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\newcommand{\map}{\textbf{Map}} +\newcommand{\Mon}{\textbf{Mon}} \newcommand{\Map}{\operatorname{\mathbf{Map}}} \newcommand{\Id}{\mathrm{Id}} \newcommand{\FVect}{\mathbf{FVect}} @@ -1852,7 +1853,7 @@ y\left(X_1^{X_2}\right)[-]=\text{Hom}_C(-,X_1^{X_2})&\cong\text{Hom}_C(-\times X \iam{Andreas} $\text{Hom}_C(y(T)\times y(X_2),y(X_1))\cong \text{Hom}_C(y(X_2),y(X_1)^{y(T)})\cong \text{Hom}_C(X_2,X_1^{T})$ - +\iam{Nicola} My point was that $y(x)$ for $x\in C_0$ is a pre-sheaf $y(x):C^{op}\to \Set$, so writing $\text{Hom}_C(y(X_2),y(X_1)^{y(T)})$ or $\text{Hom}_C(-\times y(X_2),X_1)$ is imprecise. \section{Week 8} I'm sorry that there were no new exercises this week, since we were still covering the Yoneda lemma. In the meanwhile, we can prove the following two lemmas, which are about forming two-variable functors and natural transformations from their data. @@ -1970,7 +1971,26 @@ with $FU(M) = M$ as $FU = \text{id}_{\textbf{Mon}}$. Clearly, $F(g) \circ F(e) = This completes the proof of the claim. +\iam{Nicola} You correctly point out the the unit of the adjunction is corresponds to the function that ``inserts generators'' at a set-theoretic level, but what about the co-unit? As a general fact, this should be \emph{the} homomorphism which takes the free algebra on the elements on $M$ and imposes on them the constraints given by the multiplication in $M$. + +For instance let us take $\mathbb{Z}_2$. Its underlying set has just two elements $U(\mathbb{Z}_2)=\{e,a\}$ so the free algebra on this set is given by words with two letters and concatenation. More explicitly, $FU(\mathbb{Z}_2)$ is the monoid whose underlying set is $$\{\emptyset, e, a, ee, aa, ea, ae, aaa, \dots\}$$ +and whose multiplication is given by concatenation of words, so that, for instance, $$eaeaa\cdot ea=eaeaaea$$ +Now, one can work out that the co-unit acts by taking the quotient the free monoid by the following relations (which is the multiplication table of $\mathbb{Z}_2$) +\begin{align*} +ea=a\\ +ae=a\\ +ee=e\\ +aa=e +\end{align*} +so that it defines a homomorphism $FU(\mathbb{Z}_2)\to \mathbb{Z}_2$. +This is a very general observation: all monoids admit this surjection from the free monoid on their underlying set. This allows us to both \emph{present} monoids by free monoids, with the upshot that free monoids are easy to deal with, and to do so in a functorial way. In the sense that we can lift a monoid homomorphism $f:M\to N$ to the induced homomorphism on the respective free monoids, and as such define the following commuting 2-diagram +\begin{center} +\begin{tikzcd} + & \mathbf{Proj} \arrow[rd, hook] \arrow[d, "\epsilon", Rightarrow] & \\ +\Mon \arrow[ru, "F\circ U"] \arrow[rr, "\id"'] & {\,} & \Mon +\end{tikzcd} +\end{center} \subsection{} Let $2$ denote the discrete 2-element category, and consider the diagonal functor \begin{align*} diff --git a/Exercises/Exercises.tex.bak b/Exercises/Exercises.tex.bak index 8913c2a..698d25e 100644 --- a/Exercises/Exercises.tex.bak +++ b/Exercises/Exercises.tex.bak @@ -65,11 +65,13 @@ \newtheorem{thm}{Theorem}[section] \newtheorem*{cor}{Corollary} \newtheorem*{claim}{Claim} +\newtheorem*{subclaim}{Subclaim} \newtheorem{prop}[thm]{Proposition} \newtheorem{lem}[thm]{Lemma} \theoremstyle{definition} \newtheorem{defn}[thm]{Definition} +\newtheorem*{defn*}{Definition} \newtheorem{digression}[thm]{Digression} \newtheorem{conj}[thm]{Conjecture} \newtheorem{const}[thm]{Construction} @@ -1850,7 +1852,7 @@ y\left(X_1^{X_2}\right)[-]=\text{Hom}_C(-,X_1^{X_2})&\cong\text{Hom}_C(-\times X \iam{Andreas} $\text{Hom}_C(y(T)\times y(X_2),y(X_1))\cong \text{Hom}_C(y(X_2),y(X_1)^{y(T)})\cong \text{Hom}_C(X_2,X_1^{T})$ - +\iam{Nicola} My point was that $y(x)$ for $x\in C_0$ is a pre-sheaf $y(x):C^{op}\to \Set$, so writing $\text{Hom}_C(y(X_2),y(X_1)^{y(T)})$ or $\text{Hom}_C(-\times y(X_2),X_1)$ is imprecise. \section{Week 8} I'm sorry that there were no new exercises this week, since we were still covering the Yoneda lemma. In the meanwhile, we can prove the following two lemmas, which are about forming two-variable functors and natural transformations from their data. @@ -1890,14 +1892,99 @@ define functors for all $a\in A_0$ and $b\in B_0$; \end{itemize} \end{lem} -\begin{proof} + +\begin{proof} +\iam{Muhammed} We start proving the backward direction. Assume, $(f,g): (A,B) \to (A', B')$ is a morphism in $\mathcal{C} \times \mathcal{D}$. Then, $(f,g)$ factorizes in +$$ +(A,B) \xrightarrow{(f, \text{id}_{B})} (A, B') \xrightarrow{(\text{id}_{A'} , g)} (A', B'). +$$ +Now, by assumption, $\alpha_{A, -}$ and $\alpha_{-, B}$ are natural transformations. This yields +\begin{center} +\begin{tikzcd} +{F(A,B)} \arrow[rr, "{\alpha_{A,B}}"] \arrow[d, "{F(f, \text{id}_{B})}"] \arrow[dd, "{F(f,g)}"', bend right=49] & & {G(A,B)} \arrow[d, "{G(f, \text{id}_{B})}"'] \arrow[dd, "{G(f,g)}", bend left=49] \\ +{F(A', B)} \arrow[rr, "{\alpha_{A', B}}"] \arrow[d, "{F(\text{id}_{A'}, g)}"] & & {G(A',B)} \arrow[d, "{G(\text{id}_{A'}, g)}"'] \\ +{F(A', B')} \arrow[rr, "{\alpha_{A', B'}}"'] & & {G(A', B')} +\end{tikzcd} +\end{center} +where the inner squares commute by naturality of $\alpha_{A, -}$ and $\alpha_{-, B}$ and bequest their commutativity to the outer square. Since this is true for any objects $A,A \in \mathcal{C}$ and $B, B' \in \mathcal{D}$, it follows readily that $\alpha$ is a natural transformation.\\ +The forward direction, on the other hand, is trivial. Since $\alpha$ is a natural transformation, it holds that, for any morphism $(f,g)$ on $\mathcal{C} \times {D}$ as above, the outer square commutes by assumption. Choices for $(f,g)$ as $(f,id_{B})$ and $(\id_{A},g)$ yields the naturality of $\alpha_{A,-}$ and $\alpha_{-, B}$. \end{proof} \section{Week 9} \subsection{} The forgetful functor $U:\mathbf{Mon}\to \Set$ has a left-adjoint \emph{free monoid} functor $F:\Set\to \mathbf{Mon}$. Work out the details of this adjunction (its unit and co-unit). Show that free monoids are projective in $\mathbf{Mon}$, and that every monoid $M$ admits a surjective map from a free monoid $F(S)\twoheadrightarrow M$.\hfill \break +\iam{Muhammed} +Let us start by noting that $U:\mathbf{Mon}\to \Set$ is left-adjoint to the free monoid functor $F:\Set\to \mathbf{Mon}$ by virtue of the universal property of free monoids, i.e. let $M$ be a monoid and $f: A \to U(M)$ a set-theoretic map. Then, there exists a unique homomorphism between monoids $g: F(A) \to M$ s.t. the following diagram in $\mathbf{Set}$ commutes: +\begin{center} +% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAFUAxACgEEBKEAF9S6TLnyEUAJnJVajFm3bcAQoJFjseAkTIBGefWatEIXsPkwoAc3hFQAMwBOEALZIyIHBCT7qDFhgpiBQdHAAFtYg1MZKZio2GqIgLu5Ist6+iF5xIQA6+fg4dMIpaR6ImT5+sYohjjEgDHQARjAMAAriOlIgzlg2ETiWQkA +\begin{tikzcd} +UF(A) \arrow[rr, "U(g)", dashed] & & U(M) \\ +A \arrow[u, "\iota"] \arrow[rru, "f"'] & & +\end{tikzcd} +\end{center} +where $\iota: A \to UF(A)$ is the insertion of generators. In other words, we have the isomorphism of sets +$$ +\alpha_{A,M}: \text{Hom}_{\textbf{Set}}(F(A), M) \cong \text{Hom}_{\textbf{Set}}(A, U(M)) +$$ +or equivalently, the natural transformation +$$ +\alpha: \text{Hom}_{\textbf{Set}}(F(-), -) \cong \text{Hom}_{\textbf{Set}}(-, U(-)) +$$ +The characterization of adjunctions in terms of unit and co-unit denoted by $\eta$ and $\varepsilon$ respectively, produces (via Yoneds) two morphisms +\begin{align*} +\eta_{A}=\alpha_{A, F(A)}(\text{id}_{F(A)}): A \to UF(A) \quad \text{ and } \quad \varepsilon_{M} = \alpha^{-1}_{U(M),M}(\text{id}_{U(M)}): FU(M) \to M. +\end{align*} +for $A \in \textbf{Set}_{0}$ and $M \in \mathbf{Mon}_{0}$. Clearly, $\eta_{-}: \mathbf{Set} \to \mathbf{Set}$ is the insertion of generators.\\[0.1cm] + +\begin{claim} Free monoids are projective in $\mathbf{Mon}$. +\end{claim} + +Recall the definition of a projective object. +{\small \begin{defn*} +An object $P$ is said to be \textit{projective} if it satisfies the following UMP: For any epimorphism $e: E \twoheadrightarrow X$ and any morphism $P \to X$, there exists a unique $g: P \to E$ such that $eg = f$. +\end{defn*} +} + + +\textbf{Step 1.} All sets are projective (by axiom of choice).\\ + +\textbf{Step 2.} Let $A, S$ be sets and $M$ a monoid. By step 1, $A$ is projective, thus +\begin{center} +\begin{tikzcd} + & S \arrow[d, "e", two heads] \\ +A \arrow[r, "f"'] \arrow[ru, "g", dashed] & U(M) +\end{tikzcd} +\end{center} +commutes. We use the left-adjointness of $U$ to $F$ to obtain an associated commutative diagram (via $\alpha$) which is of the form +\begin{center} +\begin{tikzcd} + & F(S) \arrow[d, "F(e)", two heads] \\ +F(A) \arrow[r, "F(f)"'] \arrow[ru, "F(g)", dashed] & FU(M) +\end{tikzcd} +\end{center} +with $FU(M) = M$ as $FU = \text{id}_{\textbf{Mon}}$. Clearly, $F(g) \circ F(e) = F(g \circ e) = F(f)$, hence $F(A) \in \textbf{Mon}_{0}$ is a projective. \\ + +\textbf{Step 3.} The existence of an epimorphism $F(S) \twoheadrightarrow M$ follows from the above diagram (and more generally by the fact, that epimorphisms are preserved by left-adjoint functors).\\ +This completes the proof of the claim. + + +\iam{Nicola} You correctly point out the the unit of the adjunction is corresponds to the function that ``inserts generators'' at a set-theoretic level, but what about the co-unit? As a general fact, this should be \emph{the} homomorphism which takes the free algebra on the elements on $M$ and imposes on them the constraints given by the multiplication in $M$. + +For instance let us take $\mathbb{Z}_2$. Its underlying set has just two elements $U(\mathbb{Z}_2)=\{e,a\}$ so the free algebra on this set is given by words with two letters and concatenation. More explicitly, $FU(\mathbb{Z}_2)$ is the monoid whose underlying set is $$\{\emptyset, e, a, ee, aa, ea, ae, aaa, \dots\}$$ +and whose multiplication is given by concatenation of words, so that, for instance, $$eaeaa\cdot ea=eaeaaea$$ +Now, one can work out that the co-unit acts by taking the quotient the free monoid by the following relations (which is the multiplication table of $\mathbb{Z}_2$) +\begin{align*} +ea=a\\ +ae=a\\ +ee=e\\ +aa=e +\end{align*} +so that it defines a homomorphism $FU(\mathbb{Z}_2)\to \mathbb{Z}_2$. + +This is a very general observation: all monoids admit this surjection from the free monoid on their underlying set. This allows us to both \emph{present} monoids by free monoids, with the upshot that free monoids are easy to deal with, and to do so in a functorial way, in that we can lift a monoid homomorphism $f:M\to N$ to the induced homomorphism on the respective free monoids. + \subsection{} Let $2$ denote the discrete 2-element category, and consider the diagonal functor \begin{align*} \Delta:\;&C\longrightarrow [2,C]\\ @@ -1924,16 +2011,9 @@ $$P\cong \Hom(-,2):\Set^{op}\longrightarrow \Set$$ \subsection{} Let $U:C\downarrow a\to C$ denote the obvious forgetful functor from the \emph{slice category} of $C$ over $a\in C_0$, to $C$. Does this functor have a right adjoint? A left adjoint? Also, deduce from duality any statement about the \emph{coslice category} $a\downarrow C$.\hfill \break -\subsection{} Let $R$ be an object in $\mathbf{Ring}$, the category of rings and ring homomorphisms, and let $\mathbf{Ring}_*$ denote the category of \emph{pointed} rings; whose objects are rings $R$ together with a distinguished element $r\in R$, and morphisms $f:(R,r)\to (R$ are ring homomorphism $f:R\to R'$ which are also a map of the distinguished points, meaning that $f(r)=r'$. - +\subsection{} Let $R$ be an object in $\mathbf{Ring}$, the category of rings and ring homomorphisms, and let $\mathbf{Ring}_*$ denote the category of \emph{pointed} rings; whose objects are rings $R$ together with a distinguished element $r\in R$, and morphisms $f:(R,r)\to (R',r')$ are ring homomorphism $f:R\to R'$ which also map the distinguished points, meaning that $f(r)=r'$. -\begin{comment} -\subsection{} We can treat any set $S$ as a discrete category with only identity morphisms, this defines an obvious functor $D:\mathbf{Set}\to \Cat$. -\begin{enumerate} -\item Describe explicitly the Yoneda embedding $y:D(S)\hookrightarrow \Set^{D(S)}$; -\item -\end{enumerate} -\end{comment} +There is an evident forgetful functor $U:\mathbf{Ring}_*\to \mathbf{Ring}$, prove that this functor has a left adjoint, sending a ring $R$, to its ring of polynomial expressions $R[x]$. Describe explicitly the unit and co-unit of this adjunction. (\emph{Hint}: this is posed as Example 9.10 in Awodey)\hfill \break \end{document} diff --git a/Exercises/Exercises.toc b/Exercises/Exercises.toc index a23107c..3afe7cf 100644 --- a/Exercises/Exercises.toc +++ b/Exercises/Exercises.toc @@ -1,90 +1,90 @@ -\contentsline {section}{\tocsection {}{1}{Instructions}}{1}{section.1}% -\contentsline {subsection}{\tocsubsection {}{1.1}{}}{1}{subsection.1.1}% -\contentsline {subsection}{\tocsubsection {}{1.2}{}}{2}{subsection.1.2}% -\contentsline {section}{\tocsection {}{2}{Week 1}}{2}{section.2}% -\contentsline {subsection}{\tocsubsection {}{2.1}{}}{2}{subsection.2.1}% -\contentsline {subsection}{\tocsubsection {}{2.2}{}}{3}{subsection.2.2}% -\contentsline {subsection}{\tocsubsection {}{2.3}{}}{3}{subsection.2.3}% -\contentsline {subsection}{\tocsubsection {}{2.4}{}}{4}{subsection.2.4}% -\contentsline {subsection}{\tocsubsection {}{2.5}{}}{4}{subsection.2.5}% -\contentsline {subsection}{\tocsubsection {}{2.6}{}}{4}{subsection.2.6}% -\contentsline {subsection}{\tocsubsection {}{2.7}{}}{4}{subsection.2.7}% -\contentsline {subsection}{\tocsubsection {}{2.8}{}}{4}{subsection.2.8}% -\contentsline {subsection}{\tocsubsection {}{2.9}{}}{5}{subsection.2.9}% -\contentsline {subsection}{\tocsubsection {}{2.10}{}}{5}{subsection.2.10}% -\contentsline {subsection}{\tocsubsection {}{2.11}{}}{6}{subsection.2.11}% -\contentsline {section}{\tocsection {}{3}{Discussion: Week 1}}{6}{section.3}% -\contentsline {section}{\tocsection {}{4}{Week 2}}{7}{section.4}% -\contentsline {subsection}{\tocsubsection {}{4.1}{}}{7}{subsection.4.1}% -\contentsline {subsection}{\tocsubsection {}{4.2}{}}{7}{subsection.4.2}% -\contentsline {subsection}{\tocsubsection {}{4.3}{}}{7}{subsection.4.3}% -\contentsline {subsection}{\tocsubsection {}{4.4}{}}{9}{subsection.4.4}% -\contentsline {subsection}{\tocsubsection {}{4.5}{}}{9}{subsection.4.5}% -\contentsline {subsection}{\tocsubsection {}{4.6}{}}{9}{subsection.4.6}% -\contentsline {subsection}{\tocsubsection 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