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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*{category of monoids} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{monoidal_categories}{}\paragraph*{{Monoidal categories}}\label{monoidal_categories} [[!include monoidal categories - contents]] \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{contents}{}\section*{{Contents}}\label{contents} \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{properties}{Properties}\dotfill \pageref*{properties} \linebreak \noindent\hyperlink{local_presentability}{Local presentability}\dotfill \pageref*{local_presentability} \linebreak \noindent\hyperlink{FreeMonoids}{Free and relative free monoids}\dotfill \pageref*{FreeMonoids} \linebreak \noindent\hyperlink{pushouts}{Pushouts}\dotfill \pageref*{pushouts} \linebreak \noindent\hyperlink{PushoutOfCommutativeMonoids}{Of commutative monoids}\dotfill \pageref*{PushoutOfCommutativeMonoids} \linebreak \noindent\hyperlink{PushoutOfNoncommutativeMonoids}{Of noncommutative monoids}\dotfill \pageref*{PushoutOfNoncommutativeMonoids} \linebreak \noindent\hyperlink{FilteredColimits}{Filtered colimits}\dotfill \pageref*{FilteredColimits} \linebreak \noindent\hyperlink{structure_induced_from_monoidal_functors}{Structure induced from monoidal functors}\dotfill \pageref*{structure_induced_from_monoidal_functors} \linebreak \noindent\hyperlink{model_structure}{Model structure}\dotfill \pageref*{model_structure} \linebreak \noindent\hyperlink{enrichment_over_}{Enrichment over $CMon$}\dotfill \pageref*{enrichment_over_} \linebreak \noindent\hyperlink{related_categories}{Related categories}\dotfill \pageref*{related_categories} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{defn} \label{}\hypertarget{}{} For $C$ a [[monoidal category]], the \textbf{category of [[monoid in a monoidal category|monoids]]} $Mon(C)$ in $C$ is the [[category]] whose \begin{itemize}% \item [[objects]] are [[monoid in a monoidal category|monoids]] in $C$; \item [[morphism]]s are morphisms in $C$ of the underlying objects that respect the monoid structure. \end{itemize} Similarly for the \textbf{category of [[commutative monoid in a symmetric monoidal category|commutative monoids]]} $CMon(C)$, if $C$ is symmetric monoidal. \end{defn} \begin{remark} \label{}\hypertarget{}{} If [[Assoc]] denotes the [[associative operad]] in $C$, then $Mon(C) = Alg_C Assoc$ is the category of [[algebras over an operad]] for $Assoc$. \end{remark} \begin{remark} \label{}\hypertarget{}{} Every category of monoids comes with a [[forgetful functor]] $U \colon Mon(C) \to C$ which is [[faithful functor|faithful]] and [[conservative functor|conservative]]. In many cases it is [[monadic functor|monadic]]. \end{remark} \hypertarget{properties}{}\subsection*{{Properties}}\label{properties} The properties of the [[category]] of monoids $Mon (C)$, especially with respect to [[colimit]]s, are markedly different according to whether or not the [[tensor product]] of $C$ preserves colimits in each variable. (This is automatically the case if $C$ is [[closed monoidal category|closed]].) Most ``algebraic'' situations have this property, but others do not. For instance, the category of [[monads]] on a fixed category $A$ is $Mon (C)$, where $C= [A,A]$ is the category of [[endofunctors]] of $A$ with composition as its monoidal structure. This monoidal product preserves colimits in one variable (since colimits in $[A,A]$ are computed pointwise), but not in the other (since most endofunctors do not preserve colimits). So far, the material on this page focuses on the case where $\otimes$ does preserve colimits in both variables, although some of the references at the end discuss the more general case. \hypertarget{local_presentability}{}\subsubsection*{{Local presentability}}\label{local_presentability} \begin{theorem} \label{}\hypertarget{}{} Let $C$ be a [[closed monoidal category|closed]] [[symmetric monoidal category]] with countable [[coproducts]] which is [[locally presentable category|locally presentable]]. Then \begin{enumerate}% \item $U : Mon(C) \to C$ is a finitary [[monadic functor]]. \item If $C$ is a $\lambda$-[[locally presentable category]] then so is $Mon(C)$. \end{enumerate} \end{theorem} This appears in (\hyperlink{Porst}{Porst, page 7}). \hypertarget{FreeMonoids}{}\subsubsection*{{Free and relative free monoids}}\label{FreeMonoids} \begin{prop} \label{}\hypertarget{}{} Let $C$ be a [[monoidal category]] with countable [[coproduct]]s that are preserved by the [[tensor product]]. Then the forgetful functor $U_C$ has a [[left adjoint]] $F_C : C \to Mon(C)$. On an object $X \in C$ the underlying object of $F_C X$ is \begin{displaymath} U_C F_C X = \coprod_{n \in \mathbb{N}} X^{\otimes n} = I_C \coprod X \coprod (X \otimes X) \coprod \cdots \end{displaymath} in $C$, with the monoidal structure given by tensor product/juxtaposition. \end{prop} \begin{proof} A morphism $f : F_C X \to A$ in $Mon(C)$ with components $f_k : X^{\otimes k} \to U_C A$ is entirely fixed by its component $\tilde f = f_1 : X \to U_C A$ on $X$, because by the homomorphism property and the special free nature of the product in $F_C X$ \begin{displaymath} \itexarray{ X^{\times k} \otimes X^{\otimes (n-k)} &\stackrel{f_k \otimes f_{n-k}}{\to}& A \otimes A \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_A}} \\ X^{\otimes n} &\stackrel{f_{n}}{\to}& A } \end{displaymath} it follows that \begin{displaymath} f_n : X^{\otimes n} \stackrel{f_1^{\otimes n}}{\to} A^{\otimes n} \stackrel{\mu_A}{\to} A \,. \end{displaymath} Conversely, every choice for $f_1$ extends to a morphism $f$ in $Mon(C)$ this way. \end{proof} \begin{example} \label{}\hypertarget{}{} Free algebras of the form $F(A)$ are called \textbf{[[tensor algebra]]}s, at least for $C =$ [[Vect]] and similar. The elements of the free algebra $F(A)$ are somtimes called [[lists]], at least for $C =$ [[Set]] and similar. \end{example} \hypertarget{pushouts}{}\subsubsection*{{Pushouts}}\label{pushouts} We discuss forming [[pushouts]] in a category of monoids. The case \begin{itemize}% \item \hyperlink{PushoutOfCommutativeMonoids}{For commutative monoids} \end{itemize} has a simple description. The case \begin{itemize}% \item \hyperlink{PushoutOfNoncommutativeMonoids}{For non-commutative monoids} \end{itemize} is more involved. \hypertarget{PushoutOfCommutativeMonoids}{}\paragraph*{{Of commutative monoids}}\label{PushoutOfCommutativeMonoids} \begin{prop} \label{PushoutsInCMonGivenByTensorProduct}\hypertarget{PushoutsInCMonGivenByTensorProduct}{} Suppose that $\mathcal{C}$ is \begin{itemize}% \item a [[symmetric monoidal category]]; \item with [[reflexive coequalizers]] \item that are preserved by the [[tensor product]] functors $A \otimes (-) \colon \mathcal{C} \to \mathcal{C}$ for all objects $A$ in $\mathcal{C}$. \end{itemize} Then for $f \colon A \to B$ and $g \colon A \to C$ two morphisms in the category $CMon(\mathcal{C})$ of \emph{[[commutative monoid in a symmetric monoidal category|commutative monoids]]} in $\mathcal{C}$, the underlying object in $\mathcal{C}$ of the [[pushout]] in $CMon(\mathcal{C})$ coincides with that of the pushout in the category $A$[[Mod]] of $A$-[[modules]] \begin{displaymath} U(B \coprod_A C) \simeq B \otimes_A C \,. \end{displaymath} Here $B$ and $C$ are regarded as equipped with the canonical $A$-module structure induced by the morphisms $f$ and $g$, respectively. \end{prop} This appears for instance as (\hyperlink{Johnstone}{Johnstone, page 478, cor. 1.1.9}). \hypertarget{PushoutOfNoncommutativeMonoids}{}\paragraph*{{Of noncommutative monoids}}\label{PushoutOfNoncommutativeMonoids} \begin{prop} \label{PushOutOfMonoidsAlongFreeMorphisms}\hypertarget{PushOutOfMonoidsAlongFreeMorphisms}{} If $C$ is [[cocomplete category|cocomplete]] and its tensor product preserves colimits on both sides, then the category $Mon(C)$ of monoids has all [[pushout]]s \begin{displaymath} \itexarray{ F(K) &\stackrel{F(f)}{\to}& F(L) \\ {}^{\mathllap{p}}\downarrow && \downarrow \\ X &\to& P } \end{displaymath} along morphisms $F(f) : F(K) \to F(L)$, for $f : K \to L$ a morphism in $C$ and $F : C \to Mon(C)$ the free monoid functor from above. Moreover, these pushouts in $Mon(C)$ are computed in $C$ as the [[colimit]] over a sequence \begin{displaymath} P \simeq \lim_{\to}( X := P_0 \to P_1 \to P_2 \to \cdots ) \end{displaymath} of objects $(P_n)_{n \in \mathbb{N}}$, which are each given by pushouts in $C$ inductively as follows. Assume $P_{n-1}$ has been defined. Write $Sub(\mathbf{n})$ for the [[poset]] of [[subset]]s of the $n$-element set $\mathbf{n}$ (this is the poset of paths along the edges of an $n$-dimensional cube). Define a [[diagram]] \begin{displaymath} K : Sub \mathbf{n} \to C \end{displaymath} by setting on subsets $S \subset \mathbf{n}$ \begin{displaymath} K_S := X \otimes V_1 \otimes X \otimes V_2 \otimes \cdots \otimes V_n \otimes X \end{displaymath} where \begin{displaymath} V_i := \left\{ \itexarray{ K & if not i \in S \\ L & if i \in S } \right. \end{displaymath} and by assigning to a morphism $S_1 \subset S_2$ the morphism which is the [[tensor product]] of identities on $X$, identities on $L$ and the given morphism $f : K \to L$. Write $K^-$ for the same diagram minus the terminal object $S = \mathbf{n}$. Now take $P_n$ to be the [[pushout]] \begin{displaymath} \itexarray{ \lim_{\to} K^- &\to& K \mathbf{n} \\ \downarrow && \downarrow \\ P_{n-1} &\to& P_n } \,, \end{displaymath} where the top morphism is the canonical one induced by the commutativity of the diagram $K$, and where the left morphism is defined in terms of components $K^-(S)$ of the colimit for $S \subset \mathbf{n}$ a proper subset by the tensor product morphisms of the form \begin{displaymath} (\cdots X \otimes K \otimes \cdots \otimes L \otimes \cdots) \stackrel{\cdots \otimes \mu_X \circ (Id \otimes p) \otimes \cdots \otimes Id_L \otimes \cdots}{\to} (X \otimes L)^{|S|} \otimes X \to P_{|S|} \to P_{n-1} \,. \end{displaymath} This gives the underlying object of the monoid $P$. Take the monoid structure on it as follows. The unit of $P$ is the composite \begin{displaymath} e_P : I_C \stackrel{e_X}{\to} X \to P \end{displaymath} with the unit of $X$. The product we take to be the image in the colimit of compatible morphisms $P_k \otimes P_k \to P_{k + l}$ defined by induction on $lk + l$ as follows. we observe that we have a pushout diagram \begin{displaymath} \itexarray{ Q_k \otimes (X \otimes L)^{\otimes l} \coprod_{Q_k \otimes Q_l} (X \otimes L)^{\otimes l} \otimes X \otimes Q_l &\to& (X \otimes L)^{\otimes k} \otimes X \otimes (X \otimes L)^{\otimes l} \otimes X \\ \downarrow && \downarrow \\ P_{k-1} \otimes P_l \coprod_{P_{k-1} \otimes P_{l-1}} P_k \otimes P_{l-1} &\to& P_k \otimes P_l } \,, \end{displaymath} where $Q_n := (\lim_{\to} K)_n$ is the colimit as in the above at stage $n$. There is a morphism from the bottom left object to $P_{k+l}$ given by the induction assumption. Moreover we have a morphism from the top right object to $P_{k+1}$ obtained by first multiplying the two adjacent factors of $X$ and then applying the morphism $(X \otimes L)^{\otimes k+l} \otimes X \to P_{k+l}$. These are compatible and hence give the desired morphism $P_k \otimes P_k \to P_{k+l}$. \end{prop} This construction is spelled out for instance in the proof of \hyperlink{SchwedeShipley}{SchwedeShipley, lemma 6.2} \begin{proof} First we need to discuss that this definition is actually consistent, in that the morphism $\lim_\to K^- \to P_{n-1}$ is well defined and the monoid structure on $P$ is well defined. (\ldots{}) That $X \to P$ is a morphism of monoids follows then essentially by the definition of the monoid structure on $P$. Finally we need to check the universal property of the cocone $P$ obtained this way: (\ldots{}) \end{proof} \hypertarget{FilteredColimits}{}\subsubsection*{{Filtered colimits}}\label{FilteredColimits} \begin{prop} \label{}\hypertarget{}{} For $C$ a [[closed monoidal category|closed]] [[symmetric monoidal category]] the [[forgetful functor]] \begin{displaymath} U : CMon(C) \to C \end{displaymath} from commutative monoids to $C$ [[creates colimits|created]] [[filtered colimit]]s. \end{prop} This appears for instance as (\hyperlink{Johnstone}{Johnstone, C1.1 lemma 1.1.8}). \hypertarget{structure_induced_from_monoidal_functors}{}\subsubsection*{{Structure induced from monoidal functors}}\label{structure_induced_from_monoidal_functors} If $F : C\to D$ is a [[lax monoidal functor]], then it induces canonically a functor between categories of monoids \begin{displaymath} Mon(F) : Mon(C) \to Mon(D) \,. \end{displaymath} This is one good way to remember the difference between \emph{lax} and \emph{colax} monoidal functors. \hypertarget{model_structure}{}\subsubsection*{{Model structure}}\label{model_structure} If $C$ is a [[monoidal model category]], then $Mon(C)$ may inherit itself the structure of a [[model category]]. See [[model structure on monoids in a monoidal model category]]. \hypertarget{enrichment_over_}{}\subsubsection*{{Enrichment over $CMon$}}\label{enrichment_over_} Some categories are \emph{implicitly} [[enriched category|enriched]] over commutative monoids, in particular [[semiadditive categories]]. Also [[Ab]]-[[enriched categories]] (and hence in particular [[abelian categories]]) of course have an underlying $CMon$-enrichment. \hypertarget{related_categories}{}\subsection*{{Related categories}}\label{related_categories} \begin{itemize}% \item [[Grp]], [[Ab]] \item [[Ring]], [[CRing]] \item [[Rng]] \end{itemize} \hypertarget{references}{}\subsection*{{References}}\label{references} A general discussion of categories of monoids in [[symmetric monoidal categories]] is in \begin{itemize}% \item [[Hans Porst]], \emph{On Categories of Monoids, Comonoids and bimonoids} (\href{http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.144.4291&rep=rep1&type=pdf}{pdf}) \end{itemize} Free monoid constructions are discussed in \begin{itemize}% \item [[Eduardo Dubuc]], \emph{Free monoids} Algebra J. 29, 208--228 (1974) \item [[Max Kelly]], \emph{A unified treatment of transfinite constructions for free algebras, free monoids,colimits, associated sheaves, and so on} Bull. Austral. Math. Soc. 22(1), 1--83 (1980) \item [[Stephen Lack]], \emph{Note on the construction of free monoids} Appl Categor Struct (2010) 18:17--29 \end{itemize} The detailed discussion of pushouts along free monoid morphisms is in the proof of lemma 6.2 of \begin{itemize}% \item [[Stefan Schwede]], [[Brooke Shipley]], \emph{Algebras and modules in monoidal model categories} Proc. London Math. Soc. (2000) 80(2): 491-511 (\href{http://www.math.uic.edu/~bshipley/monoidal.pdf}{pdf}) \end{itemize} Some remarks on commutative monoids are in section C1.1 of \begin{itemize}% \item [[Peter Johnstone]], \emph{[[Sketches of an Elephant]]} \end{itemize} Discussion of the [[closed monoidal category]] structure on a category of algebras of a [[commutative algebraic theory]] is in \begin{itemize}% \item [[Peter Freyd]], \emph{Algebra valued functors in general and tensor products in particular}, Colloq. Math. 14 (1966), 89-106. \end{itemize} [[!redirects categories of monoids]] [[!redirects category of commutative monoids]] [[!redirects Mon]] [[!redirects CMon]] category: category \end{document}