# Eunice: SW Categories (Wednesday, June 15) ## Recollections :::{.remark} Let $k$ be a field and $\Var\slice k$ be the category of finite type separated schemes over $\spec k$, and recall \[ \K_0(\Var\slice k) = \freezmod{\ts{[X] \st X\in \Var\slice k}}/\sim \] where the relations are given by $[Y] = [X] + [Y\sm X]$ for $X\injects Y$ a closed immersion. Inna constructs a spectrum for this using assemblers, but an issue is that it is hard to map out of this spectrum. Today we'll consider Jonathan's construction: a different construction of the spectrum, and an adding theorem which provides a delooping and plays a role in exhibiting an $\EE_\infty$ structure. ::: :::{.slogan} \(\K\dash\)theory is the universal machine that splits cofiber sequences. ::: :::{.remark} For $\Var\slice k$, what are the cofiber sequences? Cofibrations will be closed immersions, and cofiber sequences will be pushouts along the terminal object. We want $X\injects Y\from Y\sm X$, but the Waldhausen construction doesn't necessarily go through. The idea is to define *subtraction sequences* $\injects \bullet \ofrom$. ::: ## Waldhausen $S_\bullet\dash$construction :::{.remark} Input: a Waldhausen category $\cC$ with a zero object, cofibrations $\injects$, and weak equivalences $\iso$ such that $\pt \injects X$ is a cofibration and the pushouts of the following two rows are weak equivalences: > Todo: missing diagram. The output is $\K(\cC)$, the algebraic \(\K\dash\)theory spectrum. First construct a space whose $\pi_1$ is $\K_0$, then take loops to shift. Construct a complex: - 0-simplices: $\pt$ - 1-simplices: for all $X$, $\pt \mapsvia{X} \pt$ - 2-simplices: for each $X\injects Y \surjects Y/X$, a triangle \begin{tikzcd} & \pt \\ \\ \pt && \pt \arrow["Y"', from=3-1, to=3-3] \arrow["X"', from=3-3, to=1-2] \arrow["{Y\sm X}", from=3-1, to=1-2] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMCwyLCJcXHB0Il0sWzIsMiwiXFxwdCJdLFsxLDAsIlxccHQiXSxbMCwxLCJZIiwyXSxbMSwyLCJYIiwyXSxbMCwyLCJZXFxzbSBYIl1d) - If we have \begin{tikzcd} X && Y && Z \\ \\ && {Y\sm X} && {Z\sm X} \\ \\ &&&& {Z\sm Y} \arrow[hook, from=1-1, to=1-3] \arrow[hook, from=1-3, to=1-5] \arrow[hook, from=3-3, to=3-5] \arrow[two heads, from=1-3, to=3-3] \arrow[two heads, from=1-5, to=3-5] \arrow[two heads, from=3-5, to=5-5] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-5, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) - Since \[ Z = X + Z\sm X &= X + (Y\sm X + Z\sm Y) \\ Y + Z\sm Y &= (X + Y\sm X) + Z\sm Y ,\] we'll force these to be homotopic with a 3-simplex: \begin{tikzcd} && \bullet \\ \\ \bullet &&&& \bullet \\ \\ && \bullet \arrow["Z", from=3-1, to=5-3] \arrow["Y", from=5-3, to=3-5] \arrow["{?}"{pos=0.3}, from=3-1, to=3-5] \arrow["{?}"'{pos=0.7}, from=5-3, to=1-3] \arrow["Z\sm Y", from=3-1, to=1-3] \arrow["{Y\sm X}"', from=3-5, to=1-3] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwyLCJcXGJ1bGxldCJdLFsyLDQsIlxcYnVsbGV0Il0sWzQsMiwiXFxidWxsZXQiXSxbMiwwLCJcXGJ1bGxldCJdLFswLDEsIloiXSxbMSwyLCJZIl0sWzAsMiwiPyIsMCx7ImxhYmVsX3Bvc2l0aW9uIjozMH1dLFsxLDMsIj8iLDIseyJsYWJlbF9wb3NpdGlvbiI6NzB9XSxbMCwzLCJaXFxzbVkiXSxbMiwzLCJZXFxzbSBYIiwyXV0=) To formalize this, define an arrow category $\Ar[n]$ whose objects are pairs $(i, j)$ with $0\leq i\leq j \leq n$ with morphisms $(i, j) \to (i', j')$ iff $i \leq i', j\leq i'$. Let $S_n\cC \leq \Fun(\Ar[n], \cC)$ be the full subcategory where consecutive horizontal maps are cofibration sequences and the squares form pushouts: \begin{tikzcd} \pt & \bullet & \bullet & \bullet & \bullet \\ & \pt & \bullet & \bullet & \bullet \\ && \pt & \bullet & \bullet \\ &&& \pt & \bullet \\ &&&& \pt \arrow[hook, from=1-1, to=1-2] \arrow[hook, from=1-2, to=1-3] \arrow[hook, from=1-3, to=1-4] \arrow[hook, from=1-4, to=1-5] \arrow[hook, from=2-2, to=2-3] \arrow[hook, from=2-3, to=2-4] \arrow[hook, from=2-4, to=2-5] \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[from=1-2, to=2-2] \arrow[from=1-3, to=2-3] \arrow[from=1-4, to=2-4] \arrow[from=1-5, to=2-5] \arrow[from=2-3, to=3-3] \arrow[from=2-4, to=3-4] \arrow[from=2-5, to=3-5] \arrow[from=3-4, to=4-4] \arrow[from=3-5, to=4-5] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-3, to=1-2] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-4, to=1-3] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=2-5, to=1-4] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-4, to=2-3] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-5, to=2-4] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=4-5, to=3-4] \arrow[from=4-5, to=5-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) Define $\sdot\Cc$ to be the simplicial category $[n] \to S_n\cC$ whose faces and degeneracies are obtained by inserting/deleting identities. Define $wS_n\cC$ the simplicial subcategory generated by weak equivalences, and define the bisimplicial set $\wdot \sdot \cC = \ts{N w\sdot \cC}$. The algebraic \(\K\dash\)theory spectrum is defined as \[ \K(\cC) \da \Loop \abs{\wdot\sdot\cC} .\] The additivity theorem implies $\K(\cC)$ is an infinite loop space and thus a spectrum. ::: :::{.remark} The $\tilde\sdot$ construction: the input will be a category with *subtraction* (i.e. subtraction sequences and pullbacks) and outputs a spectrum $\K(\cC)$. Some categories that this applies to: subtraction and SW categories, where a product structure on the category yields an $\EE_\infty$ structure on the spectrum. ::: :::{.remark} Subtracion categories: - Initial objects $\emptyset$ - Cofibrations $\injects$ - Fibrations $\oto$ - Subtraction sequences satisfying some nice properties, e.g. $A\to A\disjoint B \from B$. Denote this $\injects\oto$. ::: :::{.example title="?"} $\Sch\slice X, \Var\slice X$ where cofibrations are closed immersions, fibrations are closed immersions. ::: :::{.remark} A subtractive category is a category with subtractions such that the following squares exists and ar cartesian: \begin{tikzcd} \bullet && \bullet \\ \\ \bullet && \bullet \arrow[hook, from=1-1, to=1-3] \arrow[hook, from=1-1, to=3-1] \arrow[hook, from=3-1, to=3-3] \arrow[hook, from=1-3, to=3-3] \arrow["\lrcorner"{anchor=center, pos=0.125, rotate=180}, draw=none, from=3-3, to=1-1] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=WzAsNCxbMCwwLCJcXGJ1bGxldCJdLFsyLDAsIlxcYnVsbGV0Il0sWzAsMiwiXFxidWxsZXQiXSxbMiwyLCJcXGJ1bGxldCJdLFswLDEsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzAsMiwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XSxbMiwzLCIiLDIseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsxLDMsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzMsMCwiIiwxLHsic3R5bGUiOnsibmFtZSI6ImNvcm5lciJ9fV1d) Examples include $\Sch\slice X, \Var\slice X$, but not $\smooth\Sch\slice X$ since pushouts induce singularties. Define a category $F_1^+\cC$ whose objects are subtraction sequences and morphisms are cartesian squares. Note that $F_1^+$ is subtractive, and functors $X\injects Y\injects Z \to X,Y,Z$ respectively are exact from $F_1^+\cC \to \cC$. ::: :::{.remark} SW-categories: subtractive Waldhausen categories + conditions (isomorphism are weak equivalences, compatibility with subtraction sequences). The $\tilde\sdot$ construction: define $\tilde \Ar[n]$ as the category \begin{tikzcd} \emptyset & \bullet & \bullet & \bullet & \bullet \\ & \emptyset & \bullet & \bullet & \bullet \\ && \emptyset & \bullet & \bullet \\ &&& \emptyset & \bullet \\ &&&& \emptyset \arrow[hook, from=1-1, to=1-2] \arrow[hook, from=1-2, to=1-3] \arrow[hook, from=1-3, to=1-4] \arrow[hook, from=1-4, to=1-5] \arrow[hook, from=2-2, to=2-3] \arrow[hook, from=2-3, to=2-4] \arrow[hook, from=2-4, to=2-5] \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["o", from=2-2, to=1-2] \arrow["o", from=2-3, to=1-3] \arrow["o", from=2-4, to=1-4] \arrow["o", from=2-5, to=1-5] \arrow["o", from=3-3, to=2-3] \arrow["o", from=3-4, to=2-4] \arrow["o", from=3-5, to=2-5] \arrow["o", from=4-4, to=3-4] \arrow["o", from=4-5, to=3-5] \arrow["o", from=5-5, to=4-5] \end{tikzcd} > [Link to Diagram](https://q.uiver.app/?q=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) The morphisms are morphisms in $\Fun(\tilde\Ar[n], \cC)$ such that the top row squares are cartesian. $\tilde \sdot\cC$ is a subtractive category with levelwise fibrations/cofibrations/subtraction sequences, and $[n] \to \tilde S_n\cC \in \mathsf{s}\Cat$ is a simplicial category. > Todo; fix. Let \[ \K(\cC)(\ell) = \realize{\nerve{w \tilde \sdot \cdots \tilde\sdot \cC }} \] where the $\tilde\sdot$ construction is applied $\ell$ times. ::: :::{.remark} Explicitly $\tilde S^{(\ell)}_{\bullet} \cC \leq \Fun(\tilde\Ar[n_1] \times \cdots \tilde\Ar[n_k], \cC)$ such that each slide is valid and restrictions to top rows for each $\tilde \Ar[n_i]$ are Cartesian. We get sequence of spaces $\K(\cC)(k)$ for $k\in \ZZ_{\geq 0}$ which assemble to a symmetric sequence, which will in fact assemble to a spectrum. ::: :::{.theorem title="?"} The map \[ (s, q): F_1^+\cC \to \cC \times \cC \\ X\injects Y\injectsfrom Z \mapsto (S, Z) \] induces a homotopy equivalence of simplicial sets $\tilde\sdot (F_1\cC) \homotopic ?$. ::: :::{.remark} We want $\Loop\K(\cC)(n) \to \K(\cC)(n+1)$. For $\complex{X}$ a simplicial object, there is path fibration $P\complex{X} \to X_0$ by taking the constant path. > Todo: missed diagram. Applying this to $\complex{X} = w\tilde\sdot\cC \in \mathsf{s}\Cat$. We get a sequence whose composite is constant and whose middle term is contractible: \[ w\tilde\sdot\cC \to Pw\tilde\sdot\cC \to \wdot \cC .\] This yields a map $\realize{\nerve{w\cC}} \to \Loop \realize{\nerve{w\cC}}$, which is not a weak equivalence but becomes such after another application of $\sdot$. \todo[inline]{DZG: Pretty sure I wrote this down incorrectly.} ::: :::{.remark} The multiplicative structure is induced by cartesian product of varieties. ::: :::{.remark} We'll try to lift classical motivic measures to maps $\K(\Var\slice k) \to R\in \Sp$. There is a notion of $W\dash$exact functors for functors $\cat{C} \to \cat{W}$. :::