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b/Exercises/Exercises.tex index 51b139e..9755d5b 100644 --- a/Exercises/Exercises.tex +++ b/Exercises/Exercises.tex @@ -1292,9 +1292,36 @@ Then let $E$ act as F and $D$ be the inclusion, again one can see the properties \subsection{} Let $\mathcal{C}\cong \mathcal{D}$ be an equivalence of categories. Prove that $\mathcal{C}$ has products if and only if $\mathcal{D}$ does.\hfill \break -\iam{Marius} (Work in progress... Needs to be streamlined) Let $\mathcal{C} \simeq \mathcal{D}$ be equivalent categories with equivalence $F: \mathcal{C} \simeq \mathcal{D}$. We know by the theorem we proved in class that $F$ is fully faithful and essentially surjective. In other words $\Hom(c,x) \cong \Hom(Fc,Fx)$ and any $d \in D$ is isomorphic to some $Fc$. Consequently, because isomorphic objects have isomorphic Hom-Sets, $\Hom(d,y) \cong \Hom(Fc,Fx)$ for a suitable choice of $c,x$. Furthermore, if $\alpha: d \overset{\sim}{\rightarrow} Fc$ then the following commutes for each $f: A \rightarrow B$ in $\mathcal{D}$ by associativity of composition, +\iam{Marius}(Second argument reducing the equivalent case to the isomorphic case using skeletons.) + +Define a skeleton $\text{sk}(\Ca)$ of a category $\Ca$ as follows: For each isomorphism class of objects in $\Ca$, choose one representative and add this to $\text{Ob(sk}(\Ca))$. Then $\text{sk}(\Ca) \simeq \Ca$ under the inclusion functor and $c \cong d$ implies $c = d$ for objects $c,d \in \text{sk}(\Ca)$. Note that the above requires a choice principle and that skeletons are only unique up to isomorphism. + +We next observe that equivalent categories have isomorphic skeletons. To see this let $F: \Ca \rightarrow \mathcal{D}$ be an equivalence. This induces an equivalence $\bar{F}: \text{sk}(\Ca) \rightarrow \mathcal{D}$ for an arbitrary skeleton of $\Ca$, with $\bar{F}$ simply being the restriction of $F$ under the inclusion functor of the skeleton, which we shall from now on simply denote $F$. Now consider the induced map $\tilde{F}: \text{sk}(\Ca) \rightarrow \text{sk}(\mathcal{D})$, where $\tilde{F}c$ is set to be the representing object of the isoclass of $Fc$. Moreover, an arrow $Ff : Fc \rightarrow Fd$ induces an arrow $\tilde{F}f: \tilde{F}c \rightarrow \tilde Fd$ by post- and pre-composing with appropriate isomorphisms $\alpha_{Fc}: Fc \rightarrow \tilde Fc$ by setting $\tilde Ff := \alpha_{Fd} Ff \alpha_{Fc}^{-1}$. Functoriality is preserved by this assignment: +$$\tilde F\text{id}_c = +\alpha_{Fc} F\text{id}_c \alpha_{Fc}^{-1} = +\alpha_{Fc} \alpha_{Fc}^{-1} = +\text{id}_{\tilde Fc} +$$ and +$$\tilde F(gf) = +\alpha_{Fe} F(gf) \alpha_{Fc}^{-1} = +\alpha_{Fe} Fg Ff \alpha_{Fc}^{-1} = +\alpha_{Fe} Fg \alpha_{Fd}^{-1} \alpha_{Fd} Ff \alpha_{Fc}^{-1} = +\tilde Fg \tilde Ff. +$$ + +We can do the same for the pseudo-inverse $G$ of $F$ to obtain an induced functor $\tilde G : \text{sk}(\mathcal{D}) \rightarrow \text{sk}(\Ca)$. Now because $GF \cong id_\Ca$, we have $\tilde G \tilde F = id_{\text{sk}(\Ca)}$, because our construction preserved the isoclass of the source and target objects, and because isomorphic objects are equal in skeletal categories. Similarly $\tilde F \tilde G = id_{\text{sk}(\mathcal{D})}$. We have thus shown that equivalent categories have isomorphic skeletons. + +Using the above, we observe that if $\Ca$ has all limits of a certain type, then so does $\text{sk}(\Ca)$: For any diagram $D: J \rightarrow \text{sk}(\Ca)$, consider the diagram in $\Ca$ obtained by composition with the inclusion $\bar D: J \rightarrow \text{sk}(\Ca) \hookrightarrow \Ca$. This new diagram has a limit $\lim{\bar D}$ in $\Ca$. Moreover, $\lim{\bar D} \cong A$, for an object $A \in \text{sk}(\Ca)$. Hence, composing with this isomorphism we see that $A$ also fulfils the same universal property as $\lim{\bar D}$. + +The converse also holds. Suppose $\text{sk}(\Ca)$ has all limits od shape $J$ and let $D: J \rightarrow \Ca$ be a diagram in $\Ca$. For each $c \in \Ca$ choose an isomorphism $\alpha_c: c \rightarrow \bar c$ to its representative $\bar c$ in $\text{sk}(\Ca)$. In this way we obtain a projection functor $P: \Ca \rightarrow \text{sk}(\Ca)$ which maps each object in an isoclass to its representative in $\text{sk}(\Ca)$, and each morphism $f: c \rightarrow c'$ to the composition $\alpha_{c'} f \alpha_c^{-1}$. Using this we get an equivalent diagram in $\text{sk}(\Ca)$ by composing $D$ with the projection functor $P$. By assumption, this diagram has a limit $\lim(PD)$ in $\text{sk}(\Ca)$. We claim that this is also a limit of $D$. Given any cone over $D$, compose with the selected isomorphisms $\{\alpha_d\}_{d \in D}$ to obtain a cone over $PD$ (the way we have defined $P(f)$ ensures that this is indeed a cone). Hence we get a unique morphism $u$ from the apex of the cone into $\lim(PD)$ making the sides the cone over $PD$ commute. Finally, if we view $\lim(PD)$ as a cone over $D$ by composing its legs with $\{\alpha_d^{-1}\}_{d \in D}$, we see that $u$ also makes the sides of this cone over $D$ commute. Hence $\lim(PD) = \lim(D)$, with the caveat that we must shift the base of the cone via a family of isomorphisms. + +We are now ready to conclude. Let $\Ca \simeq \mathcal{D}$ and suppose $\Ca$ has all limits of shape J. Then so does $\text{sk}(\Ca)$. Since $\Ca$ and $\mathcal{D}$ have isomorphic skeletons, $\text{sk}(\mathcal{D})$ has all limits of shape J as well. Finally, this implies that $\mathcal{D}$ also has these limits, as desired. By duality the same holds for colimits. In particular, a category have products if and only if all the categories equivalent to it do. + +\iam{Marius} (In this first attempt, I tried to solve the exercise by looking at hom functors. Implicitly, I am using the fact that the functor category $\mathbf{Set}^{C^\text{op}}$ has products (because $C$ does) to get products in $\mathbf{Set}^{D^\text{op}}$, which implies $D$ has products. But the argument is hard to follow, and I am not confident it is entirely correct. I might attempt to rephrase this idea more abstractly using the property that the yoneda embedding reflects limits. If, however, an equivalence of categories simply induces an equivalence of yoneda embeddings, then this move does not seem to generate any leverage compared to the above approach.) + +Let $\mathcal{C} \simeq \mathcal{D}$ be equivalent categories with equivalence $F: \mathcal{C} \simeq \mathcal{D}$. We know by the theorem we proved in class that $F$ is fully faithful and essentially surjective. In other words $\Hom(c,x) \cong \Hom(Fc,Fx)$ and any $d \in \mathcal{D}$ is isomorphic to some $Fc$. Consequently, because isomorphic objects have isomorphic Hom-Sets, $\Hom(d,y) \cong \Hom(Fc,Fx)$ for a suitable choice of $c,x$. Furthermore, if $\alpha: d \overset{\sim}{\rightarrow} Fc$ then the following commutes for each $f: A \rightarrow B$ in $\mathcal{D}$ by associativity of composition, \begin{center} -% https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAdDnGADx2AAJCAFsAvgAoopAIIBKEGNLpMufIRTkK1Ok1bsuPfkNGSAYgGNZCpSux4CRLcR0MWbRJ258Bw8RMtSACEbZRAMe3UiMhcaN31PQx8Tf2kQxR0YKABzeCJQADMAJ1EkMhAcCCQtXXcDDiY0AAt6AH0AKkUw4tLEACYaSqQAZji9Dy9Glo6uwpKRMsGq-rG6zwKAPU7bEB6FxBqhxFHahN2tjLEgA + % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBoAGAXVJADcBDAGwFcYkQAdDnGADx2AAJCAFsAvgAoopAIIBKEGNLpMufIRTkK1Ok1bsuPfkNGSAYgGNZCpSux4CRLcR0MWbRJ258Bw8RMtSACEbZRAMe3UiMhcaN31PQx8Tf2kQxR0YKABzeCJQADMAJ1EkMhAcCCQtXXcDDiY0AAt6AH0AKkUw4tLEACYaSqQAZji9Dy9Glo6uwpKRMsGq-rG6zwKAPU7bEB6FxBqhxFHahN2tjLEgA \begin{tikzcd} {\text{Hom}(A,Fc)} \arrow[r, "\alpha_*"] \arrow[d, "f_*"] & {\text{Hom}(A,d)} \arrow[d, "f_*"] \\ {\text{Hom}(B,Fc)} \arrow[r, "\alpha_*"] & {\text{Hom}(B,d)} @@ -1313,8 +1340,9 @@ Finally, for any $c,x,y \in \mathcal{C}$ the following commutes because $F(fg) = F(f)F(g)$ by functoriality. We thus have a natural isomorphism $\Hom(-,c) \cong \Hom(F-,Fc)$ for all $c \in C$. Observe that because $F$ is fully faithful, its image $F(\mathcal{C})$ forms a full subcategory of $\mathcal D$ (each image object has an identity by functoriality, if $h \in \Hom_{F(C)}(FA,FB)$, then $h = Ff$ for $f \in \Hom(A,B)$ by fullness, in particular $F(\mathcal{C})$ is closed under composition because $\mathcal{D}$ is). By the above natural isomorphisms and the UMP of the product -$$\Hom_C(F-,FA) \times \Hom_C(F-,FB) \cong \Hom_C(-,A) \times \Hom_C(-,B) \cong \Hom_C(-, A \times B) \cong \Hom_C(F-, F(A \times B)).$$ -Because $\Hom_C(Fc,FA) = \Hom_{F(C)}(Fc,FA)$ for all $c \in \mathcal{C}$ we have $\Hom(F-,Fc) = \Hom_D(-, Fc) \circ F$. Using the above, and the observation that $\Hom_D(-,FA) \circ F \times \Hom_D(-,FB) \circ F = (\Hom_D(-,FA) \times \Hom_D(-,FB)) \circ F$ yields +$$\Hom_C(F-,FA) \times \Hom_C(F-,FB) \cong \Hom_C(-,A) \times \Hom_C(-,B) \cong \Hom_C(-, A \times B) \cong \Hom_C(F-, F(A \times B)),$$ +where the product is to be understood in the functor category. +Because $\Hom_C(FA,Fc) = \Hom_{F(C)}(FA,Fc)$ for all $c \in \mathcal{C}$ we have $\Hom(F-,Fc) = \Hom_D(-, Fc) \circ F$. Using the above, and the observation that $\Hom_D(-,FA) \circ F \times \Hom_D(-,FB) \circ F = (\Hom_D(-,FA) \times \Hom_D(-,FB)) \circ F$ yields $$(\Hom_D(-,FA) \times \Hom_D(-,FB)) \circ F \cong \Hom_D(-, F(A \times B)) \circ F.$$ Finally, composing on the right with $G$ (functors preserve natural isos) and using $FG \cong id_D$ gives $$\Hom_D(-,FA) \times \Hom_D(-,FB) \cong \Hom_D(-, F(A \times B)).$$ @@ -1325,11 +1353,14 @@ Therefore any pair of objects in $\mathcal{D}$ have a product. \subsection{} Regarding the previous question, what sort of properties are \emph{not} preserved by equivalence? Try to give an example of a property of a category that is preserved by isomorphism but not equivalence. \hfill \break -\iam{Marius} Our discussion regarding $\mathbf{Fin_{Set}}$ in week 1 shows that smallness is not preserved by equivalence of categories. As we saw, $\mathbf{Fin_{Set}}$ is not small, but equivalent to the small category $\mathbf{Fin_{Ord}}$ of finite ordinals. Here's a fun quote from the nLab entry on the \href{https://ncatlab.org/nlab/show/principle+of+equivalence#terminology}{principle of equivalence}: +\iam{Marius} Our discussion regarding $\mathbf{Fin_{Set}}$ in week 1 shows that smallness is not preserved by equivalence of categories. As we saw, $\mathbf{Fin_{Set}}$ is not small, but equivalent to the small category $\mathbf{Fin_{Ord}}$ of finite ordinals. +Further, we have shown in the previous exercise that every category is equivalent to a skeletal one. Therefore, being skeletal is not preserved by equivalence either. + +Here's a fun quote from the nLab entry on the \href{https://ncatlab.org/nlab/show/principle+of+equivalence#terminology}{principle of equivalence}: \emph{Floating around the web (and maybe the nLab) is the idea of half-jokingly referring to a breaking of equivalence invariance as “evilâ€. This is probably meant as a pedagogical way of amplifying that it is to be avoided.} -More generally, since equivalent categories have isomorphic skeletons, and because isomorphic objects have the same "arrow-theoretic" properties, it follows that any property that can be phrased in terms of arrows and their composition is preserved by equivalence of categories. Conversely, extrinsic properties of categories (e.g. smallness) that cannot be captured by properties of arrows may break equivalence invariance. As a silly example, consider the property "Every object of $\mathcal{C}$ is a group". Clearly, this property fails to be preserved under equivalence since we can construct an abstract category isomorphic to $\mathbf{Group}$ whose objects are not groups. +More generally, since equivalent categories have isomorphic skeletons, and because isomorphic objects have the same "arrow-theoretic" properties, it follows that any property that can be phrased in terms of arrows and their composition is preserved by equivalence of categories. Conversely, extrinsic properties of categories (e.g. smallness, or the fact that isoclasses have a certain size) that cannot be captured by properties of arrows may break equivalence invariance. As a silly example, consider the property "Every object of $\mathcal{C}$ is a group". Clearly, this property fails to be preserved under equivalence since we can construct an abstract category isomorphic to $\mathbf{Group}$ whose objects are not groups. \subsection{} Show that equivalence of categories is an equivalence relation.\hfill \break diff --git a/Exercises/Exercises.toc b/Exercises/Exercises.toc index 2c236c5..5909588 100644 --- a/Exercises/Exercises.toc +++ b/Exercises/Exercises.toc @@ -63,7 +63,7 @@ \contentsline {subsection}{\tocsubsection {}{10.3}{}}{26}{subsection.10.3}% \contentsline {subsection}{\tocsubsection {}{10.4}{}}{26}{subsection.10.4}% \contentsline {subsection}{\tocsubsection {}{10.5}{}}{26}{subsection.10.5}% -\contentsline {subsection}{\tocsubsection {}{10.6}{}}{26}{subsection.10.6}% -\contentsline {subsection}{\tocsubsection {}{10.7}{}}{27}{subsection.10.7}% -\contentsline {subsection}{\tocsubsection {}{10.8}{}}{27}{subsection.10.8}% -\contentsline {subsection}{\tocsubsection {}{10.9}{}}{27}{subsection.10.9}% +\contentsline {subsection}{\tocsubsection {}{10.6}{}}{27}{subsection.10.6}% +\contentsline {subsection}{\tocsubsection {}{10.7}{}}{28}{subsection.10.7}% +\contentsline {subsection}{\tocsubsection {}{10.8}{}}{29}{subsection.10.8}% +\contentsline {subsection}{\tocsubsection {}{10.9}{}}{29}{subsection.10.9}% -- GitLab