# Danica: Three Constructions of Algebraic \(\K\dash\)theory (Thursday, June 16) :::{.remark} History: - Lower algebraic \(\K\dash\)theory for rings $A$, in particular $\K_i$ for $i=0,1,2$ (1957-67) - Higher \(\K\dash\)theory: - For rings $A$: Quillen's plus construction (1971) - For small exact categories $\mce$: Quillen's $Q\dash$construction (1972) - For Waldhausen categories $\cC$: Waldhausen's $\sdot\dash$construction (1978) ::: :::{.remark} For $A$ a ring let $P(A)$ be the monoid $\amod^{\fg, \proj}$ and let $\K_0$ be the group completion of $(\cC, \oplus)$. ::: :::{.example title="?"} For $F$ a field, $P(F) = \NN$ and thus $\K_0(F) = \ZZ$. ::: :::{.theorem title="Bass-Schanuel"} \[ \K_1(A) \cong \GL_\infty(A)/ E(A) \] where $\GL_\infty(A) = \Union_{n\geq 0} \GL_n(A)$ and $E(A)$ is a certain group of elementary matrices. ::: :::{.remark} How would they have known that $\K_0$ and $\K_1$ were related? This naturally extends a long exact sequence. ::: ## The Plus Construction :::{.remark} Idea: want a space $\K(A)$ where $\pi_1\K(A) = \GL(A)/E(A)$. Note that $\pi_1 \BGL(A) \cong \GL(A)$, so attach cells to $\BGL(A)$ to get the correct fundamental group. In particular, we want \[ H_*(\K(A); M) \cong H_*(\BGL(A); M) .\] Quillen sets $\K(A) \da \BGL(A)^+$, which is $\BGL(A)$ with the correct cells attached. Remarkably, this can be done! Quillen then defines \[ \K_n(A) \da \pi_n \BGL(A)^+ .\] ::: ## The $Q\dash$construction :::{.definition title="Exact categories"} Recall that $\cC$ is small iff $\Ob(\cC), \Mor(\cC)$ are sets. AN **exact category** is a pair $(\mce, \mcs)$ where $\mce$ is an additive category and $\mcs$ is a family of sequence sin $\mce$ of the form \[ 0 \to M'\to M\to M''\to 0 \qquad \star .\] where $\mce$ is a full subcategory of an abelian category. Here additive means admitting a zero, products, and morphisms sets are abelian groups. Full means $\mce(X, Y) = \mca(X, Y)$ when $X, Y\in \mce$ and $\mce\leq \mca$. Being abelian means admitting kernels/cokernels, epis are cokernels, monos are kernels. We'll write this as \[ M' \injects M\surjects M'' .\] where the first arrow is an *admissible monomorphism* and the second is an *admissible epimorphism*. ::: :::{.example title="?"} The ur-example is $P(\cat A)$ for a category $(\cat A, \oplus)$. ::: :::{.remark} Let $\mce$ be a small exact category, then \[ \K_0(\mce) \da \freemod{[M] \st M\in \Ob(\mce)}\modulo \gens{ [M] = [M'] + [M''] \, \forall \star\in \mce} .\] Define a category $Q\mce$ where $\Ob(Q\mce) \da \Ob(\mce)$ and for morphisms, use that $(X\to Y)\in \mce(X, Y)$ yields a class of diagrams: \begin{tikzcd} & M \\ \\ X && Y & {} \arrow[hook, from=1-2, to=3-3] \arrow[two heads, from=1-2, to=3-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMSwwLCJNIl0sWzAsMiwiWCJdLFszLDJdLFsyLDIsIlkiXSxbMCwzLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFswLDEsIiIsMix7InN0eWxlIjp7ImhlYWQiOnsibmFtZSI6ImVwaSJ9fX1dXQ==) The $Q\dash$construction is then \[ \K_n(\mce) = \pi_{n+1} \B Q\mce .\] ::: :::{.remark} When do these coincide? Temporarily distinguish the plus and $Q$ constructions, we then want \[ \K_n^+(A) \iso \K_n^Q(P(A)) .\] How this is proved: - Quillen shows that if $S\da \iso(P(A))$, then $\Loop\B QP(A) \homotopic \B(S\inv S)$ for some construction $S\inv$. - Show $\B(S\inv S) \homotopic \ZZ\times \BGL(A)^+$ - Show $\pi_n \Loop \B QP(A) \cong \pi_n \B(S\inv S)$. - The LHS is $\pi_{n+1} \B QP(A)$ (the $Q\dash$construction) and the RHS is $\pi_n\BGL(A)^+$ for $n>0$ (the plus construction). ::: ## The $\sdot$ construction :::{.remark} Recall that Waldhausen categories are those categories $\cC$ with cofibrations $\surjects$ and weak equivalences. There are subcategories $\cof(\cC)$ and $w\cC$. Important examples include $\cof(\Top)$. Note that we can extract cokernels $B/A \da \coker(A\injects B)$ for cofibrations $A\injects B$. The goal is to now build a simplicial Waldhausen category $\sdot\cC = \simplicial{S_0\cC}{S_1 \cC}{\cdots}$. Take $S_0\cC = 0, S_1 \cC = \cC$,and use staircase diagrams to construct higher objects. Then define \[ \K_n(\cC) \da \pi_{n+1} \realize{w\sdot \cC} .\] ::: :::{.remark} How does this compare to $Q$? A small exact category can be regarded as a small Waldhausen category in the following way: - Cofibrations are admissible monomorphisms, - Weak equivalences are isomorphisms, This allows us to define $i\sdot \mce$, the category of isomorphisms, and it turns out that $i\sdot \mce \iso w\sdot \mce$ and a simplicial subdivision argument will show that the two constructions coincide. An example for the 2-simplex $\Delta^2$ and its subdivision $(\Delta^2)^{e}$: ![](figures/2022-06-16_10-48-29.png) It turns out that $\realize{\complex X} \homotopic \realize{\complex{X}^{e}}$. The doubling map is realized s \[ (\wait)^e: \complex{X} \to \complex{X}^e \\ [n] &\mapsto [2n+1] \\ (0<1<\cdots [Link to Diagram](https://q.uiver.app/?q=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) In general, look for the squares in the staircase: \begin{tikzcd} &&&& {A_{55}} &&&& \textcolor{rgb,255:red,92;green,92;blue,214}{A_3} \\ &&& {A_{34}} & {A_{35}} \\ && \textcolor{rgb,255:red,92;green,92;blue,214}{A_{23}} & {A_{24}} & \textcolor{rgb,255:red,92;green,92;blue,214}{A_{25}} & \leadsto && {A_{13}} && {A_4} \\ & {A_{12}} & {A_{13}} & {A_{14}} & {A_{15}} \\ {A_1} & {A_2} & \textcolor{rgb,255:red,92;green,92;blue,214}{A_3} & {A_4} & \textcolor{rgb,255:red,92;green,92;blue,214}{A_5} && \textcolor{rgb,255:red,92;green,92;blue,214}{A_{23}} && {A_{14}} && \textcolor{rgb,255:red,92;green,92;blue,214}{A_5} \arrow[hook, from=5-1, to=5-2] \arrow[hook, from=5-2, to=5-3] \arrow[hook, from=4-2, to=4-3] \arrow[two heads, from=5-2, to=4-2] \arrow[two heads, from=5-3, to=4-3] \arrow[two heads, from=4-3, to=3-3] \arrow[two heads, from=3-8, to=5-7] \arrow[hook, from=3-8, to=5-9] \arrow[two heads, from=3-10, to=5-9] \arrow[hook, from=3-10, to=5-11] \arrow[two heads, from=1-9, to=3-8] \arrow[hook, from=1-9, to=3-10] \arrow[hook, from=5-3, to=5-4] \arrow[hook, from=5-4, to=5-5] \arrow[hook, from=4-3, to=4-4] \arrow[hook, from=4-4, to=4-5] \arrow[hook, from=3-3, to=3-4] \arrow[hook, from=3-4, to=3-5] \arrow[hook, from=2-4, to=2-5] \arrow[two heads, from=5-4, to=4-4] \arrow[two heads, from=5-5, to=4-5] \arrow[two heads, from=4-4, to=3-4] \arrow[two heads, from=4-5, to=3-5] \arrow[two heads, from=3-4, to=2-4] \arrow[two heads, from=3-5, to=2-5] \arrow[two heads, from=2-5, to=1-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) Then $s_\cdot \mce \to Q_\cdot \mce$ extends $i S_\cdot^e \mce \to iQ_\cdot\mce$, inducing a homotopy equivalence on realizations. This yields \[ \pi_{n=1}\realize{iS_\cdot \mce} \iso \pi_{n+1} \B Q\mce .\] :::