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\newtheorem{prop}{Proposition} \newtheorem{cor}{Corollary} \newtheorem*{utheorem}{Theorem} \newtheorem*{ulemma}{Lemma} \newtheorem*{uprop}{Proposition} \newtheorem*{ucor}{Corollary} \theoremstyle{definition} \newtheorem{defn}{Definition} \newtheorem{example}{Example} \newtheorem*{udefn}{Definition} \newtheorem*{uexample}{Example} \theoremstyle{remark} \newtheorem{remark}{Remark} \newtheorem{note}{Note} \newtheorem*{uremark}{Remark} \newtheorem*{unote}{Note} %------------------------------------------------------------------- \begin{document} %------------------------------------------------------------------- \section*{associative unital algebra} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{higher_algebra}{}\paragraph*{{Higher algebra}}\label{higher_algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{over_ordinary_rings}{Over ordinary rings}\dotfill \pageref*{over_ordinary_rings} \linebreak \noindent\hyperlink{OverMonoidsInAMonoidalCategory}{Over monoids in a monoidal category}\dotfill \pageref*{OverMonoidsInAMonoidalCategory} \linebreak \noindent\hyperlink{variants}{Variants}\dotfill \pageref*{variants} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{tannaka_duality}{Tannaka duality}\dotfill \pageref*{tannaka_duality} \linebreak \noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \hypertarget{over_ordinary_rings}{}\subsubsection*{{Over ordinary rings}}\label{over_ordinary_rings} For $R$ a [[commutative ring]], an \textbf{associative unital $R$-algebra} is equivalently \begin{itemize}% \item a [[monoid in a monoidal category|monoid]] [[internalization|internal to]] $R$[[Mod]] equipped with the [[tensor product of modules]] $\otimes$; \item a [[pointed object|pointed]] one-object category [[enriched category|enriched over]] $(R Mod, \otimes)$; \item a pointed $R$-[[algebroid]] with one object; \item an $R$-[[module]] $V$ equipped with [[linear maps]] $p : V \otimes V \to V$ and $i : R \to V$ satisfying the associative and unit laws; \item a [[ring]] $A$ [[under category|under]] $R$ such that the corresponding map $R \to A$ lands in the [[center]] of $A$. \end{itemize} If there is no danger for confusion, one often says simply `associative algebra', or even only `[[algebra]]'. More generally, a (merely) \textbf{associative algebra} need not have $i: R \to V$; that is, it is a [[semigroup]] instead of a monoid. Less generally, a \textbf{[[commutative algebra]]} (where associative and unital are usually assumed) is an [[commutative monoid in a symmetric monoidal category]] in $Vect$. \hypertarget{OverMonoidsInAMonoidalCategory}{}\subsubsection*{{Over monoids in a monoidal category}}\label{OverMonoidsInAMonoidalCategory} \begin{defn} \label{MonoidsInMonoidalCategory}\hypertarget{MonoidsInMonoidalCategory}{} Given a [[monoidal category]] $(\mathcal{C}, \otimes, 1)$, then a \textbf{[[monoid in a monoidal category|monoid internal to]]} $(\mathcal{C}, \otimes, 1)$ is \begin{enumerate}% \item an [[object]] $A \in \mathcal{C}$; \item a morphism $e \;\colon\; 1 \longrightarrow A$ (called the \emph{[[unit]]}) \item a morphism $\mu \;\colon\; A \otimes A \longrightarrow A$ (called the \emph{product}); \end{enumerate} such that \begin{enumerate}% \item ([[associativity]]) the following [[commuting diagram|diagram commutes]] \begin{displaymath} \itexarray{ (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 } \,, \end{displaymath} where $a$ is the associator isomorphism of $\mathcal{C}$; \item ([[unitality]]) the following [[commuting diagram|diagram commutes]]: \begin{displaymath} \itexarray{ 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 } \,, \end{displaymath} where $\ell$ and $r$ are the left and right unitor isomorphisms of $\mathcal{C}$. \end{enumerate} 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 \textbf{[[commutative monoid in a symmetric monoidal category|commutative monoid in]]} $(\mathcal{C}, \otimes, 1, B)$ if in addition \begin{itemize}% \item (commutativity) the following [[commuting diagram|diagram commutes]] \begin{displaymath} \itexarray{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,. \end{displaymath} \end{itemize} A [[homomorphism]] of monoids $(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)$ is a morphism \begin{displaymath} f \;\colon\; A_1 \longrightarrow A_2 \end{displaymath} in $\mathcal{C}$, such that the following two [[commuting diagram|diagrams commute]] \begin{displaymath} \itexarray{ 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 } \end{displaymath} and \begin{displaymath} \itexarray{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,. \end{displaymath} 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. \end{defn} \begin{defn} \label{ModulesInMonoidalCategory}\hypertarget{ModulesInMonoidalCategory}{} 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 \textbf{left [[module object]]} in $(\mathcal{C}, \otimes, 1)$ over $(A,\mu,e)$ is \begin{enumerate}% \item an [[object]] $N \in \mathcal{C}$; \item a [[morphism]] $\rho \;\colon\; A \otimes N \longrightarrow N$ (called the \emph{[[action]]}); \end{enumerate} such that \begin{enumerate}% \item ([[unitality]]) the following [[commuting diagram|diagram commutes]]: \begin{displaymath} \itexarray{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,, \end{displaymath} where $\ell$ is the left unitor isomorphism of $\mathcal{C}$. \item (action property) the following [[commuting diagram|diagram commutes]] \begin{displaymath} \itexarray{ (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 } \,, \end{displaymath} \end{enumerate} A [[homomorphism]] of left $A$-module objects \begin{displaymath} (N_1, \rho_1) \longrightarrow (N_2, \rho_2) \end{displaymath} is a morphism \begin{displaymath} f\;\colon\; N_1 \longrightarrow N_2 \end{displaymath} in $\mathcal{C}$, such that the following [[commuting diagram|diagram commutes]]: \begin{displaymath} \itexarray{ 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 } \,. \end{displaymath} For the resulting \textbf{[[category of modules]]} of left $A$-modules in $\mathcal{C}$ with $A$-module homomorphisms between them, we write \begin{displaymath} A Mod(\mathcal{C}) \,. \end{displaymath} This is naturally a (pointed) [[topologically enriched category]] itself. \end{defn} \begin{defn} \label{TensorProductOfModulesOverCommutativeMonoidObject}\hypertarget{TensorProductOfModulesOverCommutativeMonoidObject}{} 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 \textbf{[[tensor product of modules]]} $N_1 \otimes_A N_2$ is, if it exists, the [[coequalizer]] \begin{displaymath} 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 \end{displaymath} \end{defn} \begin{prop} \label{MonoidalCategoryOfModules}\hypertarget{MonoidalCategoryOfModules}{} 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. \end{prop} \begin{defn} \label{AAlgebra}\hypertarget{AAlgebra}{} 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 \textbf{$A$-[[associative algebra|algebra]]}. \end{defn} \begin{prop} \label{}\hypertarget{}{} 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]] \begin{displaymath} A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/} \end{displaymath} 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 $A \longrightarrow E$. \end{prop} (e.g. \hyperlink{EKMM97}{EKMM 97, VII lemma 1.3}) \begin{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 $\mu_E$ \begin{displaymath} \itexarray{ 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 } \,. \end{displaymath} 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 \begin{displaymath} (\star) \;\;\;\;\; \;\;\;\;\; \itexarray{ 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 } \end{displaymath} [[commuting diagram|commutes]]. Moreover it satisfies the unit property \begin{displaymath} \itexarray{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,. \end{displaymath} By forgetting the tensor product over $A$, the latter gives \begin{displaymath} \itexarray{ 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 \;\;\;\;\;\;\;\; \itexarray{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,, \end{displaymath} 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]] \begin{displaymath} \itexarray{ 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 } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \itexarray{ 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 } \,. \end{displaymath} 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 \begin{displaymath} \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,. \end{displaymath} 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. \end{proof} \hypertarget{variants}{}\subsection*{{Variants}}\label{variants} \begin{itemize}% \item A [[cosimplicial algebra]] is a [[cosimplicial object]] in the category of algebras. \item A [[dg-algebra]] is a [[monoid]] not in [[Vect]] but in the category of (co)[[chain complex]]es. \item 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]]. \end{itemize} \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item [[function algebra]] \end{itemize} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} \hypertarget{tannaka_duality}{}\subsubsection*{{Tannaka duality}}\label{tannaka_duality} [[!include structure on algebras and their module categories - table]] \hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts} \begin{itemize}% \item [[noncommutative algebra]] \item [[nonassociative algebra]] \item [[nonunital algebra]] \item [[finitely generated algebra]], [[finitely presented algebra]] \item [[power-associative algebra]] \item [[augmented algebra]] \item [[unitisation of C\emph{-algebras]]} \item [[differential algebra]] \item [[differential graded algebra]], [[A-infinity algebra]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item [[Anthony Elmendorf]], [[Igor Kriz]], [[Michael Mandell]], [[Peter May]], \emph{[[Rings, modules and algebras in stable homotopy theory]]}, AMS Mathematical Surveys and Monographs Volume 47 (1997) (\href{http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf}{pdf}) \end{itemize} [[!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]] \end{document}