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\def \Ass{\mathop{\rm Ass}}
\def \Fun{\mathop{\rm Fun}}
\def \Nat{\mathop{\rm Nat}}
\def \Jac{\mathop{\rm Jac}}
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{\bf Math 8212\hfill Commutative and Homological Algebra 2\hfill Spring 2022}
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{\bf Homework Assignment 2} Due Saturday 3/5/2022, uploaded to Gradescope.
Each question part is worth 1 point. There are 12 question parts. Assume that all categories are small. We define $\Fun(\calC,\calD)$ to be the category whose objects are functors $\calC\to\calD$ and whose morphisms are natural transformations.
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1.
Suppose that $F:\calC\to\calD$ is an equivalence of categories.
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(a) Show that, for all objects $x,y\in\Ob\calC$, the functor $F$ provides a bijection
$$
\Hom_\calC(x,y)\leftrightarrow\Hom_\calD(F(x),F(y)),
$$
that preserves composition, so that $\End_\calC(x)\cong \End_\calD(F(x))$ as monoids.
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(b) Show that $x\cong y$ in $\calC$ if and only if $F(x)\cong F(y)$ in $\calD$, so that $F$ provides a bijection between the isomorphism classes of $\calC$, and of $\calD$.
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(c) Let $\calE$ be a further category. Show that the functor categories $\Fun(\calC,\calE)$ and $\Fun(\calD,\calE)$ are naturally equivalent.
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2. Let $\calC$ be a category and let $x,y\in\Ob\calC$. Prove that if $x\cong y$ then $\Hom_\calC(x,-)$ and $\Hom_\calC(y,-)$ are naturally isomorphic functors $\calC\to\Set$.
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3. Let $F,G:\calC\to\calD$ be functors and $\eta:F\to G$ a natural transformation.
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(a) Show that if, for all $x\in\Ob\calC$, the mapping $\eta_x:F(x)\to G(x)$ is an isomorphism in $\calD$, then $\eta$ is a natural isomorphism (meaning that it has a 2-sided inverse natural transformation $\theta:G\to F$).
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(b) Suppose that $F$ is an equivalence of categories and that $F$ is naturally isomorphic to $G$, so $F\simeq G$. Show that $G$ is an equivalence of categories.
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4. Let $G$ be a group, which we regard as a category $\calG$ with a single object, and with the elements of $G$ as morphisms. Let $F:\calG\to\calG$ be a functor.
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(a) Show that $F$ is naturally isomorphic to the identity functor $1_\calG:\calG\to \calG$ if and only if the mapping $F:G\to G$, induced by $F$ on the set of morphisms, is an inner automorphism; that is, an automorphism of the form $c_g: G\to G$ for some $g\in G$, where $c_g(h)=ghg^{-1}$ for all $h\in G$.
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(b) Show that self equivalences of $\calG$ are automorphisms of $\calG$.
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(c) Show that the group of natural isomorphism classes of self equivalences of $\calG$ is isomorphic to $\Aut(G)/\Inn(G)$. (In the context of group theory, $\Inn(G)$ denotes the set of inner automorphisms of $G$, and $\Out(G):=\Aut(G)/\Inn(G)$ is called the group of {\it outer} (or {\it non-inner}) automorphisms.)
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5. Let $I$ be the poset with two elements 0 and 1, and with $0<1$. If $P$ and $Q$ are posets we can regard them as categories $\calP$ and $\calQ$ whose objects are the elements of the posets, and where there is a unique morphism $x\to y$ if and only if $x\le y$.
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(a) Show that if $P$ and $Q$ are posets then a functor $\calP\to\calQ$ is `the same thing as' an order-preserving map. (Don't worry about any fancy interpretation of `the same thing as'!)
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(b) Now consider two functors $F,G:\calP\to\calQ$, which we may regard as order-preserving maps $f,g:P\to Q$ by part (a). Show that the following three conditions are equivalent:
(i) there exists a natural transformation $F\to G$,
(ii) $f(x)\le g(x)$ for all $x\in P$,
(iii) there is an order-preserving map $h:P\times I\to Q$ such that $h(x,0)=f(x)$ and $h(x,1)=g(x)$ for all $x\in\calP$. Here $P\times I$ denotes the product poset with order relation $(a_1,b_1)\le (a_2,b_2)$ if and only if $a_1\le a_2$ and $b_1\le b_2$, where $a_i\in P$ and $b_i\in I$.
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6. Let $1_{R{\rm-mod}}: R\hbox{-mod}\to R\hbox{-mod}$ denote the identity functor. Let $\Nat(1_{R{\rm-mod}},1_{R{\rm-mod}})$ denote the set of natural transformations from this functor to itself, noting that this set has the structure of a ring (multiplication is composition and addition comes because we can add homomorphisms of $R$-modules, so that for two natural transformations $\theta,\psi$ at an object $x$ we have $(\theta+\psi)_x = \theta_x + \psi_x$). Show that $\Nat(1_{R{\rm-mod}},1_{R{\rm-mod}})\cong Z(R)$.
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{\bf Extra question: do not upload to Gradescope.}
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7. Let $\calC$ be a small category and let $F,G:\calC\to\Set$ be functors. Show that a natural transformation of functors $\tau:F\to G$ is an epimorphism in $\Fun(\calC,\Set)$ if and only if for every object $x$ of $\calC$, $\tau_x:F(x)\to G(x)$ is a surjection; and it is a monomorphism if and only if for every object $x$ of $\calC$, $\tau_x:F(x)\to G(x)$ is a 1-1 map.
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8. Write out a proof that if $G$ is the right adjoint of a functor $F$ with the property that $F$ preserves monomorphisms, then $G$ sends injective objects to injective objects.
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9. Let $F:\calC\to\calD$ and $G:\calD\to\calC$ be functors with $F$ left adjoint to $G$, and with adjunction unit $\eta$ and counit $\epsilon$. Write out a proof that the second triangular identity holds, namely the following triangle commutes:
$$
\matrix{G&&\umapright{1_G}&&G\cr
&{\searrow\atop \eta_G}&&{\nearrow\atop G\epsilon}&\cr
&&GFG&&\cr}
$$
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10. Assume the axiom of choice in this question, or else make some assumption such as: everything is finite. Let $\calC$ be a category, and for each isomorphism class $\hat x$ of objects $x$, choose a fixed representative $u_{\hat x}$. For each object $x$ choose a fixed isomorphism $i_x:x\to u_{\hat x}$. Let $\calD$ be the full subcategory whose objects are the $u_{\hat x}$ where $x\in\Ob\calC$. `Full' means that for each pair of objects $y,z$ of $\calD$ we have $\Hom_\calD(y,z) = \Hom_\calC(y,z)$. Define $F(x) = \hat x$, and for each morphism $\alpha:x\to y$ define $F(\alpha) : F(x)\to F(y)$ to be $i_y\alpha i_x^{-1}$.
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(a) Show that $F$ is a functor.
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(b) Show that $F$ and the inclusion functor $\hbox{inc} :\calD\to \calC$ are inverse equivalences of categories $\calD\simeq \calC$. (It will help to assume that when $x=u_{\hat x}$, the chosen isomorphism is the identity $1_x$.)
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(c) Deduce that the category $\Set$ of finite sets is equivalent to the category with objects $\NN:=\{0, 1, 2,\ldots\}$ and where $\Hom(n,m)$ is the set of all mappings of sets from ${\bf n}:=\{1,\ldots, n\}$ to ${\bf m}:=\{1,\ldots, m\}$. We take ${\bf 0}=\emptyset$.
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(d) Deduce also the following: let $K$ be a field. Show that the category Vec of finite dimensional vectors spaces over $K$ is equivalent to the category $\calC$ with objects $\NN:=\{0, 1, 2,\ldots\}$, where $\Hom_\calC(n,m)$ is the set $M_{m,n}(K)$ of $m\times n$ matrices with entries in $K$, and where composition of morphisms is matrix multiplication. In case $m$ or $n$ is zero, give a definition of $\Hom_\calC(n,m)$ that will make this question make sense.
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11. Let $\calC$ be a small category. A {\it self-equivalence} of $\calC$ is an equivalence of categories $F:\calC\to\calC$.
Show that the set of natural isomorphism classes of self equivalences of $\calC$ is a group, with multiplication induced by composition of functors.
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1.
Let $\calC_n$ denote the category with $n$ objects, labeled $a_1,\ldots,a_n$, and where there is a unique homomorphism $a_i\to a_j$ for every ordered pair of numbers $(i,j)$. Note that this defines the composition of morphisms in the category. Show that the category algebra $R\calC_n$ is isomorphic to the algebra of $n\times n$-matrices $M_n(R)$.
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2.
Let $x$ be an object of a finite category $\calC$.
(a) Show that the subset $R\calC\cdot 1_x$ of the category algebra $R\calC$ is the span of the morphisms whose domain is $x$, and that $1_x\cdot R\calC$ is the span of the morphisms whose codomain is $x$.
(b) Show that $R\calC = \bigoplus_{x\in\Ob\calC} R\calC\cdot 1_x$ as left $R\calC$-modules.
(c) Let $R\Hom_\calC(x,-)$ denote the functor $\calC\to R\hbox{-mod}$ that sends an object $y$ to the free $R$-module with the set of homomorphisms $\Hom_\calC(x,y)$ as a basis. Under the correspondence between representations of $\calC$ over $R$ and $R\calC$-modules, show that the functor $R\Hom_\calC(x,-)$ corresponds to the left $R\calC$ module $R\calC\cdot 1_x$ and that $R\Hom(-,x)$ corresponds to the right $R\calC$ module $1_x\cdot R\calC$.
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3. What is the dimension of the category algebra $K\calP$ when $\calP$ is the poset with Hasse diagram
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Find the dimensions of the spaces $K\calP\cdot 1_x$.
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