# Friday, February 12 :::{.remark} Last time: right-exact functors have left-derived functors where a SES induces a LES. The functors are *natural* with respect to the connecting morphisms in the sense that certain squares commute. Weibel refers to \( \ts{ L_i F }_{i\geq 0} \) as a **homological $\delta\dash$functor**, i.e. anything that takes SESs to LESs which are natural with respect to connecting morphism. ::: ## Aside: Natural Transformations :::{.definition title="Natural Transformation"} Given functors \( F, G, \cat{C} \to \cat{D} \), a **natural transformation** $\eta: F \implies G$ is the following data: - For all $C\in \cat{C}$ there is a map $F(C) \mapsvia{\eta_C} G(C) \in \Mor(\cat{D})$, sometimes referred to as $\eta(C)$. - If \( C \mapsvia{f} C' \in \Mor(\cat{C}) \), there is a diagram \begin{tikzcd} FC && {FC'} \\ \\ GC && {GC'} \arrow["Gf", from=3-1, to=3-3] \arrow["Ff", from=1-1, to=1-3] \arrow["{\eta_C}"{description}, from=1-1, to=3-1] \arrow["{\eta_{C'}}"{description}, from=1-3, to=3-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJGQyJdLFsyLDAsIkZDJyJdLFswLDIsIkdDIl0sWzIsMiwiR0MnIl0sWzIsMywiR2YiXSxbMCwxLCJGZiJdLFswLDIsIlxcZXRhX0MiLDFdLFsxLDMsIlxcZXRhX3tDJ30iLDFdXQ==) - $\eta$ is a **natural isomorphism** if all of the $\eta_C$ are isomorphisms, and we write $F \cong C$. ::: :::{.definition title="Equivalence of Categories"} A functor $F: \cat{C} \to \cat{D}$ is an **equivalence of categories** if and only if there exists a $G: \cat{D} \to \cat{C}$ such that $GF \cong \one_{\cat{C}}$ and $FG \cong \one_{\cat{D}}$. ::: :::{.example title="?"} A category $\cat{C}$ is **small** if $\Ob(\cat{C})$ is a set, then take $\cat{C} \da \Cat$ whose objects are categories and morphisms are functors. Note that in all categories, all collections of morphisms should be sets, and the small condition guarantees it. In this case, natural transformations $\eta: F\to G$ is an additional structure yielding morphisms of morphisms. These are called **2-morphisms**, and in this entire structure is a **2-category**, and our previous notion is referred to as a **1-category**. ::: :::{.theorem title="Left-derived functors of a right-exact functor form a universal $\delta\dash$functor"} Assume \( \cat{A}, \cat{B} \) are abelian and \( F:\cat{A} \to \cat{B} \) is a right-exact additive functor where \( \cat{A} \) has enough projectives. Then the family \( \ts{ L_i F } _{i\geq 0} \) is a *universal $\delta\dash$functor* where $L_0 F \cong F$ is a natural isomorphism. ::: :::{.remark} Here *universal* means that if \( \ts{ T_i } _{i\geq 0} \) is also a $\delta\dash$functor with a natural *transformation* (not necessarily an isomorphism) \( \varphi_0: T_0 \to F \), then there exist unique morphism of \( \delta\dash \)functors \( \ts{ \varphi_i: T_i \to L_i F } _{i\geq 0} \), i.e. a family of natural transformations that commute with the respective \( \delta \) maps coming from both the $T_i$ and the $L_i F$, which extend \( \varphi_0 \). This will be important later on when we try to show Ext and Tor are functors in either slot. ::: :::{.proof title="?"} Assume \( \ts{ T_i } _{i\geq 0} \) and \( \varphi_0 \) are given, and assume inductively that $n>0$ and we've defined \( \varphi_i: T_i \to F \) for $0\leq i< n$ which commute with the \( \delta \) maps. Step 1: given \( A\in \cat{A} \), fix a reference exact sequence: pick a projective mapping onto $A$ and its kernel to obtain \[ 0 \to K \to P \to A \to 0 .\] Applying the functors $T_i$ and $L_i F$ yields \begin{tikzcd} && \textcolor{rgb,255:red,214;green,92;blue,214}{x} && k && 0 \\ && {T_nA} && {T_{n-1}K} && {T_{n-1}P} \\ \\ \\ \textcolor{rgb,255:red,214;green,92;blue,214}{L_n FP = 0} && {L_nFA} && {L_{n-1}FK} && {L_{n-1}FP} \\ && \textcolor{rgb,255:red,214;green,92;blue,214}{\exists ! y \da \phi_{n-1}(x)} && \ell && 0 \arrow["{\phi_{n-1}(K)}", from=2-5, to=5-5] \arrow["{\phi_{n-1}(P)}", from=2-7, to=5-7] \arrow["{\exists \phi_{n-1}(A)}", dashed, from=2-3, to=5-3] \arrow["\delta"{description}, from=2-3, to=2-5] \arrow[from=2-5, to=2-7] \arrow[from=5-1, to=5-3] \arrow["\delta"', hook, from=5-3, to=5-5] \arrow[from=5-5, to=5-7] \arrow[dotted, from=1-3, to=1-5] \arrow[dotted, from=1-5, to=1-7] \arrow[curve={height=18pt}, dotted, from=1-5, to=6-5] \arrow[curve={height=18pt}, dotted, from=1-7, to=6-7] \arrow[color={rgb,255:red,214;green,92;blue,214}, dotted, from=6-3, to=6-5] \arrow[dotted, from=6-5, to=6-7] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTMsWzIsMSwiVF9uQSJdLFs0LDEsIlRfe24tMX1LIl0sWzYsMSwiVF97bi0xfVAiXSxbMiw0LCJMX25GQSJdLFs0LDQsIkxfe24tMX1GSyJdLFs2LDQsIkxfe24tMX1GUCJdLFswLDQsIkxfbiBGUCA9IDAiLFszMDAsNjAsNjAsMV1dLFsyLDAsIngiLFszMDAsNjAsNjAsMV1dLFs0LDAsImsiXSxbNCw1LCJcXGVsbCJdLFs2LDAsIjAiXSxbNiw1LCIwIl0sWzIsNSwiXFxleGlzdHMgISB5IFxcZGEgXFxwaGlfe24tMX0oeCkiLFszMDAsNjAsNjAsMV1dLFsxLDQsIlxccGhpX3tuLTF9KEspIl0sWzIsNSwiXFxwaGlfe24tMX0oUCkiXSxbMCwzLCJcXGV4aXN0cyBcXHBoaV97bi0xfShBKSIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dLFswLDEsIlxcZGVsdGEiLDFdLFsxLDJdLFs2LDNdLFszLDQsIlxcZGVsdGEiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFs0LDVdLFs3LDgsIiIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRvdHRlZCJ9fX1dLFs4LDEwLCIiLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkb3R0ZWQifX19XSxbOCw5LCIiLDAseyJjdXJ2ZSI6Mywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzEwLDExLCIiLDAseyJjdXJ2ZSI6Mywic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZG90dGVkIn19fV0sWzEyLDksIiIsMix7ImNvbG91ciI6WzMwMCw2MCw2MF0sInN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRvdHRlZCJ9fX1dLFs5LDExLCIiLDIseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkb3R0ZWQifX19XV0=) So define \( \varphi_{n}(A)(x) \da y \), which makes the LHS square commute by construction. Note that $L_n FP$ vanishes (as do all its higher derived functors) since $P$ is projective. :::{.warnings} The map \( \varphi_n(A) \) could depend on the choice of $P$! ::: We now want to show that \( \varphi_n \) is a natural transformation. Supposing $f:A' \to A$, we need to show \( \varphi_n \) commutes with $f$. \begin{tikzcd} 0 & {K'} & {P'} & A & 0 \\ \\ 0 & K & P & A & 0 \arrow["f", from=1-4, to=3-4] \arrow["{\exists g}", dashed, from=1-3, to=3-3] \arrow[from=3-1, to=3-2] \arrow[from=3-2, to=3-3] \arrow[from=3-3, to=3-4] \arrow[from=3-4, to=3-5] \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \arrow["{\exists h}", dashed, from=1-2, to=3-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMCwiMCJdLFsxLDAsIksnIl0sWzIsMCwiUCciXSxbMywwLCJBIl0sWzMsMiwiQSJdLFsyLDIsIlAiXSxbMSwyLCJLIl0sWzAsMiwiMCJdLFs0LDIsIjAiXSxbNCwwLCIwIl0sWzMsNCwiZiJdLFsyLDUsIlxcZXhpc3RzIGciLDAseyJzdHlsZSI6eyJib2R5Ijp7Im5hbWUiOiJkYXNoZWQifX19XSxbNyw2XSxbNiw1XSxbNSw0XSxbNCw4XSxbMCwxXSxbMSwyXSxbMiwzXSxbMyw5XSxbMSw2LCJcXGV4aXN0cyBoIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV1d) Since $P'$ is projective, we can lift $f$ to $P'\to P$, and then define $h$ to be the restriction of $g$ to $K' \to K$. \begin{tikzcd} {T_n A'} &&&& {T_nA} \\ & {T_{n-1}K'} && {T_{n-1}K} \\ \\ & {L_{n-1}FK'} && {L_{n-1}FK} \\ {L_n FA'} &&&& {L_nF(A)} \arrow["{T_nf}", from=1-1, to=1-5] \arrow["{L_nFf}", from=5-1, to=5-5] \arrow["{\phi_n(A)}", from=1-5, to=5-5] \arrow["{\phi_n(A')}", from=1-1, to=5-1] \arrow["{T_{n-1}h}", from=2-2, to=2-4] \arrow["{\delta'}"{description}, from=1-1, to=2-2] \arrow["\delta"{description}, from=1-5, to=2-4] \arrow["{L_{n-1}Fh}", from=4-2, to=4-4] \arrow["{\phi_{n-1}}", from=2-2, to=4-2] \arrow["{\phi_{n-1}}", from=2-4, to=4-4] \arrow["{\delta'}"{description}, from=5-1, to=4-2] \arrow["\delta"{description}, from=5-5, to=4-4] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsOCxbMCwwLCJUX24gQSciXSxbNCwwLCJUX25BIl0sWzAsNCwiTF9uIEZBJyJdLFs0LDQsIkxfbkYoQSkiXSxbMSwxLCJUX3tuLTF9SyciXSxbMywxLCJUX3tuLTF9SyJdLFsxLDMsIkxfe24tMX1GSyciXSxbMywzLCJMX3tuLTF9RksiXSxbMCwxLCJUX25mIl0sWzIsMywiTF9uRmYiXSxbMSwzLCJcXHBoaV9uKEEpIl0sWzAsMiwiXFxwaGlfbihBJykiXSxbNCw1LCJUX3tuLTF9aCJdLFswLDQsIlxcZGVsdGEnIiwxXSxbMSw1LCJcXGRlbHRhIiwxXSxbNiw3LCJMX3tuLTF9RmgiXSxbNCw2LCJcXHBoaV97bi0xfSJdLFs1LDcsIlxccGhpX3tuLTF9Il0sWzIsNiwiXFxkZWx0YSciLDFdLFszLDcsIlxcZGVsdGEiLDFdXQ==) Note that all of the quadrilaterals here commute. The middle top and bottom come from naturality of $T_*, L_*F$ with respect to $\delta$, the RHS/LHS due to the construction of the \( \varphi_n \), and $\phi_{n-1}$ is natural by the inductive hypothesis. Now in order to traverse $T_nA' \to T_n A \to L_n F (A)$, we can pass the path through one commuting square at a time to make it equal to $T_nA' \to L_n FA' \to L_n FA$, so the outer square commutes. We have \[ \delta \varphi_n(A) T_n F = \delta L_n Ff \varphi_n(A') ,\] and since \( \delta \) is monic (using the previous vanishing due to projectivity), so we can cancel on the left and this yields the definition of naturality. :::{.corollary title="?"} The definition of \( \varphi_n(A) \) does not depend on the choice of $P$. Taking $A' = A$ in the previous argument with $f = \one$, suppose $P'\neq P$. Then $T_n f = \one = L_n Ff$ and setting \( \varphi_n'(A)\) to be the map coming from $P'$, we get \( \varphi_n'(A) = \varphi_n(A) \) using the following diagram: \begin{tikzcd} 0 & {K'} & {P'} & A & 0 \\ \\ 0 & K & P & A & 0 \arrow["\one", from=1-4, to=3-4] \arrow["{\exists g}", dashed, from=1-3, to=3-3] \arrow[from=3-1, to=3-2] \arrow[from=3-2, to=3-3] \arrow[from=3-3, to=3-4] \arrow[from=3-4, to=3-5] \arrow[from=1-1, to=1-2] \arrow[from=1-2, to=1-3] \arrow[from=1-3, to=1-4] \arrow[from=1-4, to=1-5] \arrow["{\exists h}", dashed, from=1-2, to=3-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMTAsWzAsMCwiMCJdLFsxLDAsIksnIl0sWzIsMCwiUCciXSxbMywwLCJBIl0sWzMsMiwiQSJdLFsyLDIsIlAiXSxbMSwyLCJLIl0sWzAsMiwiMCJdLFs0LDIsIjAiXSxbNCwwLCIwIl0sWzMsNCwiXFxvbmUiXSxbMiw1LCJcXGV4aXN0cyBnIiwwLHsic3R5bGUiOnsiYm9keSI6eyJuYW1lIjoiZGFzaGVkIn19fV0sWzcsNl0sWzYsNV0sWzUsNF0sWzQsOF0sWzAsMV0sWzEsMl0sWzIsM10sWzMsOV0sWzEsNiwiXFxleGlzdHMgaCIsMCx7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==) ::: So the \( \varphi_n(A) \) are uniquely defined. We now want to show that \( \varphi_n \) commutes with the \( \delta_n \) coming from an *arbitrary* SES instead of a fixed reference SES. \begin{tikzcd} {T_n A} &&&&&&& {T_{n-1}C} \\ &&& {T_* \text{ a } \delta \text{ functor}} \\ {T_nA} &&&& {T_{n-1}K} \\ &&& {\text{reference}} &&& {\phi_{n-1} \text{natural}} \\ \\ {L_nFA} &&&& {L_{n-1}FA} \\ &&& {L_*F \text{ a } \delta \text{ functor}} \\ {L_n FA} &&&&&&& {L_{n-1}FC} \arrow["{\phi_n}", from=3-1, to=6-1] \arrow["{\delta_{(2)}}", from=3-1, to=3-5] \arrow["{\delta_{(1)}}", from=6-1, to=6-5] \arrow["{\phi_{n-1}}"', from=3-5, to=6-5] \arrow["{=}", from=1-1, to=3-1] \arrow["{=}", from=8-1, to=6-1] \arrow["{\delta_{(1)}}"', from=8-1, to=8-8] \arrow["{\delta_{(2)}}"', from=1-1, to=1-8] \arrow["{\phi_{n-1}}"{description}, from=1-8, to=8-8] \arrow["{T_{n-1}h}"{description}, from=3-5, to=1-8] \arrow["{L_{n-1}Fh}"{description}, from=6-5, to=8-8] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) This diagram commutes for the reasons indicated, and commutativity of the outside square implies that \( \varphi_n \) commutes with the \( \delta \) coming from any SES. > See section 2.4 in Weibel. :::