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+-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ###Context### #### Higher algebra +--{: .hide} [[!include higher algebra - contents]] =-- =-- =-- #Contents# * table of contents {:toc} ## Idea Similar to the way [[modules]] generalize [[abelian groups]] by adding the operation of taking non-integer multiples, an $R$-algebra can be thought of as a generalization of a ring $S$, where the operation of taking integer multiples (seen as iterated addition) has been extended to taking arbitrary multiples with coefficients in $R$. In the trivial case, a $\mathbb{Z}$-algebra is simply a ring. ## Definition ### Over ordinary rings {#OverOrdinaryRings} For $R$ a [[commutative ring]], an **associative unital $R$-algebra** is equivalently: * a [[monoid in a monoidal category|monoid]] [[internalization|internal to]] the [[category]] [[Mod|$R Mod$]] of $R$-[[modules]] equipped with the [[tensor product of modules]] $\otimes$; * a [[pointed object|pointed]] single-[[object]] [[enriched category|category enriched over]] $(R Mod, \otimes)$; * a pointed $R$-[[algebroid]] with a single object; * an $R$-[[module]] $A$ equipped with $R$-[[linear maps]] $p \colon A \otimes A \longrightarrow A$ and $i \colon R \to A$ satisfying [[associativity]] and [[unitality]]; * a [[ring]] $A$ [[under category|under]] $R$ such that the corresponding map $R \to A$ lands in the [[center]] of $A$. If there is no danger for confusion, one often says simply 'associative algebra', or even only '[[algebra]]'. More generally: * a (merely) **associative algebra** need not have a [[unit]] $i \colon R \to A$; that is, it is a [[semigroup]] instead of a [[monoid]]; * an [[ring over a ring|$R$-ring]] is a [[monoid object]] [[internalization|in]] the category [[Bimod|$R BiMod$]] of $R$-[[bimodules]] equipped with, crucially, the [[tensor product of bimodules]]. Less generally, a **[[commutative algebra]]** (where [[associativity]] and [[unitality]] are usually assumed) is a [[commutative monoid in a symmetric monoidal category|commutative monoid objecy]] [[internalization|in]] [[Mod|$R Mod$]]. For a given ring the algebras form a category, [[Alg]], and the commutative algebras a subcategory, [[CommAlg]]. ### Over semi-rings Note that everywhere rings can be replaced by [[semi-rings]] in the previous paragraph. For example a commutative associative unital $\mathbb{Q}^{+}$-algebra is nothing more than a commutative semi-ring $R$ with a [[semi-ring homomorphism]] $\mathbb{Q}^{+} \rightarrow R$. ### Over monoids in a monoidal category {#OverMonoidsInAMonoidalCategory} +-- {: .num_defn #MonoidsInMonoidalCategory} ###### Definition Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, then a **[[monoid in a monoidal category|monoid internal to]]** $(\mathcal{C}, \otimes, 1)$ is 1. an [[object]] $A \in \mathcal{C}$; 1. a morphism $e \;\colon\; 1 \longrightarrow A$ (called the _[[unit]]_) 1. a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the _product_); such that 1. ([[associativity]]) the following [[commuting diagram|diagram commutes]] $$ \array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{id \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes id}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,, $$ where $a$ is the associator isomorphism of $\mathcal{C}$; 1. ([[unitality]]) the following [[commuting diagram|diagram commutes]]: $$ \array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,, $$ where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$. Moreover, if $(\mathcal{C}, \otimes , 1)$ has the structure of a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1, B)$ with symmetric [[braiding]] $\tau$, then a monoid $(A,\mu, e)$ as above is called a **[[commutative monoid in a symmetric monoidal category|commutative monoid in]]** $(\mathcal{C}, \otimes, 1, B)$ if in addition * (commutativity) the following [[commuting diagram|diagram commutes]] $$ \array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,. $$ A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism $$ f \;\colon\; A_1 \longrightarrow A_2 $$ in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]] $$ \array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 } $$ and $$ \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,. $$ Write $Mon(\mathcal{C}, \otimes,1)$ for the [[category of monoids]] in $\mathcal{C}$, and $CMon(\mathcal{C}, \otimes, 1)$ for its subcategory of commutative monoids. =-- +-- {: .num_defn #ModulesInMonoidalCategory} ###### Definition Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, and given $(A,\mu,e)$ a [[monoid in a monoidal category|monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), 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]]) 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. (action property) the following [[commuting diagram|diagram commutes]] $$ \array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,, $$ A [[homomorphism]] of left $A$-module objects $$ (N_1, \rho_1) \longrightarrow (N_2, \rho_2) $$ is a morphism $$ f\;\colon\; N_1 \longrightarrow N_2 $$ in $\mathcal{C}$, such that the following [[commuting diagram|diagram commutes]]: $$ \array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,. $$ For the resulting **[[category of modules]]** of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write $$ A Mod(\mathcal{C}) \,. $$ This is naturally a (pointed) [[topologically enriched category]] itself. =-- +-- {: .num_defn #TensorProductOfModulesOverCommutativeMonoidObject} ###### Definition Given a (pointed) [[topologically enriched category|topological]] [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$, given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}), and given $(N_1, \rho_1)$ and $(N_2, \rho_2)$ two left $A$-[[module objects]] (def.\ref{MonoidsInMonoidalCategory}), then the **[[tensor product of modules]]** $N_1 \otimes_A N_2$ is, if it exists, the [[coequalizer]] $$ N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coequ}{\longrightarrow} N_1 \otimes_A N_2 $$ =-- +-- {: .num_prop #MonoidalCategoryOfModules} ###### Proposition Given a [[symmetric monoidal category]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{SymmetricMonoidalCategory}), and given $(A,\mu,e)$ a [[commutative monoid in a symmetric monoidal category|commutative monoid in]] $(\mathcal{C}, \otimes, 1)$ (def. \ref{MonoidsInMonoidalCategory}). If all [[coequalizers]] exist in $\mathcal{C}$, then the [[tensor product of modules]] $\otimes_A$ from def. \ref{TensorProductOfModulesOverCommutativeMonoidObject} makes the [[category of modules]] $A Mod(\mathcal{C})$ into a [[symmetric monoidal category]], $(A Mod, \otimes_A, A)$ with [[tensor unit]] the object $A$ itself. =-- +-- {: .num_defn #AAlgebra} ###### Definition Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ as in prop. \ref{MonoidalCategoryOfModules}, then a [[monoid in a monoidal category|monoid]] $(E, \mu, e)$ in $(A Mod , \otimes_A , A)$ (def. \ref{MonoidsInMonoidalCategory}) is called an **$A$-[[associative algebra|algebra]]**. =-- +-- {: .num_prop } ###### Proposition Given a [[monoidal category|monoidal]] [[category of modules]] $(A Mod , \otimes_A , A)$ in a [[monoidal category]] $(\mathcal{C},\otimes, 1)$ as in prop. \ref{MonoidalCategoryOfModules}, and an $A$-algebra $(E,\mu,e)$ (def. \ref{AAlgebra}), then there is an [[equivalence of categories]] $$ A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/} $$ between the [[category of commutative monoids]] in $A Mod$ and the [[coslice category]] of commutative monoids in $\mathcal{C}$ under $A$, hence between commutative $A$-algebras in $\mathcal{C}$ and commutative monoids $E$ in $\mathcal{C}$ that are equipped with a homomorphism of monoids $A \longrightarrow E$. =-- (e.g. [EKMM 97, VII lemma 1.3](#EKMM97)) +-- {: .proof} ###### Proof In one direction, consider a $A$-algebra $E$ with unit $e_E \;\colon\; A \longrightarrow E$ and product $\mu_{E/A} \colon E \otimes_A E \longrightarrow E$. There is the underlying product $\mu_E$ $$ \array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,. $$ By considering a diagram of such coequalizer diagrams with middle vertical morphism $e_E\circ e_A$, one find that this is a unit for $\mu_E$ and that $(E, \mu_E, e_E \circ e_A)$ is a commutative monoid in $(\mathcal{C}, \otimes, 1)$. Then consider the two conditions on the unit $e_E \colon A \longrightarrow E$. First of all this is an $A$-module homomorphism, which means that $$ (\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E } $$ [[commuting diagram|commutes]]. Moreover it satisfies the unit property $$ \array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,. $$ By forgetting the tensor product over $A$, the latter gives $$ \array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,, $$ where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be [[pasting|pasted]] to the square $(\star)$ above, to yield a [[commuting square]] $$ \array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,. $$ This shows that the unit $e_A$ is a homomorphism of monoids $(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)$. Now for the converse direction, assume that $(A,\mu_A, e_A)$ and $(E, \mu_E, e'_E)$ are two commutative monoids in $(\mathcal{C}, \otimes, 1)$ with $e_E \;\colon\; A \to E$ a monoid homomorphism. Then $E$ inherits a left $A$-[[module]] structure by $$ \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,. $$ By commutativity and associativity it follows that $\mu_E$ coequalizes the two induced morphisms $E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E$. Hence the [[universal property]] of the [[coequalizer]] gives a factorization through some $\mu_{E/A}\colon E \otimes_A E \longrightarrow E$. This shows that $(E, \mu_{E/A}, e_E)$ is a commutative $A$-algebra. Finally one checks that these two constructions are inverses to each other, up to isomorphism. =-- ## Variants * A [[cosimplicial algebra]] is a [[cosimplicial object]] in the category of algebras. * A [[dg-algebra]] is a [[monoid]] not in [[Vect]] but in the category of (co)[[chain complex]]es. * A [[smooth algebra]] is an associative $\mathbb{R}$-algebra that has not only the usual binary product induced from the product $\mathbb{R}\times \mathbb{R} \to \mathbb{R}$, but has a $n$-ary product operation for every [[smooth function]] $\mathbb{R}^n \to \mathbb{R}$. This may be understood as a special case of an [[algebra over a Lawvere theory]], here the [[Lawvere theory]] [[CartSp]]. ## Examples * [[function algebra]] ## Properties ### Tannaka duality [[!include structure on algebras and their module categories - table]] ## Related concepts * [[noncommutative algebra]] * [[nonassociative algebra]] * [[nonunital algebra]] * [[finitely generated algebra]], [[finitely presented algebra]] * [[power-associative algebra]] * [[augmented algebra]] * [[unitisation of C*-algebras]] * [[differential algebra]] * [[differential graded algebra]], [[A-infinity algebra]] ## References See most references on *[[algebra]]*. See also: * Wikipedia, *[Associative algebra](https://en.wikipedia.org/wiki/Associative_algebra)* Discussion in the generality of [[brave new algebra]]: * {#EKMM97} [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], _[[Rings, modules and algebras in stable homotopy theory]]_, AMS Mathematical Surveys and Monographs Volume 47 (1997) ([pdf](http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf)) [[!redirects associative unital algebra]] [[!redirects associative unital algebras]] [[!redirects unital associative algebra]] [[!redirects unital associative algebras]] [[!redirects associative algebra]] [[!redirects associative algebras]] [[!redirects unital algebra]] [[!redirects unital algebras]]