-- {: .num_defn #ActionOfDiscreteGroupOnSet}
###### Definition

For $S \in $ [[Set]] and $G$ a [[discrete group]], a **$G$-[[action]]** of $G$ on $S$ is a [[function]]

$$
  \lambda \colon G \times S \to S
$$

such that

1. the neutral element acts trivially

   $$
     \array{
        * \times S &&\stackrel{\simeq}{\to}&& S
        \\
        & {}_{(e,id_S)}\searrow && \nearrow_{\mathrlap{\lambda}}
        \\
        && G \times S
     }
   $$

1. the action property holds: for all $g_1, g_2 \in G$ and $s \in S$ we have $\lambda(g_1,\lambda(g_2, s)) = \lambda(g_1 \cdot g_2, s)$.

=--

### Abelian groups with $G$-action as modules over the group ring
 {#AbelianGroupsWithGAction}

If a [[discrete group]] acts, as in def. \ref{ActionOfDiscreteGroupOnSet},
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. \ref{ModuleOverARing}.

+-- {: .num_defn }
###### Definition

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

1. whose underlying [[abelian group]] is the [[free abelian group]] on the set underlying $G$;

1. whose multiplication is given on [[basis]] elements by the group operation in $G$.

=--

+-- {: .num_remark }
###### Remark

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

$$
  r = \sum_{g \in G} r_g g
$$

with $r_g \in \mathbb{Z}$. Addition is given by addition of the [[coefficients]] $r_g$ and multiplication is given by the formula

$$
  \begin{aligned}
    r \cdot \tilde r
    & =
    \sum_{g \in G} \sum_{\tilde g \in G} (r_g \tilde r_{\tilde g}) g \cdot \tilde g
    \\
    & =
    \sum_{q \in G}
    \left(
      \sum_{g \tilde g = q} r_g \tilde r_{\tilde g}
    \right)
    q
  \end{aligned}
  \,.
$$

=--

+-- {: .num_prop }
###### Proposition

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 $A$.

=--

+-- {: .proof}
###### Proof

Since the underlying abelian group of $\mathbb{Z}[G]$ is a [[free abelian group|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 $U(A)$.

=--

+-- {: .num_remark }
###### Remark

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](Ext#RelationToGroupCohomology)_.

=--

### more examples

* In a [[symmetric monoidal category|symmetric monoidal]] [[category of chain complexes]] equipped with the [[tensor product of chain complexes]], then a [[monoid]] is a [[dg-algebra]], and a module is a _[[dg-module]]_. See there for more.

## Related concepts

* [[model structure on modules in a monoidal model category]]

## 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](http://thesesups.ups-tlse.fr/540/)

A summary is in section 4.1 of

* [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_, [arXiv:1410.1716](http://arxiv.org/abs/1410.1716).

See also [MO/180673](http://mathoverflow.net/questions/180673/category-of-modules-over-commutative-monoid-in-symmetric-monoidal-category), 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).

[[!redirects modules over a monoid]]
[[!redirects modules over monoids]]
[[!redirects modules over a monoid]]


module over a monoidal functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

To the extent that a [[monoidal functor]] is analogous to a [[monoid]], a _module over a monoidal functor_ is analogous to a [[module]] over (hence an [[action]] of) that monoid.

## Definition

+-- {: .num_defn}
###### Definition

Let

1. $\mathcal{C},\mathcal{D}$ be [[monoidal categories]], hence equipped with tensor product functors $\otimes_{\mathcal{C}}\colon \mathcal{C} \times \mathcal{C}\to \mathcal{C}$ and $\otimes_{\mathcal{D}}\colon \mathcal{D} \times \mathcal{D}\to \mathcal{D}$;

1. $\mathcal{N}$ be a left [[module category]] over $\mathcal{C}$, hence equipped with a compatible [[action]] functor $\rho \colon \mathcal{C}\times\mathcal{N} \to \mathcal{N}$;

1. $F \colon \mathcal{C}\to \mathcal{D}$ a [[lax monoidal functor]].

Then a _left module_ over $F$ is

1. a functor $N \colon \mathcal{N} \longrightarrow \mathcal{D}$

1. a [[natural transformation]]

   $\alpha \colon F(-)  \otimes_{\mathcal{D}} N(-)  \longrightarrow N(\rho(-,-))$

satisfying the evident [[categorification]] of the [[action]]-property.
Analogously for right modules and bimodules. (e.g. [Yetter 01, def. 39](#Yetter01)).

=--

## Example

* For $(\mathcal{C},\otimes)$ a [[symmetric monoidal category]] and the [[functor category]] $([\mathcal{C},Set],\otimes_{Day})$ equipped with the induced [[Day convolution]] product, then a [[monoid object]] with respect to $\otimes_{Day}$ is equivalently a [[lax monoidal functor]] from $\mathcal{C}$ to itself, and a [[module object]] over that monoid is equivalently a module over that functor in the above sense. See [there](Day+convolution#Monoids) for more.

## References

* {#Yetter01} [[David Yetter]], _Abelian categories of modules over a (Lax) monoidal functor_ ([arXiv:math/0112307](http://arxiv.org/abs/math/0112307))

[[!redirects modules over a monoidal functor]]

[[!redirects modules over monoidal functors]]


module over a quantale

# Modules over quantales and quantaloids

(People use also expressions: quantale module, quantic module)

## Idea

A [[quantale]] is an analogue of a noncommutative ring and a noncommutative generalization of a locale; it is a semigroup in the monoidal category of [[sup-lattice]]s; therefore, for a quantale $Q$, a left $Q$-module should be a sup-lattice $M$ together with an action $Q\otimes M\to M$ satisfying the usual axioms.  In the special case when the quantale is unital and commutative and hence a locale, such a module is called a "localic module", or module over a locale.  The multiobject generalization is called a quantaloid module.

## References

* Isar Stubbe, _Q-modules are Q-suplattices_, Theory and Appl. of Cat. 19, 2007, No. 4, pp 50-60 [abs](http://www.tac.mta.ca/tac/volumes/19/4/19-04abs.html)
* [[Pedro Resende]], _Groupoid sheaves as quantale sheaves_, J. Pure Appl. Algebra 216 (2012), 41–70; [arxiv/0807.4848](http://arxiv.org/abs/0807.4848v3)
[doi](http://dx.doi.org/10.1016/j.jpaa.2011.05.002)
* Hans Heymans, _Sheaves on quantales as generalized metric spaces_, Doctor thesis, Universiteit Antwerpen 2010 [pdf](http://www.ua.ac.be/download.aspx?c=hans.heymans&n=31611&ct=32634&e=282776)

The localic case is studied in

* [[André Joyal]], M. Tierney, _An extension of the Galois theory of Grothendieck_, Mem. Amer. Math. Soc. 51 (1984), no. 309, vii+71 pp.
* [[Pedro Resende]], E. Rodrigues, _Sheaves as modules_,  Appl. Categ. Structures 18 (2010) 199-217; [arXiv:0711.4401](http://arxiv.org/abs/0711.4401)

[[!redirects module over a locale]]
[[!redirects modules over quantales]]
[[!redirects modules over a quantale]]
[[!redirects localic module]]
[[!redirects quantaloid module]]
[[!redirects module over a quantaloid]]


[[!redirects ∞-module over an ∞-algebra over an (∞,1)-operad]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

(...)

## Related concepts

* [[∞-module]]

* [[(∞,1)-operad]], [[model structure on operads]]

  * [[algebra over an (∞,1)-operad]], [[model structure on algebras over an operad]]

    * **module over an algebra over an (∞,1)-operad**, [[model structure on modules over an algebra over an operad]]

## References

Section 3.3.3 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}


module over an algebra over an operad

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## Idea

The notion of [[module]] over an [[associative algebra]] has a generalization to a notion of modules over an algebra that is an [[algebra over an operad]].

Note that sometimes an algebra over an operad is called a module over the operad, so here we have a module over a module.  (Whether algebras/modules over operads are more like algebras or more like modules depends on your point of view, so both terms are used.)

## Definition

Let $\mathcal{E}$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]], $P$ an [[operad]] in $\mathcal{E}$ and $A$ a $P$-[[algebra over an operad]].

A **module** over $A$ consists of

* an [[object]] $N \in \mathcal{E}$;

* for all $1 \leq k \leq n \in \mathbb{N}$ a [[morphism]]

  $$
    \mu_{n,k} : P(n) \otimes A^{\otimes^{k-1}} \otimes N \otimes
      A^{\otimes^{n-k}} \to N
  $$

  in $\mathcal{E}$ (the [[action]] morphims)

* such that this data satisfies

  * [[unitality]] (...)

  * [[associativity]] (...)

  * equivariance (...)

## Properties

Under suitable conditions there is a [[model structure on modules over an algebra over an operad]].

## Examples

$A_\infty$-modules, etc.

(...)

## Related concepts

* [[model structure on modules over an algebra over an operad]]

## References

A review of modules over algebras over operads is at the beginning of

* [[Clemens Berger]], [[Ieke Moerdijk]], _On the derived category of an algebra over an operad_ ([arXiv](http://arxiv.org/abs/0801.2664))

[[!redirects module over an algebra over an operad]]
[[!redirects modules over an algebra over an operad]]
[[!redirects modules over algebras over an operad]]
[[!redirects modules over algebras over operads]]


module over an enriched category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Higher algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Higher linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The [[action]] on a [[module]] over a [[monoid]] $A$ in a [[closed monoidal category]] $V$ may be equivalently encoded in terms of a $V$-[[enriched functor]]

   $$
     \rho : \mathbf{B}A^{op} \to V
   $$

from the [[delooping]] one-object $V$-[[enriched category]] $\mathbf{B}A$, corresponding to $A$, to $V$ itself.

   More generally it makes sense to replace $\mathbf{B}A$ by any $V$-[[enriched category]] $C$ -- regarded as the [[horizontal categorification]] of a monoid, a "monoid-oid" -- and think of a $V$-enriched presheaf $\rho : C \to V$ as a __module over the category__ $C$.

   From this perspective a $C$-$D$-[[bimodule]] is a $V$-[[enriched functor]] $C^{op}\times D \to V$, which is in this context known as a [[profunctor]] from $C$ to $D$. The notion of the [[bicategory]] $V Mod$ of $V$-enriched categories, $V$-profunctors between these and transformations between those is then a generalization of the category of monoids in $V$ and bimodules between them.

## Definition

Let $V$ be a [[closed monoidal category]]. Let $C$ be a $V$-[[enriched category]].

+-- {: .num_defn}
###### Definition
A **left module** over $V$ is a $V$-[[presheaf]] on $C$, i.e. a [[enriched functor|functor]] of $V$-[[enriched categories]]
  $$ M : C^{op} \longrightarrow V. $$
Dually a **right module** is a $V$-[[enriched functor]] $M : C \to V$.
=--

Let $C$ and $D$ be $V$-[[enriched categories]].

+-- {: .num_defn}
###### Definition
A $C$-$D$-**[[bimodule]]** is a $V$-[[enriched functor]]
  $$ C^{op} \otimes D \longrightarrow V, $$
i.e. a left $C \otimes D^{op}$-module.
=--

$C$-$D$-[[bimodules]] are also known as _[[profunctors]]_ or _[[distributor]]s_ from $C$ to $D$.

## Examples

### Modules over a ring

For $R$ a [[ring]], write $\mathbf{B}R$ for the [[Ab-enriched category]] with a single object and [[hom-object]] $R = \mathbf{B}R(\bullet, \bullet)$.

Then a left $R$-module $N$ is equivalently an [[Ab]]-[[enriched functor]]
$$
  N : \mathbf{B}R^{op} \to Ab
  \,.
$$

This makes manifest that the category $R$[[Mod]] is
an [[Ab-enriched category]], namely the
[[Ab]]-[[enriched functor category]]

$$
  R Mod \simeq [\mathbf{B}R,Ab]
  \,.
$$

The right $R$-modules can be considered as $Ab$-functors $\mathbf{B}R\to Ab$. Then the usual [[tensor product of abelian groups]] $M\otimes N$ of left and right $R$-modules can be considered as a functor
  $$ \mathbf{B}R^{op}\otimes \mathbf{B}R\to Ab. $$
The [[coend]] $\int^R M\otimes N$ computes then to $M\otimes_R N$.

### $G$-Sets
 {#GSetsAsPresheaves}

Classically the notion of module is always regarded internal to [[Ab]], so that a module is always an abelian group with extra structure. But noticing that such abelian ring modules are just enriched presheaves in [[Ab]]-[[enriched category theory]], it makes sense to consider enriched presheaves in general $V$-enriched category theory as a natural generalization of the notion of module.

For that generalization the case of [[Set]]-[[enriched category theory]] plays a special basic role:

a [[group]] $G$ (with no extra structure, i.e. just a [[set]] with group structure) is a monoid in [[Set]]. A module over $G$ in the sense of [[Set]]-[[enriched functor]] (just an ordinary [[functor]])

$$
  \mathbf{B}G \to Set
$$

is nothing but a **$G$-set**: a [[set]] equipped with a $G$-action.

$\mathbf{B}G$ is the [[small category]] that is the [[delooping]] [[groupoid]] of $G$, which has a single object and $Hom_{\mathbf{B}G}(\bullet,\bullet) = G$. The functor $\mathbf{B}G \to Set$ takes the single object to some set $S$ and takes each morphism $(\bullet \stackrel{g}{\to} \bullet)$ to an [[automorphism]] $\rho(g) : S \to S$ of that set, such that composition is respected. This is just a [[representation]] of $G$ on the set $S$.

Of course for this story to work, $G$ need not be a group, but could be any [[monoid]].

## Related concepts

* [[enriched presheaf category]]

## References

See the references at [[enriched category theory]] and [[profunctor]].

[[!redirects modules over an enriched category]]
[[!redirects module over enriched category]]
[[!redirects modules over enriched categories]]
[[!redirects enriched presheaf]]
[[!redirects enriched presheaves]]


module over an operad

## Definition

An [[operad]] is a [[monoid]] in the [[monoidal category]]
of [[symmetric sequences]] equipped with the [[substitution product]].

A __module over an operad__ is just a [[right module]] over this [[monoid]].

Right modules are very different from left modules,
the latter are essentially [[algebras over an operad]].

## Related concepts

* [[operad]]

* [[algebra over an operad]]

* [[module over an algebra over an operad]]

## References

* [[V. A. Smirnov]].  ON THE COCHAIN COMPLEX OF TOPOLOGICAL SPACES.  Mathematics of the USSR-Sbornik 43:1 (1982), 133–144.  [doi](http://dx.doi.org/10.1070/sm1982v043n01abeh002437).

* [[Martin Markl]], _Models for operads_, Comm. Algebra 24 (1996), no. 4, 1471–1500.  [arXiv:hep-th/9411208v1](https://arxiv.org/abs/hep-th/9411208v1).


module sheaf

A [[sheaf]] of [[modules]] (hence in particular an [[abelian sheaf]]). In a  [[ringed topos]] usually understood as being modules over the [[structure sheaf]].

Special important types include

* [[coherent sheaf]]

* [[quasicoherent sheaf]].

[[!redirects module sheaves]]


module spectrum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Stable Homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _module spectrum_ is a [[module]] over a [[ring spectrum]]. As for [[ring spectra]], this has different interpretations according to how much [[coherence]] is considered:

A _homotopy module spectrum_ over a [[homotopy commutative ring spectrum]] is a [[module object]] in the [[stable homotopy category]] equipped with its derived [[smash product of spectra]].

On the other hand if "ring spectrum" means [[E-∞ ring]] (or [[A-∞ ring]]) then a module over a ring spectrum is  a [[module over an algebra over an (∞,1)-operad]] for the [[commutative operad]]:

for $R$ an [[E-∞ ring]] (an [[algebra over an (∞,1)-operad|∞-algebra]] over [[Comm]]), an $R$-module spectrum is a [[spectrum]] equipped with an $R$-[[action]].

## Properties

### General

By the discussion an [[tangent (∞,1)-category]] we may realize $E_\infty$-modules over $R$ as objects in the [[stabilization]] of the [[over-(∞,1)-category]] over $R$:

+-- {: .num_prop}
###### Proposition

Let $E_\infty := Alg^{Comm}(\infty Grpd)$ be the [[(∞,1)-category]] of [[E-∞ ring]]s and let $R \in E_\infty$. Then the [[stabilization]] of the [[over-(∞,1)-category]] over $R$

$$
  Stab(E_\infty/R) \simeq R Mod(Spec)
$$

is equivalent to the category of $R$-module spectra.

=--

This is ([Lurie, cor. 1.5.15](#Lurie)).

### Stable Dold-Kan correspondence
 {#StableDoldKanCorrespondence}

For $R$ an ordinary [[ring]], write $H R$ for the corresponding [[Eilenberg-MacLane spectrum]].

+-- {: .num_theorem #StableDoldKan}
###### Theorem

For $R$ any [[ring]] (or [[ringoid]], even) there is a [[Quillen equivalence]]

$$
  H R Mod \simeq Ch_\bullet(R Mod)
$$

between model structure on  $H R$-module spectra and the [[model structure on chain complexes]] (unbounded) of ordinary $R$-[[module]]s.

This presents a corresponding [[equivalence of (∞,1)-categories]]. If $R$ is a commutative ring, then this is an equivalence of [[symmetric monoidal (∞,1)-categories]].

=--

This equivalence on the level of [[homotopy categories]] is due to ([Robinson](#Robinson)). The refinement to a Quillen equivalence is ([SchwedeShipley, theorem 5.1.6](#SchwedeShipley)). See also the discussion at _[[stable model categories]]_. A direct description as an equivalence of $(\infty,1)$-categories appears as ([Lurie, remark 7.1.1.16](#Lurie)). If $R$ is a commutative ring, then this equivalence underlies an equivalence of symmetric monoidal $(\infty,1)$-categories by ([Lurie, theorem 7.1.2.13](#Lurie)).

+-- {: .num_remark}
###### Remark

This is a stable version of the [[Dold-Kan correspondence]]. See at _[[stable Dold-Kan correspondence]]_ for more.

=--

See at __[[algebra spectrum]]_ for the corresponding statement for $H R$-algebra spectra and [[dg-algebras]].

+-- {: .num_example}
###### Example

For $X$ a [[topological space]] and $R$ a ring, let $C_\bullet(X, R)$ be the standard [[chain complex]] for [[singular homology]] $H_\bullet(X, R)$ of $X$ with coefficients in $R$.

Under the stable Dold-Kan correspondence, prop. \ref{StableDoldKan}, this ought to be identified with the [[smash product]] $(\Sigma^\infty_+ X) \wedge H R$ of the [[suspension spectrum]] of $X$ with the [[Eilenberg-MacLane spectrum]]. Notice that by the general theory of [[generalized homology]] the [[homotopy groups]] of the latter are again singular homology

$$
  \pi_\bullet( (\Sigma^\infty_+ X) \wedge H R) \simeq H_\bullet(X, R)
  \,.
$$

While the correspondence $(\Sigma^\infty_+ X) \wedge H R \sim C_\bullet(X,R)$ under the above equivalence is suggestive, maybe nobody has really checked it in detail. It is sort of stated as true for instance on <a href="http://arxiv.org/PS_cache/arxiv/pdf/0906/0906.5198v1.pdf#page=15">p. 15 </a> of ([BCT](#BCT)).

=--

### As fibers for $(\infty,1)$-vector bundles

An ordinary [[vector bundle]] is a [[bundle]] of $k$-[[module]]s for $k$ some [[ring]] (which should be a [[field]], or otherwise we'd rather say "module bundle"). Generalizing $k$ here from a ring to a [[ring spectrum]], we may hence regard $K$-module spectra as [[(∞,1)-vector space]]s, and [[associated ∞-bundle|∞-bundles]] of these as [[(∞,1)-vector bundle]]s. See there for more details.

## Related concepts

[[!include homological and higher algebra -- table]]

* [[comodule spectrum]]

* [[ring spectrum]], [[algebra spectrum]]

* [[(∞,1)-category of (∞,1)-modules]]

## References

The notion of homotopy module spectra ([[module objects]] in the [[stable homotopy category]]):

* {#Adams74} [[Frank Adams]], part III, section 10 of _[[Stable homotopy and generalised homology]]_, 1974

Review:

* [[Urs Schreiber]], _[[Introduction to Stable homotopy theory]] -- [Homotopy ring spectra](Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra)_, 2016

Discussion in terms of [[symmetric spectra]]:

* {#Schwede12} [[Stefan Schwede]], chapter IV _[[Symmetric spectra]]_, 2012 ([pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf))

and in terms of [[(infinity,1)-category theory]] in

* {#Lurie} [[Jacob Lurie]], chapter 7  of _[[Higher Algebra]]_.

The equivalence between the [[homotopy categories]] of  $H R$-module spectra and $Ch_\bullet(R Mod)$ is due to

* {#Robinson} Alan Robinson, _The extraordinary derived category_ , Math. Z. 196 (2) (1987) 231–238.

The refinement of this statement to a [[Quillen equivalence]] is due to

* {#SchwedeShipley} [[Stefan Schwede]], [[Brooke Shipley]], _Stable model categories are categories of modules_ , Topology 42 (2003), 103-153 ([pdf](http://www.math.uic.edu/~bshipley/classTopFinal.pdf))

Applications to [[string topology]] are discussed in

* {#BCT} [[Andrew Blumberg]], [[Ralph Cohen]], [[Constantin Teleman]], _Open-closed field theories, string topology and Hochschild homology_ ([arXiv:0906.5198](http://arxiv.org/abs/0906.5198))

See the section on string topology at [[sigma model]] for more on this.

[[!redirects module spectra]]
[[!redirects module-spectra]]
[[!redirects module-spectrum]]

[[!redirects homotopy module spectrum]]
[[!redirects homotopy module spectra]]


modules over a ring are equivalent to quasicoherent sheaves over its spectrum

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###Context###
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A basic result in the study of [[coherent sheaf|coherent]] and [[quasicoherent sheaves]] (of [[modules]]) over [[affine schemes]] says that for a [[commutative ring]] $R$ the [[quasicoherent modules]] over the [[spectrum of a ring|Zariski spectrum]] $Spec(R)$ are equivalent to $R$-modules [[Serre 1955](#Serre55)].

## Statement in the formalism of locally ringed spaces

Recall that given a commutative unital ring $R$, there is an adjunction

$$
  \widetilde{(-)}
  \,\dashv\,
  \Gamma
  \;\colon\;
  \mathcal{O}(Spec\; R)\text{-}Mod
    \leftrightarrows
  R\text{-}Mod
  \,,
$$

between the [[category of modules]] over its affine spectrum, i.e., the [[local ring]] object $\mathcal{O}(Spec\; R)$ in the category of [[sheaves]] over the [[topological space]] $Spec(R)$, and the category of ordinary [[modules]] over $R$.

In the [[right adjoint]] direction, for an $\mathcal{O}(Spec\; R)$-module $\mathcal{F}$, we have the [[presheaf]] value $\mathcal{F}(Spec\; R)$, the module of [[global sections]], usually denoted $\Gamma(\mathcal{F})$.

In the [[left adjoint]] direction, for a (left) $R$-module $M$, we have a sheaf $\tilde{M}$ over $Spec(R)$, whose value $\tilde{M}(D_f)$ at a typical open $D_f$ of $Spec(R)$ is given by the localization $R[f^{-1}]\otimes_R M$, and where the restriction maps are given by canonical maps between these localizations. This gives a sheaf of modules over the sheaf of rings $\mathcal{O}(Spec\; R)$.

The [[unit of an adjunction|unit of this adjunction]], with components $M \to \Gamma(\tilde{M})$, is the canonical isomorphism $M \cong R \otimes_R M$. The counit, with components $\widetilde{\Gamma(\mathcal{F})} \to \mathcal{F}$, is the presheaf map whose value at a typical open $D_f$ is the canonical map

$$
  R[f^{-1}]
    \otimes_R
  \mathcal{F}(Spec\; R)
    \longrightarrow
  \mathcal{F}(D_f)
$$

induced by the restriction map

$$
  \mathcal{F}(Spec\; R)
    \longrightarrow
  \mathcal{F}(D_f)
$$

of $R$-modules (with $R$ acting on the [[codomain]] by [[restriction of scalars]] along the ring map $R \to R[f^{-1}]$), noting that $R[f^{-1}] \otimes_R (-)$ is left adjoint to the functor that restricts scalars along $R \to R[f^{-1}]$.

The theorem of [Serre (1955)](#Serre55) is that this [[adjoint functor|adjunction]] restricts to an [[adjoint equivalence]] of categories, which we denote as

$$
  \widetilde{(-)} \dashv \Gamma
  \;\colon\;
  Qcoh(Spec R)
    \leftrightarrows
  R\text{-}Mod
$$

between [[quasicoherent sheaf|quasicoherent modules]] over $\mathcal{O}_{Spec R}$, and ordinary $R$-modules. In particular, an $\mathcal{O}_{Spec\; R}$-module $\mathcal{F}$ is quasicoherent if and only if the counit map

$$
  \varepsilon_\mathcal{F}
    \;\colon\;
  \widetilde{\Gamma(\mathcal{F})}
    \longrightarrow
  \mathcal{F}
$$

is an [[isomorphism]].

Furthermore, if $R$ is [[Noetherian ring|Noetherian]], the adjoint equivalence restricts further to an equivalence between [[coherent sheaves|coherent modules]] over $\mathcal{O}_{Spec R}$ and finitely generated modules over $R$.

## Statement in the formalism of functor of points

In the formalism of [[functor of points]], the equivalence turns into a definition: affine schemes are defined as the [[opposite category]] of the category of [[commutative rings]] (with the functor $Spec$ now being tautologically defined as the identity functor), and the category of quasicoherent modules over $Spec R$ is now defined as the category of $R$-modules.  This assignment defines a [[stack]] of categories over the site of affine schemes with the [[Zariski topology]].

The functor of points approach carries over to quasicoherent modules over non-affine schemes: given such a scheme $X$, a quasicoherent module over $X$ is a morphism of [[stacks]] from $X$ to the stack of quasicoherent modules defined above.  In concrete terms, this boils down to picking an open cover of $X$ and defining a quasicoherent module using [[cocycle data]].

## Related concepts

* [[Serre-Swan theorem]]

* [[duality between algebra and geometry]]

## References

The result is originally due to:

* {#Serre55} [[Jean-Pierre Serre]], §48 & §49 in: *Faisceaux Algebriques Coherents*, Ann. Math. **61** 2 (1955) 197. [[doi:10.2307/1969915](https://doi.org/10.2307/1969915)]

It appears in many texts (often without a name, but elsewhere in the nLab it is referred to as the "affine Serre theorem"), for example:

* [[Robin Hartshorne]], Chapter II, Corollary 5.5 in *Algebraic Geometry*, Graduate Texts in Mathematics **52**, Springer (1977) [[doi.org/10.1007/978-1-4757-3849-0](https://doi.org/10.1007/978-1-4757-3849-0)]

* [[The Stacks Project]], [Lemma 26.7.5](https://stacks.math.columbia.edu/tag/01IB).

category: algebra, algebraic geometry

[[!redirects affine Serre's theorem]]
[[!redirects affine Serre's theorem]]
[[!redirects Serre's theorem on quasicoherent sheaves over affine schemes]]
[[!redirects modules over a ring are equivalent to quasicoherent modules over its spectrum]]
[[!redirects modules over a ring are equivalent to quasicoherent modules over its spectrum]]
[[!redirects quasicoherent sheaves over affine schemes]]
[[!redirects quasicoherent sheaves over affine schemes]]
[[!redirects quasicoherent module on an affine spectrum]]


moduli

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The term _moduli_ refers to

1. generally in [[mathematics]]: [[elements]] of a [[moduli space]]/[[moduli stack]]; more generally: [[generalized elements]], hence [[maps]] to a [[moduli stack]];

1. specifically in [[physics]] and notably in [[string theory]]: parameters of a [[KK compactification]], hence in particular elements of the [[moduli space of Riemannian metrics]] on a given KK-compactification [[fiber]]. See also at _[[moduli stabilization]]_.

## Related concepts

* [[moduli problem]]

* [[dilaton]]

* [[moduli stabilization]], [[landscape of string theory vacua]]

[[!redirects moduli field]]
[[!redirects moduli fields]]


moduli of higher lines -- table

**[[moduli spaces]] of [[line n-bundles with connection]] on $n$-[[dimension|dimensional]] $X$**

| $n$ | [[Calabi-Yau object|Calabi-Yau n-fold]] | [[line n-bundle]] |  moduli of [[line n-bundles]]  | moduli of [[flat infinity-connection|flat]]/degree-0 n-bundles  |  [[Artin-Mazur formal group]] of [[deformation theory|deformation moduli]] of [[line n-bundles]] | [[complex oriented cohomology theory]] | [[modular functor]]/[[self-dual higher gauge theory]] of [[higher dimensional Chern-Simons theory]] |
|-----|---|---|-----|------|-----|----|----|
| $n = 0$ |   | [[unit]] in [[structure sheaf]] | [[multiplicative group]]/[[group of units]] |  |  [[formal multiplicative group]]  | [[complex K-theory]] |  |
| $n = 1$ | [[elliptic curve]] | [[line bundle]] |  [[Picard group]]/[[Picard scheme]] | [[Jacobian]]  |  [[formal Picard group]] | [[elliptic cohomology]] | [[3d Chern-Simons theory]]/[[WZW model]]  |
| $n = 2$ | [[K3 surface]] |  [[line 2-bundle]] |  [[Brauer group]] | [[intermediate Jacobian]] | [[formal Brauer group]] |  [[K3 cohomology]] |  |
| $n = 3$ | [[Calabi-Yau 3-fold]] |  [[line 3-bundle]] |   | [[intermediate Jacobian]] |  | [[Calabi-Yau cohomology|CY3 cohomology]]  | [[7d Chern-Simons theory]]/[[M5-brane]] |
| $n$   |  |  |  |     [[intermediate Jacobian]]  |   |   |  |


moduli problem

#Contents#
* table of contents
{:toc}

## Idea

The problem of finding a suitable [[moduli space]]/[[moduli stack]] for given [[moduli]].

## Related concepts

* [[formal moduli problem]]

[[!redirects moduli problems]]