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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*{Tall-Wraith monoid} \hypertarget{context}{}\subsubsection*{{Context}}\label{context} \hypertarget{algebra}{}\paragraph*{{Algebra}}\label{algebra} [[!include higher algebra - contents]] \hypertarget{tallwraith_monoids}{}\section*{{Tall--Wraith monoids}}\label{tallwraith_monoids} \noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak \noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak \noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak \noindent\hyperlink{general}{General results and constructions}\dotfill \pageref*{general} \linebreak \noindent\hyperlink{bialgebras_over_a_monad}{Bialgebras over a monad}\dotfill \pageref*{bialgebras_over_a_monad} \linebreak \noindent\hyperlink{monoidal_product_on_bialgebras_over_a_monad}{Monoidal product on bialgebras over a monad}\dotfill \pageref*{monoidal_product_on_bialgebras_over_a_monad} \linebreak \noindent\hyperlink{tallwraith_monoids_relative_to_a_monad}{Tall-Wraith monoids relative to a monad}\dotfill \pageref*{tallwraith_monoids_relative_to_a_monad} \linebreak \noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak \hypertarget{idea}{}\subsection*{{Idea}}\label{idea} Given an [[algebraic theory]] $V$, a $V$-algebra is a model of $V$ in the category $Set$. A \emph{Tall--Wraith $V$-monoid} is \emph{the kind of thing that acts on $V$-algebras}. \hypertarget{definition}{}\subsection*{{Definition}}\label{definition} \begin{definition} \label{twmonoid}\hypertarget{twmonoid}{} Let $V$ be an [[algebraic theory]] and let $V Alg$ be the category of models of this theory in $Set$. Then a \textbf{Tall--Wraith $V$-monoid} is a [[monoid|monoid object]] in the category of co-$V$-objects in $V Alg$. \end{definition} To see why these are \emph{what acts on $V$-algebras} one needs to understand what a co-$V$-object in $V Alg$ actually is. A co-$V$-object in some category $D$ is a [[representable functor|representable covariant functor]] from $D$ to $V Alg$. To give a particular $D$-object, $d$, the structure of a co-$V$-object is to give a lift of the $Set$-valued $Hom$-functor $D(d,-)$ to $V Alg$. Thus a co-$V$-object in $V Alg$ is a representable covariant functor from $V Alg$ to itself. One can therefore consider composition of such representable covariant functors. The main result of this can be simply stated: \begin{proposition} \label{repcomp}\hypertarget{repcomp}{} The composition of representable covariant functors $V Alg \to V Alg$ is again representable. \end{proposition} This is a basic result in general algebra, and is not stated here in its full generality (although see \href{/nlab/show/Tall-Wraith+monoid#general}{below} for some reasonably general constructions). An almost corollary of this is that the category of representable covariant functors from $V Alg$ to itself is monoidal (the ``almost'' refers to the fact that you have to show that the identity functor is representable, but this is not hard). Thus for two co-$V$-algebra objects in $V$, say $R_1$ and $R_2$, there is a product $R_1 \odot R_2$ and a natural isomorphism \begin{displaymath} Hom_V(R_1 \odot R_2,A) \cong Hom_V(R_1, Hom_V(R_2,A)) \end{displaymath} for any $V$-algebra, $A$. A \textbf{Tall--Wraith $V$-monoid} is thus a triple $(P,\mu,\eta)$ with $\mu : P \odot P \to P$ and $\eta : I \to P$ (where $I$ is the free $V$-algebra on one element --- this represents the identity functor), satisfying the obvious coherence diagrams. An action of $P$ on a $V$-algebra, say $A$, is then a morphism $\rho : P \odot A \to A$ again satisfying certain coherence diagrams. Ah, but I have not told you what $P \odot A$ is! At the moment, one can take the ``product'' of two co-$V$-algebra objects in $V Alg$ but now I want to take the product of a co-$V$-algebra object with a $V$-algebra. How do I do this? I do this by observing that a $V$-algebra is a \emph{co-$Set$-algebra object in $V Alg$}! That's a complicated way of saying that $V$ represents a covariant functor $V Alg \to Set$. Precomposing this with the functor represented by $P$ yields again a covariant functor $V Alg \to Set$. This is again representable and we write its representing object $P \odot A$. As an aside, we note a consequence. As we've seen, the category of co-$V$-algebra objects in $V$ is a monoidal category, with the tensor product $\odot$. Now we're seeing this monoidal category [[action|acts]] on the category of $V$-algebras. Indeed, it acts on the categories of $V$-algebra and co-$V$-algebra objects in a reasonably arbitrary base category. One postscript to this is that although the category of co-$V$-algebra objects in $V Alg$ is not a variety of algebras, for a specific Tall--Wraith $V$-monoid $P$, the category of $P$-modules \emph{is} a variety of algebras. \hypertarget{examples}{}\subsection*{{Examples}}\label{examples} \begin{itemize}% \item If $V$ is the theory of [[commutative ring|commutative unital rings]], a $V$-algebra is a commutative unital ring, a co-$V$-algebra object in $V$ is a [[biring]] and the corresponding sort of Tall--Wraith $V$-monoid is called, in Tall and Wraith's original paper, a \emph{biring triple}. \item If $V$ is the theory of [[commutative algebra|commutative associative algebras]] over a [[field]] $k$, then a $V$-algebra is a commutative associative algebra over $k$, and the corresponding sort of Tall--Wraith $V$-monoid is called a [[plethory]]. \item If $V$ is the theory of [[abelian group|abelian groups]], than a $V$-algebra is an abelian group, and the corresponding sort of Tall--Wraith $V$-monoid is a [[ring]]. To understand the last example, we need to think about \emph{co-abelian group objects in the category of abelian groups}. Abstractly, such a thing is an abelian group object [[internalization|internal to]] $AbGp^{op}$ (though this picture gets the morphisms the wrong way around; in full abstraction then the category of co-abelian group objects in $AbGrp$ is the \emph{opposite} category of the category of abelian group objects in $AbGrp^{op}$). Concretely, such a thing is an abelian group $A$ together with group homomorphisms \begin{displaymath} \begin{aligned} \mu &: A \to A \coprod A, \\ \epsilon &: A \to I \\ \iota &: A \to A \end{aligned} \end{displaymath} where $I$ is the initial object in the category of abelian groups. These homomorphisms must satisfy certain laws: just the abelian group axioms written out diagrammatically, with all the arrows turned around. In fact, $I = \{0\}$. Thus $\epsilon$ is forced to be the map that sends everything to $0$: we have no choice here. We also have that $A \coprod A = A \oplus A$. That means that for $a \in A$, $\mu(a) = (a_1,a_2)$ for some $a_1, a_2 \in A$. Now, one of the laws says that $\epsilon$ is a counit for $\mu$. This means that $(\epsilon \oplus 1) \mu = 1$ and similarly for $1 \oplus \epsilon$. Thus $a_1 = a_2 = a$ and $\mu$ is the diagonal map. So, we have no choice here either. The diagram for $\iota$ (representing the inverse map) is a little more complicated. As $I$ is the initial object in $AbGrp$, there is a unique morphism $I \to A$ (inclusion of the zero). Composing this with $\epsilon$ yields a morphism $A \to A$ which maps every element to the zero in $A$. Using $\mu$ and $\iota$ we can construct another morphism $A \to A$ as \begin{displaymath} A \overset{\mu}\rightarrow A \coprod A \overset{1 \coprod \iota}\rightarrow A \coprod A \overset{\Delta^c}\rightarrow A \end{displaymath} where $\Delta^c$ is the co-diagonal. The relations for abelian groups say that this morphism must be the same as the zero morphism $A \to A$. Using the fact that $A \coprod A \cong A \oplus A$ and that $\mu$ is the diagonal, this says that $a + \iota(a) = 0$. Hence, by the uniqueness of inverses for abelian groups, $\iota(a) = -a$. Thus \emph{if} $(A, \mu, \iota, \epsilon)$ is a co-abelian group object in $AbGrp$ then $\mu$ is the diagonal, $\iota$ the inverse from abelian groups, and $\epsilon$ the zero morphism. However, that is still not quite the same as saying that $(A, \mu, \iota, \epsilon)$ is a co-abelian group object in $AbGrp$. Certainly, $(A, \mu, \epsilon)$ is a co-commutative co-monoid object in $AbGrp$ since $\mu$ is the diagonal, which is automatically co-commutative and co-associative, and $\epsilon$ the zero map, which is the co-unit for the diagonal. What remains is to fit $\iota$ into the structure. The first issue is that $\iota$ is not \emph{automatically} a morphism in $AbGrp$. That is to say, when defining an algebraic theory then the operations are defined on the underlying objects. It is a consequence of the relations of abelian groups that the operations lift to morphisms of abelian groups (algebraic theories where this happens for all operations are sometimes called \emph{commutative}). Thus $\iota$ is a morphism of abelian groups and so $(A, \mu, \iota, \epsilon)$ is a co-commutative co-monoid with an extra unary co-operation. In fact, it is an involution from the relations for abelian groups. The final relation is that $\iota$ is the inverse for $\mu$. The relation that $\iota$ is the inverse for addition (let us write it as, say, $\alpha$) is that \begin{displaymath} A \overset{\Delta}\rightarrow A \times A \overset{1 \times \iota}\rightarrow A \times A \overset{\alpha}\rightarrow A \end{displaymath} is the zero map $A \to I \to A$. This is precisely the relation that $\iota$ is the inverse for $\mu$ since we have the following identifications: $\mu = \Delta$, $A \coprod A = A \times A$, and $\Delta^c = \alpha$. Also, $\epsilon = 0$ and $\eta : I \to A$ is the initial morphism in $AbGrp$. Thus the fact that $\iota$ is the inverse for the diagonal+zero co-monoidal structure is due to the fact that $\iota$ is the inverse for $(\alpha,\eta)$ and $\alpha : A \oplus A \to A$ is the co-diagonal in $AbGrp$ and $\eta : I \to A$ is the unit. It is part of the general theory that the category of co-$V$-objects in $V$ is monoidal (though not, in general, symmetric). For details on this see \emph{The Hunting of the Hopf Ring}, referred to below. This monoidal structure for abelian groups turns out to be the tensor product. Thus a Tall--Wraith monoid for abelian groups is actually an ordinary monoid in the category of abelian groups: in other words, a [[ring]]! \end{itemize} \hypertarget{general}{}\subsection*{{General results and constructions}}\label{general} We now recapitulate the discussion above in a slightly more general context. \hypertarget{bialgebras_over_a_monad}{}\subsubsection*{{Bialgebras over a monad}}\label{bialgebras_over_a_monad} For now our context is that of [[monads]] $T$ on [[Set]], although all of what follows can be generalized considerably, for example to [[enriched category theory]] replacing $Set$ by a suitable [[cosmos]] $\mathbf{V}$. Notation: the category of $T$-[[Eilenberg-Moore category|algebras]] is denoted $Set^T$, with forgetful functor $U: Set^T \to Set$ and free functor $F: Set \to Set^T$, whose composite is the monad $T = U F$, and whose [[counit]] is denoted $\epsilon: F U \to 1_{Set^T}$. For each $T$-algebra $R$, there is an [[adjunction|adjoint pair]] of functors \begin{displaymath} - \cdot R \dashv \hom(R, -): Set^T \to Set \end{displaymath} with associated monad $\hom(R, - \cdot R)$. The functor $- \cdot R: Set \to Set^T$ takes a set $X$ to the $T$-algebra $X \cdot R$, an $X$-indexed [[coproduct]] of copies of $R$ in $Set^T$. We define a $T$-\emph{bialgebra} to be a $T$-algebra $R$ equipped with a morphism of monads $\phi: T \to \hom(R, -\cdot R)$. By the adjunction, the datum $\phi$ is equivalent to a left $T$-algebra structure \begin{displaymath} \alpha: T \circ \hom(R, -) \to \hom(R, -) \end{displaymath} on $\hom(R, -): Set^T \to Set$, thus giving a lifting denoted (by abuse of language) $\hom(R, -): Set^T \to Set^T$. This datum is also equivalent to a right $T$-algebra (aka right $T$-module) structure \begin{displaymath} \beta: W T \to W \end{displaymath} where $W = W_R \coloneqq - \cdot R: Set \to Set^T$. A $T$-\emph{bialgebra map} is a $T$-algebra map $f: R \to S$ such that the induced map $W_f: W_R \to W_S$ is a morphism of right $T$-modules. \begin{example} \label{}\hypertarget{}{} A good case to keep in mind is that of [[birings]], which are $T$-bialgebras for the [[Lawvere theory]] $T$ of [[commutative rings]]. The monad morphism $T \to \hom(R, -\cdot R)$ has components $T X \to \hom(R, X \cdot R)$ for each set $X$. Here $X \cdot R$ is an $X$-indexed coproduct of copies of $R$, where coproduct in the category of commutative rings $Set^T$ is given by tensor product. Thus, for example, $2 \cdot R$ is the ring $R \otimes R$. The component $T(2) \to \hom(R, 2\cdot R)$ therefore ``interprets'' each element $\theta \in T(2)$, i.e., each binary operation in the Lawvere theory, as a binary \emph{co}-operation $R \to R \otimes R$. This applies in particular to the elements $m, a \in T(2)$ which abstractly represent multiplication and addition (seen as natural operations on the category of commutative rings). \end{example} \hypertarget{monoidal_product_on_bialgebras_over_a_monad}{}\subsubsection*{{Monoidal product on bialgebras over a monad}}\label{monoidal_product_on_bialgebras_over_a_monad} We thus have several perspectives on what a $T$-bialgebra is: \begin{itemize}% \item A $T$-algebra $R$ equipped with a monad morphism $T \to \hom(R, -\cdot R)$, \item A $T$-algebra equipped with a compatible $T$-coalgebra structure (actually the same as the preceding item, but in different words), \item A $T$-algebra $R$ for which $\hom(R, -)$ is provided with a left $T$-algebra structure, \item A $T$-algebra $R$ for which $-\cdot R$ is provided with a right $T$-algebra/module structure. \end{itemize} The following proposition gives two more useful descriptions: \begin{prop} \label{}\hypertarget{}{} Let $Ladj(Set^T, Set^T)$ ($Radj(Set^T, Set^T)$) be the category of left (right) adjoint functors $\Psi: Set^T \to Set^T$. The functor $T$-$BiAlg \to Ladj(Set^T, Set^T)$ that takes $(R, \phi)$ to the right $T$-module $(W_R, \beta)$ is an equivalence. Or, what is the same, the functor $T$-$BiAlg \to Radj(Set^T, Set^T)^{op}$, taking $(R, \phi)$ to the left $T$-algebra $\hom(R, -), \alpha)$, is an equivalence. \end{prop} \begin{proof} The main thing to check is that the functor $R \mapsto \hom(R, -)$ to $Radj(Set^T, Set^T)^{op}$ is [[essentially surjective functor|essentially surjective]]. The essential point is that $\Phi$ has a left adjoint iff $U \Phi$ has a left adjoint iff $U \Phi: Set^T \to Set$ is [[representable functor|representable]]: $U \Phi \cong \hom(R, -)$ for some $T$-algebra $R$ (in which case the lift $\Phi$ of $\hom(R, -)$ through $U$ is tantamount to a $T$-algebra structure on $\hom(R, -)$). The only (mildly) tricky part is that $\Phi$ has a left adjoint if $U\Phi$ has a left adjoint $W = W_R$. To define the left adjoint $\Psi$ of $\Phi$ objectwise, we take any $T$-algebra $S$ with its canonical presentation \begin{displaymath} F U F U S \stackrel{\overset{\epsilon F U S}{\to}}{\underset{F U \epsilon}{\to}} F U S \stackrel{\epsilon}{\to} S \end{displaymath} which is a [[coequalizer]] diagram. A left adjoint $\Psi$ must preserve this coequalizer, and we must have $\Psi F \cong W$ since both sides are left adjoint to $U \Phi$. Thus we define $\Psi (S)$ to be a coequalizer \begin{displaymath} W(T U S) \stackrel{\overset{\beta U S}{\to}}{\underset{W U\epsilon S}{\to}} W(U S) \to \Psi(S) \end{displaymath} where $\beta: W T \to T$ is the $T$-module structure coming from the monad morphism $\phi: T \to \hom(R, -\cdot R)$. This objectwise definition of $\Psi$ easily extends to morphisms by [[universal property|universality]] and provides a left adjoint to $\Phi$. Remaining details are left to the reader. \end{proof} One import of this proposition is that left adjoint endofunctors on $Set^T$ compose, i.e., endofunctor composition gives a monoidal structure on $Ladj(Set^T, Set^T)$, and this monoidal structure transports across the categorical equivalence of the proposition to give a monoidal structure on $T$-$BiAlg$. The resultant monoidal product on $T$-bialgebras is denoted $\odot$. A second import of this proposition is that the canonical functor $T\text{-}Bialg \to Ladj(Set^T, Set^T)$ induces a functor which is reasonably denoted \begin{displaymath} \odot: T\text{-}Bialg \times Set^T \to Set^T, \end{displaymath} realizing an [[actegory]] structure over the monoidal category $T\text{-}Bialg$. A direct construction of the monoidal product $\odot$ can be extracted by following the proof of the proposition. If $R, S$ are $T$-bialgebras, then the underlying $T$-algebra of $S \odot R$ (corresponding to composition of $\hom(S, -) \circ \hom(R, -)$ of right adjoints $Set^T \to Set^T$) is computed as a reflexive coequalizer in $Set^T$: \begin{displaymath} T U S \cdot R \stackrel{\overset{\epsilon U S \cdot R}{\to}}{\underset{\beta U S}{\to}} U S \cdot R \to S \odot R. \end{displaymath} Here $\epsilon U S$ is the same as the $T$-algebra structure $T U S \to U S$ on $S$. Whereas $\beta X: T X \cdot R \to X \cdot R$ is a component of the $T$-module structure $W_R T \to W_R$; it is mated by the $- \cdot R \dashv \hom(R, -)$ adjunction to the component of the coalgebra structure $\phi X: T X \to \hom(R, X \cdot R)$. To extract the $T$-coalgebra structure on $S \odot R$, let us observe generally that if $F: C \to D$ is a left adjoint, then for any category $E$ there is an induced left adjoint $[1_E, F]: [E, C] \to [E, D]$ and similarly an induced left adjoint $Ladj(E, C) \to Ladj(E, D)$. Applying this to the case where $C = D = E = Set^T$ and where $F$ is the left adjoint to the lift $\hom(R, -): Set^T \to Set^T$, we find that \begin{displaymath} - \odot R: Ladj(Set^T, Set^T) \to Ladj(Set^T, Set^T) \end{displaymath} is a left adjoint, and in particular preserves $Y$-indexed copowers $Y \cdot S$. In other words, for each $X$ we have canonical isomorphisms \begin{displaymath} X \cdot (S \odot R) \cong (X \cdot S) \odot R, \qquad T X \cdot (S \odot R) \cong (T X \cdot S) \odot R \end{displaymath} so that the desired right $T$-module structure is given componentwise by a composite \begin{displaymath} \beta (S \odot R)_X \coloneqq \left(T X \cdot (S \odot R) \cong (T X \cdot S) \odot R \stackrel{(\beta S)_X \odot R}{\to} (X \cdot S) \odot R \cong X \cdot (S \odot R)\right). \end{displaymath} \hypertarget{tallwraith_monoids_relative_to_a_monad}{}\subsubsection*{{Tall-Wraith monoids relative to a monad}}\label{tallwraith_monoids_relative_to_a_monad} A \emph{Tall-Wraith monoid} over $T$ is of course a monoid in the monoidal category $(T\text{-}BiAlg, \odot)$. We note that the unit in his monoidal category is the free $T$-algebra $F(1)$, equipped with its canonical lift $id: Set^T \to Set^T$. That is, the $T$-coalgebra structure on $F(1)$ is given tautologously by \begin{displaymath} T(X) \cong U F(X) \cong Set^T(F(1), F(X)) \cong Set^T(F(1), X \cdot F(1)). \end{displaymath} So, multiplication on a Tall-Wraith monoid is a bialgebra map $m: R \odot R \to R$ and the unit is a bialgebra map $u: F(1) \to R$. Such a monoid is tantamount precisely to a monoid in $Ladj(Set^T, Set^T)$, i.e., to a \emph{left adjoint monad} on $Set^T$. In particular, for a Tall-Wraith monoid $R$, one has a category $R Alg$ of algebras over that monad, giving a monadic functor $Alg_R \to Set^T$. Now, recall that left adjoint monads are canonically [[mate|mated]] to right adjoint comonads $C$, in such a way that the category of algebras over the monad is equivalent to the category of coalgebras over the comonad. In short, Tall-Wraith monoids over $T$ are essentially the same thing as functors \begin{displaymath} G: C \to Set^T \end{displaymath} which are simultaneously \emph{monadic and comonadic}: the comonadicity means $G$ is a left adjoint and has a left adjoint $F$, so that the monad $G F: Set^T \to Set^T$ resides in $Ladj(Set^T, Set^T)$, and such monads are tantamount to Tall-Wraith monoids. \begin{example} \label{}\hypertarget{}{} In the important example where $T$ is the theory of commutative rings and $\Lambda$ is the bialgebra $\mathbb{Z}[x_1, x_2, \ldots]$, equipped with a Tall-Wraith multiplication $\Lambda \odot \Lambda \to \Lambda$ given by [[plethysm]] (a decategorified product that arises by viewing $\Lambda$ as the Grothendieck ring of the category of $Ab$-valued [[species]] together with its substitution or plethystic product), the category $Alg_\Lambda$ may be identified with the category of [[lambda-rings]]. In this case the monad $\Lambda \odot -$ has right adjoint given by $\hom(\Lambda, -)$. The \emph{right} adjoint $CRing \to \Lambda Ring$ to the forgetful functor is the big Witt functor, often denoted $W$. \end{example} \hypertarget{references}{}\subsection*{{References}}\label{references} \begin{itemize}% \item D. Tall, [[Gavin Wraith|G. Wraith]], \emph{Representable functors and operations on rings}, Proc. London Math. Soc. (3), 1970, 619--643, \href{http://www.ams.org/mathscinet-getitem?mr=265348}{MR265348}, \href{http://dx.doi.org/10.1112/plms/s3-20.4.619}{doi} \item [[James Borger]], B. Wieland, \emph{Plethystic algebra}, Adv. Math. \textbf{194} (2005), no. 2, 246--283, \href{http://dx.doi.org/10.1016/j.aim.2004.06.006}{doi}, \href{http://wwwmaths.anu.edu.au/~borger/papers/03/lambda.pdf}{pdf}, \href{http://www.ams.org/mathscinet-getitem?mr=2139914}{MR2006i:13044} \item [[Andrew Stacey|A. Stacey]] and S. Whitehouse, \emph{The Hunting of the Hopf Ring}, Homology, Homotopy and Applications \textbf{11}(2), 2009, 75--132, \href{http://intlpress.com/HHA/v11/n2/a6}{online}, \href{http://arxiv.org/abs/0711.3722}{arXiv/0711.3722}. \end{itemize} An old and long query-discussion has been archived starting \href{http://www.math.ntnu.no/~stacey/Mathforge/nForum/comments.php?DiscussionID=291&Focus=26302#Comment_26302}{here}. [[!redirects Tall-Wraith monoid]] [[!redirects Tall-Wraith monoids]] [[!redirects Tall–Wraith monoid]] [[!redirects Tall–Wraith monoids]] [[!redirects Tall--Wraith monoid]] [[!redirects Tall--Wraith monoids]] \end{document}