From 37ddb0a771a5f5dd07acefc2c04a82624172c1c6 Mon Sep 17 00:00:00 2001 From: Derived Cat Date: Fri, 26 Jul 2024 14:17:40 -0400 Subject: [PATCH] update category theory --- category_theory.tex | 273 ++++++++++++++++++++++++++++++++++++++------ group.tex | 8 +- ring.tex | 7 +- 3 files changed, 247 insertions(+), 41 deletions(-) diff --git a/category_theory.tex b/category_theory.tex index 96f63c3..381a15e 100644 --- a/category_theory.tex +++ b/category_theory.tex @@ -1955,30 +1955,56 @@ \section{Representable Functor} In addition, an alternative ad-hoc proof is conceivable. If $\diagfunctor \{*\}$ is naturally isomorphic to $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ through $\theta:\mathrm{Hom}_\mathsf{C}\left(A,-\right)\xRightarrow{\sim} \diagfunctor \{*\}$, we have no choice but to define $\theta_X(\mathrm{id}_X)=*$. Note that $\diagfunctor \{*\}(A)=\{*\}$. Yoneda lemma also implies that $\theta$ must correspond to $*\in \diagfunctor \{*\}(A)$ and accordingly $\theta$ is the unique natural isomorphism from $\mathrm{Hom}_\mathsf{C}\left(A,-\right)$ to $\diagfunctor\{*\}$. \end{prf} - +\begin{lemma}{}{isomorphism_between_comma_category_and_Grothendieck_construction} + Let $F:\mathsf{C}\to \mathsf{D}$ be a functor and $X\in \mathrm{Ob}(\mathsf{D})$. Consider functors + \[ + \begin{tikzcd}[ampersand replacement=\&,column sep=2.2cm] + \boldone \arrow[r, "\diagfunctor X"] \& {\mathsf{D}} \& \mathsf{C} \arrow[l, "F"']\\ + \boldone \arrow[r, "\diagfunctor \{*\}"] \& {\mathsf{Set}} \& \mathsf{C} \arrow[l, "{\mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)}"'] + \end{tikzcd} + \] + + We have the following category isomorphism + $$ + \left(X \downarrow F\right) + \cong + \left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)\right)=\int_{\mathsf{C}} \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right) + $$ + through the functor $(A,u)\mapsto(A,\diagfunctor u)$, whose action on commutative diagrams is illustrated as follows + \[ + \begin{tikzcd}[ampersand replacement=\&] + X \arrow[r, "u"] \arrow[rdd, "g"'] \&[+20pt] F(A) \arrow[dd, "F\left(h\right)", dashed]\&[+10pt]\&[+10pt] \{*\} \arrow[r, "\diagfunctor u"] \arrow[rdd, "\diagfunctor g"'] \&[+15pt] \mathrm{Hom}_{\mathsf{D}}\left(X,F(A)\right)\arrow[dd, "F(h)_*", dashed] \\[-2pt] + \& \&\longmapsto\&\&\\[-2pt] + \& F(B)\&\&\&\mathrm{Hom}_{\mathsf{D}}\left(X,F(B)\right) + \end{tikzcd} + \] +\end{lemma} \begin{proposition}{Equivalent Characterizations of Universal Morphism}{universal_morphism_by_representability} - Let $F:\mathsf{C}\to \mathsf{D}$ be a functor and $A\in \mathrm{Ob}(\mathsf{D})$. Then the following statements are equivalent: + Let $F:\mathsf{C}\to \mathsf{D}$ be a functor, $A,X\in \mathrm{Ob}(\mathsf{D})$ and $u:X\to F(A)$ be a morphism. Then the following statements are equivalent: \begin{enumerate}[(i)] \item $(A,u)$ is initial in the category $\left(X \downarrow F\right)$. \item $\mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)$ is representable by universal element $(A,u)$. - \item $(A,u)$ is initial in the category $\int_{\mathsf{C}} \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)=\left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)\right)$. + \item $(A,\diagfunctor u)$ is initial in the category $\int_{\mathsf{C}} \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)=\left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)\right)$. \end{enumerate} Dually, the following statements are equivalent: \begin{enumerate}[(i)] \item $(A,u)$ is terminal in the category $\left(F \downarrow X\right)$. \item $\mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)$ is representable by universal element $(A,u)$. - \item $(A,u)$ is initial in the category $\int_{\mathsf{C}^{\mathrm{op}}} \mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)=\left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)\right)$. + \item $(A,\diagfunctor u)$ is initial in the category $\int_{\mathsf{C}^{\mathrm{op}}} \mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)=\left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{D}}\left(F(-),X\right)\right)$. \end{enumerate} \end{proposition} \begin{prf} - It suffices to show (i)$\iff$ (iii). + It suffices to show (i)$\iff$ (iii). The category $\left(X \downarrow F\right)$ is isomorphic to $\int_{\mathsf{C}} \mathrm{Hom}_{\mathsf{D}}\left(X,F(-)\right)$ through the functor \[ \begin{tikzcd}[ampersand replacement=\&] - X \arrow[r, "u"] \arrow[rd, "g"'] \&[+20pt] F(A) \arrow[d, "F\left(h\right)", dashed]\&[+50pt] \{*\} \arrow[r, "\diagfunctor u"] \arrow[rd, "\diagfunctor g"'] \&[+15pt] \mathrm{Hom}_{\mathsf{D}}\left(X,F(A)\right)\arrow[d, "F(h)_*", dashed] \\[+15pt]\& F(B)\&\&\mathrm{Hom}_{\mathsf{D}}\left(X,F(B)\right) + X \arrow[r, "u"] \arrow[rdd, "g"'] \&[+20pt] F(A) \arrow[dd, "F\left(h\right)", dashed]\&[+10pt]\&[+10pt] \{*\} \arrow[r, "\diagfunctor u"] \arrow[rdd, "\diagfunctor g"'] \&[+15pt] \mathrm{Hom}_{\mathsf{D}}\left(X,F(A)\right)\arrow[dd, "F(h)_*", dashed] \\[-2pt] + \& \&\longmapsto\&\&\\[-2pt] + \& F(B)\&\&\&\mathrm{Hom}_{\mathsf{D}}\left(X,F(B)\right) \end{tikzcd} \] + We can also verify it directly. \begin{align*} (A,u)\text{ is initial in }\left(X \downarrow F\right)&\iff \forall(B,g)\in \mathrm{Ob}\left(X \downarrow F\right),\;\exists! h\in \mathrm{Hom}_{\mathsf{C}}(A,B),\;F(h)\circ u=g\\ &\iff \forall B\in \mathrm{Ob}\left(\mathsf{C}\right),\;\forall g\in \mathrm{Hom}_{\mathsf{C}}(X,F(B)),\;\exists! h\in \mathrm{Hom}_{\mathsf{C}}(A,B),\;F(h)\circ u=g\\ @@ -1989,7 +2015,7 @@ \section{Representable Functor} \end{prf} -\begin{example}{Identity Functor $\mathrm{id}_{\mathsf{Set}}$ is Representable}{} +\begin{example}{Identity Functor $\mathrm{id}_{\mathsf{Set}}$ is Representable}{identity_functor_is_representable} The identity functor $\mathrm{id}_{\mathsf{Set}}:\mathsf{Set}\to \mathsf{Set}$ is representable by $\left(\{*\},\mathrm{id}_{\{*\}}\right)$. The natural isomorphism $\phi:\mathrm{Hom}_{\mathsf{Set}}\left(\{*\},-\right)\xRightarrow{\sim} \mathrm{id}_{\mathsf{Set}}$ is defined by \begin{align*} \phi_X:\mathrm{Hom}_{\mathsf{Set}}\left(\{*\},X \right)&\xlongrightarrow{\sim} X\\ @@ -2068,7 +2094,10 @@ \section{Limit and Colimit} \mathsf{C} \arrow[r, "\diagfunctor"] \& {[\mathsf{J},\mathsf{C}]} \& \boldone \arrow[l, "\diagfunctor F"'] \end{tikzcd} \] - The comma category $\left(\diagfunctor \downarrow \diagfunctor F\right)$ is called the \textbf{cone category from $\textsf{C}$ to $F$}, denoted by $\mathsf{Cone}(\textsf{C},F)$. + The comma category $\left(\diagfunctor \downarrow \diagfunctor F\right)$ is called the \textbf{cone category from $\textsf{C}$ to $F$}, denoted by $\mathsf{Cone}(\textsf{C},F)$. According to \Cref{th:isomorphism_between_comma_category_and_Grothendieck_construction}, we have category isomorphism + \[ + \mathsf{Cone}(\textsf{C},F)=\left(\diagfunctor \downarrow \diagfunctor F\right)\cong \left( \{*\}\downarrow \mathrm{Hom}_{\mathsf{[\mathsf{J},\mathsf{C}]}}\left(\diagfunctor (-),F\right)\right)=\int_{\mathsf{C}^{\mathrm{op}}} \mathrm{Hom}_{\mathsf{[\mathsf{J},\mathsf{C}]}}\left(\diagfunctor (-),F\right). + \] \begin{itemize} \item Objects: The objects in $\mathsf{Cone}(\textsf{C},F)$ are all natural transformations \[ @@ -2170,7 +2199,7 @@ \section{Limit and Colimit} \end{definition} -\begin{proposition}{Limit Characterized by Representability}{} +\begin{proposition}{Limit Characterized by Representability}{limit_by_representability} Let $\mathsf{J},\mathsf{C}$ be categories and $F:\mathsf{J}\to\mathsf{C}$ be a functor. The limit of $F$ exists if and only if the functor $\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\diagfunctor\left(-\right), F\right)$ \begin{align*} \begin{minipage}{.7\textwidth} @@ -2206,6 +2235,44 @@ \section{Limit and Colimit} According to \Cref{th:representable_functor_by_universal_element}, $\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\diagfunctor\left(-\right), F\right)$ is representable by the universal element $(A,u)$ if and only if $(A,u)$ is initial in $\int_{\mathsf{C}^{\mathrm{op}}}\mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\diagfunctor\left(-\right), F\right)$, which is equivalent to saying that for any $C\in\mathrm{Ob}\left(\mathsf{C}\right)$ and $(h_i)\in \mathrm{Hom}_{[\mathsf{J},\mathsf{C}]}\left(\diagfunctor C, F\right)$, there is a unique morphism $f:C\to A$ in $\mathsf{C}$ such that $h_i=f^*(u_i)=u_i\circ f$ for each $i\in \mathrm{Ob}(\mathsf{J})$. This is exactly the universal property of limit. \end{prf} +\begin{lemma}{}{} + If $\mathsf{J}$ is a small category with a terminal object $t$, then for any functor $F:\mathsf{J}\to \mathsf{C}$, we have $\varinjlim F\cong F(t)$. +\end{lemma} +\begin{prf} + Since for every $j\in \mathrm{Ob}(\mathsf{J})$, there is a unique morphism $\lambda_j:j\to t$, $\left(F(t),\left(F(\lambda_j):F(j)\to F(t)\right)_{j\in \mathrm{Ob}(\mathsf{J})}\right)$ is a cocone from $F$ to $F(t)$. We need to show that it is initial. Note that $F\left(\lambda_t \right)=F(\mathrm{id}_t)=\mathrm{id}_{F(t)}$. For any cocone $\left(C,\left(h_j:F(j)\to C\right)_{j\in \mathrm{Ob}(\mathsf{J})}\right)$, there is a morphism $h_t:F(t)\to C$ such that for any $j\in \mathrm{Ob}(\mathsf{J})$, $h_t\circ F(\lambda_j) =h_j$, which means the following diagram commutes + \[ + \begin{tikzcd} + & C & \\[+10pt] + & F(t) \arrow[u, "h_t"', dashed] & \\ + F(j) \arrow[ru, "F(\lambda_j)"'] \arrow[ruu, "h_j"] \arrow[rr, "F(\lambda_j)"'] & & F(t) \arrow[lu, "\mathrm{id}"] \arrow[luu, "h_t"'] + \end{tikzcd} + \] + If there is another morphism $g:F(t)\to C$ such that for any $j\in \mathrm{Ob}(\mathsf{J})$, $g\circ F(\lambda_j) =h_j$. Take $j=t$, and we have $g=h_t$. Therefore, $\left(F(t),\left(F(\lambda_j):F(j)\to F(t)\right)_{j\in \mathrm{Ob}(\mathsf{J})}\right)$ is the colimit of $F$. + +\end{prf} + +\begin{example}{}{} + Let $\mathsf{C}$ be a category and $X\in \mathrm{Ob}(\mathsf{C})$. Consider the hom functor $\mathrm{Hom}_{\mathsf{C}}(X,-):\mathsf{C}\to\mathsf{Set}$. We have + \[ + \varinjlim \mathrm{Hom}_{\mathsf{C}}(X,-)\cong \{*\}. + \] +\end{example} +\begin{prf} + Let $h^X=\mathrm{Hom}_{\mathsf{C}}(X,-)$. + By \Cref{th:limit_by_representability}, we have + \[ + \mathrm{Hom}_{[\mathsf{C},\mathsf{Set}]}\left(h^X,\diagfunctor\left(-\right)\right)\cong \mathrm{Hom}_{\mathsf{Set}}\left(\varinjlim h^X,-\right). + \] + where $\diagfunctor:\mathsf{Set}\to[\mathsf{C},\mathsf{Set}]$ is the diagonal functor. By \hyperref[th:yoneda_lemma]{Yoneda Lemma}, we have natural isomorphism + \[ + \mathrm{Hom}_{[\mathsf{C},\mathsf{Set}]}\left(h^X,\diagfunctor\left(-\right)\right)\cong \mathrm{Hom}_{[\mathsf{C},\mathsf{Set}]}\left(h^X,-\right)\circ \diagfunctor \cong \mathrm{ev}_X \circ \diagfunctor \cong \mathrm{id}_{\mathsf{Set}}\cong \mathrm{Hom}_{\mathsf{Set}}\left(\{*\},- \right). + \] + The last isomorphism has been proved in \Cref{ex:identity_functor_is_representable}. Therefore, we have + \[ + \mathrm{Hom}_{\mathsf{Set}}\left(\varinjlim h^X,-\right) \cong \mathrm{Hom}_{\mathsf{Set}}\left(\{*\},- \right). + \] + Since Yoneda embedding is full and faithful, we have $\varinjlim h^X\cong \{*\}$. +\end{prf} \begin{definition}{$\varprojlim$ Functor}{} Let $\mathsf{J}$ be a small category and $\mathsf{C}$ be a category. If for any functor $F:\mathsf{J}\to\mathsf{C}$, $\varprojlim F$ exists, then we have a functor @@ -2232,8 +2299,8 @@ \section{Limit and Colimit} \] \end{definition} -\begin{proposition}{Diagonal is Left Adjoint to Limit $\diagfunctor\dashv \varprojlim$}{} - Let $\mathsf{J}$ be a small category and $\mathsf{C}$ be a category. Then the functor $\varinjlim:\left[\mathsf{J},\mathsf{C}\right]\to\mathsf{C}$ is right adjoint to the diagonal functor $\diagfunctor:\mathsf{C}\to\left[\mathsf{J},\mathsf{C}\right]$ +\begin{proposition}{Diagonal is Left Adjoint to Limit: $\diagfunctor\dashv \varprojlim$}{} + Let $\mathsf{J}$ be a small category and $\mathsf{C}$ be a category. If for any functor $F\in[\mathsf{J},\mathsf{C}]$, $\varprojlim F$ exists, then the functor $\varinjlim:\left[\mathsf{J},\mathsf{C}\right]\to\mathsf{C}$ is right adjoint to the diagonal functor $\diagfunctor:\mathsf{C}\to\left[\mathsf{J},\mathsf{C}\right]$ \[ \begin{tikzcd}[ampersand replacement=\&] \mathsf{C} \arrow[r, "\diagfunctor"{name=U}, bend left, start anchor=east, yshift=1.7ex, end anchor=west] \&[+12pt] @@ -2247,8 +2314,8 @@ \section{Limit and Colimit} \] \end{proposition} -\begin{proposition}{Diagonal is Right Adjoint to Colimit $\varinjlim\dashv \diagfunctor$}{} - Let $\mathsf{J}$ be a small category and $\mathsf{C}$ be a category. Then the functor $\varinjlim:\left[\mathsf{J},\mathsf{C}\right]\to\mathsf{C}$ is right adjoint to the diagonal functor $\diagfunctor:\mathsf{C}\to\left[\mathsf{J},\mathsf{C}\right]$ +\begin{proposition}{Diagonal is Right Adjoint to Colimit: $\varinjlim\dashv \diagfunctor$}{} + Let $\mathsf{J}$ be a small category and $\mathsf{C}$ be a category. If for any functor $F\in[\mathsf{J},\mathsf{C}]$, $\varprojlim F$ exists, then the functor $\varinjlim:\left[\mathsf{J},\mathsf{C}\right]\to\mathsf{C}$ is right adjoint to the diagonal functor $\diagfunctor:\mathsf{C}\to\left[\mathsf{J},\mathsf{C}\right]$ \[ \begin{tikzcd}[ampersand replacement=\&] \left[\mathsf{J},\mathsf{C}\right] \arrow[r, "\varinjlim"{name=U}, bend left, start anchor=east, yshift=1.7ex, end anchor=west] \&[+12pt] @@ -2364,21 +2431,31 @@ \section{Limit and Colimit} \begin{definition}{Preserve, Reflect, Create Limits}{} Suppose $\mathsf{J}$, $\mathsf{C}$, $\mathsf{D}$ are categories and $\mathcal{K}\subseteq\mathrm{Ob}\left([\mathsf{J}, \mathsf{C}]\right)$ is a class of diagrams valued in $\mathsf{C}$. A functor $F: \mathsf{C} \rightarrow \mathsf{D}$ is said to \begin{itemize} - \item \textbf{preserves} limits for $\mathcal{K}$ if for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in $\mathcal{K}$ and limit cone over $K$, the image of this cone defines a limit cone over the composite diagram $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$; Or equivalently, $F_*$ maps terminal objects in $\mathsf{Cone}(\mathsf{C},K)$ to terminal objects in $\mathsf{Cone}(\mathsf{D},F\circ K)$; - \item \textbf{reflects} limits for $\mathcal{K}$ if any cone over a diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in $\mathcal{K}$, whose image upon applying $F$ is a limit cone for the diagram $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$, is a limit cone over $K$; - \item \textbf{creates} limits for $\mathcal{K}$ if whenever $K: \mathsf{J} \rightarrow \mathsf{C}$ is a diagram in $\mathcal{K}$ and $F\circ K: \mathsf{J} \rightarrow \mathsf{D}$ has a limit in $\mathsf{D}$, there exists some limit cone over $F\circ K$ that can be lifted to a limit cone over $K$, and moreover $F$ reflects the limits for $\mathcal{K}$. - \end{itemize} -\end{definition} - -\begin{definition}{}{} - A functor $F: \mathsf{C} \rightarrow \mathsf{D}$ strictly creates limits for a given class of diagrams if for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in the class and limit cone over $F K: \mathsf{J} \rightarrow \mathsf{D}$, - \begin{itemize} - \item there exists a unique lift of that cone to a cone over $K$, and - \item moreover, this lift defines a limit cone in $\mathsf{C}$. + \item \textbf{preserves} limits for $\mathcal{K}$ if for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in $\mathcal{K}$ and any cone $\mu\in \mathsf{Cone}(\mathsf{C},K)$, + \[ + \text{$\mu$ is terminal in $\mathsf{Cone}(\mathsf{C},K)$ $\implies$ $F_*\mu$ is terminal in $\mathsf{Cone}(\mathsf{D},F\circ K)$} + \] + \item \textbf{reflects} limits for $\mathcal{K}$ if for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in $\mathcal{K}$ and any cone $\mu\in \mathsf{Cone}(\mathsf{C},K)$, + \[\text{$F_*\mu$ is terminal in $\mathsf{Cone}(\mathsf{D},F\circ K)$ $\implies$ $\mu$ is terminal in $\mathsf{Cone}(\mathsf{C},K)$}\] + \item \textbf{creates} limits for $\mathcal{K}$ if + \begin{itemize} + \item $F$ reflects the limits for $\mathcal{K}$, and + \item for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in $\mathcal{K}$, if $\eta$ is terminal in $\mathsf{Cone}(\mathsf{D},F\circ K)$, then there exists a cone $\mu$ in $\mathsf{Cone}(\mathsf{C},K)$ such that $F_*\mu$ is terminal in $\mathsf{Cone}(\mathsf{D},F\circ K)$. + \end{itemize} + \item \textbf{strictly creates} limits for $\mathcal{K}$ if for any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in $\mathcal{K}$ and limit cone over $F \circ K: \mathsf{J} \rightarrow \mathsf{D}$, + \begin{itemize} + \item there exists a unique lift of that cone to a cone over $K$, and + \item moreover, this lift defines a limit cone in $\mathsf{C}$. + \end{itemize} \end{itemize} - \end{definition} +\begin{proposition}{}{} + Suppose $\mathsf{J}$, $\mathsf{C}$, $\mathsf{D}$ are categories and $\mathcal{K}\subseteq\mathrm{Ob}\left([\mathsf{J}, \mathsf{C}]\right)$ is a class of diagrams valued in $\mathsf{C}$. If $F: \mathsf{C} \rightarrow \mathsf{D}$ creates limits for $\mathcal{K}$ and $\varprojlim F\circ K$ exists for each $K \in\mathcal{K}$, then $\varprojlim K$ exists for each $K \in\mathcal{K}$ and $F$ preserves them. +\end{proposition} +\begin{proof} + For any diagram $K: \mathsf{J} \rightarrow \mathsf{C}$ in $\mathcal{K}$, the hypothesis asserts that there is a cone $\mu: \diagfunctor d \Rightarrow F\circ K$ in $\mathsf{Cone}(\mathsf{D},F\circ K)$. As $F$ creates these limits, there must be a limit cone $\lambda: \diagfunctor c \Rightarrow K$ in $\mathsf{Cone}(\mathsf{C},K)$ such that $F_* \lambda: \diagfunctor F(c) \Rightarrow F\circ K$ is a limit cone. Since $F$ reflects limits, $\lambda$ is a limit cone, which means that $\mathsf{C}$ admits limits for $\mathcal{K}$. To see that $F$ preserves them, consider another limit cone $\lambda^{\prime}: \diagfunctor c^{\prime} \Rightarrow K$. The two limit cones in $\mathsf{Cone}(\mathsf{C},K)$ are isomorphic and by composing isomorphisms we see that the cone $F \lambda^{\prime}: \diagfunctor F c^{\prime} \Rightarrow F\circ K$ is isomorphic to the limit cone $\mu: \diagfunctor d \Rightarrow F\circ K$. This implies that $F \lambda^{\prime}: F c^{\prime} \Rightarrow F K$ is again a limit cone, proving that $F$ preserves these limits. +\end{proof} \begin{definition}{Conservative Functor}{} A functor $F:\mathsf{C}\to\mathsf{D}$ is \textbf{conservative} if it reflects isomorphisms. Equivalently, $F$ is a conservative functor if for any morphism $f$ in $\mathsf{C}$, @@ -2394,21 +2471,21 @@ \section{Limit and Colimit} \begin{proposition}{Fully Faithful Functor Reflects Limits}{} - A full and faithful functor reflects all limits and + Any full and faithful functor reflects all limits and colimits. \end{proposition} \begin{prf} Let $F: \mathsf{C} \rightarrow \mathsf{D}$ be a full and faithful functor. Suppose $K: \mathsf{J} \rightarrow \mathsf{C}$ is a diagram, $$ - \left(\ell_i: A \rightarrow K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathrm{Cone}\left(\mathsf{C},K\right) + \left(\ell_i: A \rightarrow K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathsf{Cone}\left(\mathsf{C},K\right) $$ - is a cone over $K$, and $\left(F(\ell_i): F(A) \rightarrow F\circ K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is a limit cone over $F\circ K$. We want to show that $\left(\ell_i\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is initial in $\mathrm{Cone}\left(\mathsf{C},K\right)$. + is a cone over $K$, and $\left(F(\ell_i): F(A) \rightarrow F\circ K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is a limit cone over $F\circ K$. We want to show that $\left(\ell_i\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is initial in $\mathsf{Cone}\left(\mathsf{C},K\right)$. - Suppose $\left(h_i:B \rightarrow K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathrm{Cone}\left(\mathsf{C},K\right)$ is another cone over $K$. Then + Suppose $\left(h_i:B \rightarrow K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathsf{Cone}\left(\mathsf{C},K\right)$ is another cone over $K$. Then \[ - \left( F(h_i): F(B)\longrightarrow F\circ K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathrm{Cone}\left(\mathsf{D},F\circ K\right) + \left( F(h_i): F(B)\longrightarrow F\circ K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathsf{Cone}\left(\mathsf{D},F\circ K\right) \] is a cone over $F\circ K$. By the universal property of $F(A)$, there exists a unique morphism $g':F(B)\to F(A)$ such that $F(\ell_i) \circ g' = F(h_i)$ for each $i\in \mathrm{Ob}(\mathsf{J})$ \[ @@ -2427,7 +2504,7 @@ \section{Limit and Colimit} \[ \forall i \in\mathrm{Ob}\left(\mathsf{J}\right),\;F\left( \ell_i \circ q\right)=F\left(\ell_i \right)\circ F\left( q\right)=F\left(h_i\right)\implies F\left( q\right)=g'\implies q=g. \] - Thus we show that $\left(\ell_i:A\to K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is initial in $\mathrm{Cone}\left(\mathsf{C},K\right)$. + Thus we show that $\left(\ell_i:A\to K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$ is initial in $\mathsf{Cone}\left(\mathsf{C},K\right)$. \end{prf} @@ -2438,13 +2515,13 @@ \section{Limit and Colimit} \begin{prf} For any $K\in\mathcal{K}$, since $\varprojlim F\circ K$ exists and $F$ creates limits for $\mathcal{K}$, there exists a limit cone \[ - \left(\ell_i:\varprojlim K\to K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in\mathrm{Cone}\left(\mathsf{C},K\right) + \left(\ell_i:\varprojlim K\to K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in\mathsf{Cone}\left(\mathsf{C},K\right) \] such that \[ - \left(F(\ell_i):F\left(\varprojlim K\right)\to F\circ K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathrm{Cone}\left(\mathsf{D},F\circ K\right) + \left(F(\ell_i):F\left(\varprojlim K\right)\to F\circ K(i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}\in \mathsf{Cone}\left(\mathsf{D},F\circ K\right) \] - is terminal in $\mathrm{Cone}\left(\mathsf{D},F\circ K\right)$, i.e it is a limit cone over $F\circ K$. Given any limit cone in $\mathrm{Cone}\left(\mathsf{C},K\right) $, it is isomorphic to $\left(\ell_i\right)_{i\in \mathrm{Ob}(\mathsf{J})}$, which implies that its image under $F$ is isomorphic to $\left(F(\ell_i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$. Thus we show $F$ preserves limits for $\mathcal{K}$. + is terminal in $\mathsf{Cone}\left(\mathsf{D},F\circ K\right)$, i.e it is a limit cone over $F\circ K$. Given any limit cone in $\mathsf{Cone}\left(\mathsf{C},K\right) $, it is isomorphic to $\left(\ell_i\right)_{i\in \mathrm{Ob}(\mathsf{J})}$, which implies that its image under $F$ is isomorphic to $\left(F(\ell_i)\right)_{i\in \mathrm{Ob}(\mathsf{J})}$. Thus we show $F$ preserves limits for $\mathcal{K}$. \end{prf} @@ -2453,7 +2530,7 @@ \section{Limit and Colimit} \end{theorem} -The next result shows that functor categories inherit limits and colimits, defined ``objectwise" in the target category: that is, given a $\mathsf{J}$-indexed diagram in $[\mathsf{A},\mathsf{C}]$ whose objects are functors $F_j : A \to C$, the value of the limit functor $\lim_{j\in J} F_j : A \to C$ at an object $a \in \mathsf{A}$ is the limit of the $\mathsf{J}$-indexed diagram in $\mathsf{C}$ whose objects are the objects $F_j(a) \in \mathrm{Ob}(C)$. +The next result shows that functor categories inherit limits and colimits, defined ``objectwise" in the target category: that is, given a $\mathsf{J}$-indexed diagram in $[\mathsf{A},\mathsf{C}]$ whose objects are functors $F_j : \mathsf{A} \to \mathsf{C}$, the value of the limit functor $\varprojlim_{j\in J} F_j : \mathsf{A} \to \mathsf{C}$ at an object $a \in \mathsf{A}$ is the limit of the $\mathsf{J}$-indexed diagram in $\mathsf{C}$ whose objects are the objects $F_j(a) \in \mathrm{Ob}(C)$. \begin{proposition}{Evaluation Functor Preserves Limits}{ev_functor_preserves_limits} Let $\mathsf{A}$ be a small category and $\mathsf{C}$ be a category. Given a diagram $F: \mathsf{J} \to\left[\mathsf{A},\mathsf{C}\right]$ with $\mathsf{J}$ small, if for any $a\in \mathsf{A}$, the diagram @@ -2469,7 +2546,7 @@ \section{Limit and Colimit} \begin{proposition}{}{} - If $\mathsf{A}$ is a small category, then the forgetful functor $U:[\mathsf{A},\mathsf{C}] \rightarrow \left[\mathsf{Disc}\left(\mathrm{Ob} \left(\mathsf{A}\right)\right), \mathsf{C}\right]\cong \prod\limits_{a\in \mathrm{Ob} \left(\mathsf{A}\right)}\mathsf{C}$ + Suppose $\mathsf{A}$ is a small category. Denote $\mathsf{Ob}(A)=\mathsf{Disc}\left(\mathrm{Ob} \left(\mathsf{A}\right)\right)$. Then the forgetful functor $U:[\mathsf{A},\mathsf{C}] \rightarrow \left[\mathsf{Ob}(A), \mathsf{C}\right]\cong \prod\limits_{a\in \mathrm{Ob} \left(\mathsf{A}\right)}\mathsf{C}$ \[ \begin{tikzcd}[ampersand replacement=\&] [\mathsf{A},\mathsf{C}] \&[-25pt]\&[+10pt]\&[-30pt]\prod\limits_{i\in \mathrm{Ob} \left(\mathsf{A}\right)}\mathsf{C}\&[-30pt]\&[-30pt] \\ [-15pt] @@ -2569,8 +2646,132 @@ \section{Limit and Colimit} This implies $\backsimeq$ contains $\approx$. Therefore, $\approx$ is the smallest equivalence relation containing $\sim^*$, which means $\approx$ coincides with $\sim$. \end{prf} +\begin{definition}{Connected Category}{} + A category $\mathsf{C}$ is \textbf{connected} if it is nonempty and for any pair of objects $X,Y\in \mathrm{Ob}(\mathsf{C})$, there exists a zigzag of morphisms $(f_1,f_2,\cdots,f_{2n})$ connecting $X$ and $Y$ as follows +\[ + \begin{tikzcd}[ampersand replacement=\&] + \& [-3em] \& Z_1 \arrow[ld, "f_1"', dashed] \arrow[rd, "f_2", dashed] \& \& Z_3 \arrow[ld, "f_3"', dashed] \arrow[rd, "f_4", dashed] \& \& Z_{2n-1} \arrow[ld, "f_{2n-1}"', dashed] \arrow[rd, "f_{2n}", dashed] \& \& [-3em] \\ + X= \& Z_0 \& \& Z_2 \& \& \cdots \& \& Z_{2n} \& =Y + \end{tikzcd} + \] +\end{definition} +\begin{definition}{Final Functor}{} + A functor $F:\mathsf{C}\to\mathsf{D}$ is \textbf{final} if for every object $d\in \mathsf{D}$, the comma category $(d\downarrow F)$ obtained from + \[ + \begin{tikzcd}[ampersand replacement=\&] + \boldone \arrow[r, "\text{const}_d"] \&[+20pt]\mathsf{D} \&[+20pt]\mathsf{C} \arrow[l, "F"'] + \end{tikzcd} + \] + is connected. That is, given any $d\in \mathrm{Ob}\left(\mathsf{D}\right)$, $(d\downarrow F)$ is nonempty, and for any morphisms $d\to F(c)$ and $d\to F(c')$ there exists a zigzag of morphisms $(f_1,f_2,\cdots,f_{2n})$ such that the following diagram commutes + \[ + \begin{tikzcd} + & [-3em] & & d \arrow[lldd, bend right=40] \arrow[rrdd, bend left=36] \arrow[ld] \arrow[rd] \arrow[dd] & & & [-3em]\\ + & & F(c_1) \arrow[ld, "f_1", dashed] \arrow[rd, "f_2"', dashed] & & F(d_{2n-1}) \arrow[ld, "f_{2n-1}", dashed] \arrow[rd, "f_{2n}"', dashed] & &\\ + F(c)= & F(c_0) & & \cdots & & F(c_{2n}) & =F(c') + \end{tikzcd} + \] +\end{definition} + +\begin{proposition}{Equivalent Characterization of Final Functor}{} + Let $F:\mathsf{C}\to\mathsf{D}$ be a functor. The following are equivalent: + \begin{enumerate}[(i)] + \item $F$ is final. + \item For any functor $G:\mathsf{D}\to\mathsf{E}$, the natural morphism between colimits + \[ + \varinjlim G \circ F \longrightarrow \varinjlim G + \] + is an isomorphism. + \end{enumerate} +\end{proposition} + +Note that $d\in \mathrm{Ob}\left(\mathsf{D}\right)$ is a final object in $\mathsf{D}$ if and only if the functor $\text{const}_d:\boldone \to \mathsf{D}$ is final. + +\begin{definition}{Initial Functor}{} + A functor $F:\mathsf{C}\to\mathsf{D}$ is \textbf{initial} if the opposite functor $F^{\text{op}}:\mathsf{C}^{\text{op}}\to\mathsf{D}^{\text{op}}$ is final. +\end{definition} + +\subsection{Product and Coproduct} +\begin{definition}{Binary Product}{} + \begin{center} + \begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}] + & X \arrow[d, dash pattern=on 4pt off 2pt, draw=arrowRed] \arrow[ld, "f_1"', draw=cyan] \arrow[rd, "f_2", draw=cyan] & \\[+10pt] + Y_1 & Y_1\times Y_2 \arrow[l, "\pi_1", draw=arrowBlue] \arrow[r, "\pi_2"', draw=arrowBlue] & Y_2 + \end{tikzcd} + \end{center} +\end{definition} + +In the diagram $Y_1\xleftarrow{f_1} X \xrightarrow{f_2} Y_2$, the information in $X$ is coarsened through $f_1$ and reinterpreted in $Y_1$, while the same information in $X$ is processed in another coarsening manner through $f_2$ and reinterpreted differently in $Y_2$. + +$Y_1\xleftarrow{\pi_1}Y_1\times Y_2 \xrightarrow{\pi_2} Y_2$ is the most refined way to combine the information in $Y_1$ and $Y_2$ such that $Y_1\times Y_2$ exactly captures the mixture of information in $Y_1$ and $Y_2$ and the projections $\pi_1$ and $\pi_2$ can coarsen the information in $Y_1\times Y_2$ exactly to recover the information in $Y_1$ and $Y_2$ respectively. + +\begin{example}{Binary Product in $\mathsf{Set}$}{} + Let $Y_1$ and $Y_2$ be sets. The binary product $Y_1\times Y_2$ can be constructed as follows + \begin{align*} + Y_1\times Y_2&=\left\{(y_1,y_2) \midv y_1\in Y_1\text{ and }y_2\in Y_2\right\}.\\ + \pi_1&:(y_1,y_2)\longmapsto y_1,\\ + \pi_2&:(y_1,y_2)\longmapsto y_2. + \end{align*} + Given any $Y_1\xleftarrow{f_1} X \xrightarrow{f_2} Y_2$, there exists a unique map + \begin{align*} + f_1\times f_2 :X&\longrightarrow Y_1\times Y_2\\ + x&\longmapsto (f_1(x),f_2(x)) + \end{align*} + such that $\pi_1\circ (f_1\times f_2)=f_1$ and $\pi_2\circ (f_1\times f_2)=f_2$. +\end{example} + +\begin{definition}{Binary Coproduct}{} + \begin{center} + \begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}] + & Y & \\[+10pt] + X_1 \arrow[r, "\iota_1"', draw=arrowBlue] \arrow[ru, "f_1", draw=cyan] & X_1 \sqcup X_2 \arrow[u, dash pattern=on 4pt off 2pt, draw=arrowRed] & X_2 \arrow[l, "\iota_2", draw=arrowBlue]\arrow[lu, "f_2"', draw=cyan] + \end{tikzcd} + \end{center} +\end{definition} + +\subsection{Fibered Product and Fibered Coproduct} +\begin{definition}{Fibered Product / Pullback}{} + \begin{center} + \begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}] + X \arrow[rd, dash pattern=on 4pt off 2pt, draw=arrowRed] \arrow[rrd, draw=cyan, bend left] \arrow[rdd, draw=cyan, bend right] &[-2em] & \\[5pt] + & Y_1\times_{A} Y_2 \arrow[d, "\pi_1"', draw=arrowBlue] \arrow[r, "\pi_2", draw=arrowBlue] & Y_2 \arrow[d,"f_2"] \\[10pt] + & Y_1 \arrow[r, "f_1"'] & A + \end{tikzcd} + \end{center} +\end{definition} + +\begin{example}{Pullback in $\mathsf{Set}$}{} + Let $Y_1$, $Y_2$, and $A$ be sets and $f_1:Y_1\to A$, $f_2:Y_2\to A$ be maps. The fibered product $Y_1\times_{A} Y_2$ can be constructed as follows + \begin{align*} + Y_1\times_{A} Y_2&=\left\{(y_1,y_2) \midv y_1\in Y_1\text{ and }y_2\in Y_2\text{ such that }f_1(y_1)=f_2(y_2)\right\}.\\ + \end{align*} + Given any $X\xrightarrow{f} Y_1 \xleftarrow{g} Y_2$, there exists a unique map + \begin{align*} + X&\longrightarrow Y_1\times_{A} Y_2\\ + x&\longmapsto (f(x),g(x)) + \end{align*} + such that the following diagram commutes + \begin{center} + \begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}] + X \arrow[rd, dash pattern=on 4pt off 2pt, draw=arrowRed] \arrow[rrd, draw=cyan, bend left] \arrow[rdd, draw=cyan, bend right] &[-2em] & [-0.4em] \\[5pt] + & Y_1\times_{A} Y_2 \arrow[d, draw=arrowBlue] \arrow[r, draw=arrowBlue] & Y_2 \arrow[d] \\[10pt] + & Y_1 \arrow[r] & A + \end{tikzcd} + \end{center} + +\end{example} + +\begin{definition}{Fibered Coproduct / Pushout}{} + \begin{center} + \begin{tikzcd}[every arrow/.append style={-latex, line width=1.2pt}] + & [-0.4em] \\[10pt] + A \arrow[d] \arrow[r] & X_2 \arrow[d, draw=arrowBlue]\arrow[rdd, draw=cyan, bend left] \\[10pt] + X_1 \arrow[rrd, draw=cyan, bend right]\arrow[r, draw=arrowBlue] & X_1 \sqcup_A X_2 \arrow[rd, dash pattern=on 4pt off 2pt, draw=arrowRed] \\ + & & Y + \end{tikzcd} + \end{center} +\end{definition} \section{Adjoint Functor} \begin{definition}{Adjoint Pair of Functors}{} diff --git a/group.tex b/group.tex index ca19a42..583f297 100644 --- a/group.tex +++ b/group.tex @@ -616,16 +616,16 @@ \subsection{Coset} \subsection{Conjugacy Action} \begin{definition}{Conjugacy Action and Inner Automorphism Group}{conjugacy_action} - Let $G$ be a group. The \textbf{conjugacy action} of $G$ on itself is defined as + Let $G$ be a group. The \textbf{conjugacy action} of $G$ on itself is defined as a group homomorphism \begin{align*} \gamma:G &\longrightarrow \mathrm{Aut}_{\mathsf{Grp}}(G)\\ g &\longmapsto (\gamma_g: x\longmapsto gxg^{-1}) \end{align*} - The \textbf{inner automorphism group} of $G$ is defined as + The \textbf{inner automorphism group} of $G$ is defined as the image of $\gamma$ $$ - \mathrm{Inn}(G)=\gamma(G)=\{ \gamma_g\mid g\in G\}, + \mathrm{Inn}(G)=\operatorname{im}\gamma=\{ \gamma_g\mid g\in G\}. $$ - which is a subgroup of $\mathrm{Aut}_{\mathsf{Grp}}(G)$. + And we have inclusion relation $\mathrm{Inn}(G)\hookrightarrow\mathrm{Aut}_{\mathsf{Grp}}(G)\hookrightarrow\mathrm{Aut}_{\mathsf{Set}}(G)$. \end{definition} \begin{definition}{Conjugate Subgroups}{} diff --git a/ring.tex b/ring.tex index 8a0a7c9..cbbd1c8 100644 --- a/ring.tex +++ b/ring.tex @@ -89,11 +89,16 @@ \section{Basic Concepts} \begin{proposition}{Examples of Reduced Ring}{} \begin{enumerate}[(i)] - \item Subrings, products, and localizations of reduced rings are again reduced rings. + \item Subrings, products, and localizations of reduced commutative rings are again reduced rings. \item Every integral domain is reduced. \item $\mathbb{Z}/n\mathbb{Z}$ is reduced if and only if $n=0$ or $n$ is square-free. \end{enumerate} \end{proposition} +\begin{proof} + \begin{enumerate}[(i)] + \item Let $R$ be a reduced ring and $S$ be a multiplicative subset of $R$. For any $\frac{f}{s}\in S^{-1}R$, if $\left(\frac{f}{s}\right)^n=\frac{f^n}{s^n}=0$, then there exists $t\in S$ such that $tf^n=0$, which implies $(tf)^n=0$. Since $R$ is reduced, we have $tf=0$, which means $\frac{f}{s}=0$. Hence $S^{-1}R$ is reduced. + \end{enumerate} +\end{proof} \begin{definition}{Local Ring}{}