# Brandon: Assemblers (Tuesday, June 14)

## Defining Assemblers

:::{.remark}
Toward \(\K\dash\)theory: consider finite decompositions of the form $B = A + C$.
Some places this comes up:

| Category | Decompositions |
|---------|------|
| $\rmod$ | SESs |
| $\Finset$ | $A\injects B \injectsfrom C \da B\sm A$ |
| $\Var\slice k$ | Similar to $\Finset$ |
| Polytopes | Same as $\Finset$ |

We'll be defining a formalism describing how to take \(\K\dash\)theory of such things: assemblers.
Assume $\cC$ has pullbacks -- this is not strictly necessary, but makes things easier.
:::

:::{.definition title="Sieves"}
A **sieve** on $C\in \cC$ is a full subcategory of $\cC\slice{C}$ closed under precomposition.
:::

:::{.remark}
A collection of morphisms in a sieve $\ts{C_i\to C}$ generate a sieve on $C$.
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:::{.definition title="?"}
For $f: B\to C$ and a sieve $S$ on $C$, the sieve $f^* S$ is the collection of pullbacks $\ts{f^* C_i}$ for all covers ${C_i\to C}\in S$.
:::

:::{.definition title="Grothendieck topologies"}
A **Grothendieck topology** on $\cC$ is a collection of sieves $J(C)$ for $C\in \cC$, which we'll refer to as *covering sieves*.
This is required to satisfy the following: 

- For all $S\in J(C)$ and all $f: B\to C$, $f^* S\in J(B)$.
- For all $S\in J(C)$ and $T$ a sieve on $C$, if $f^* T\in J(C)$ for all $C_i \mapsvia{f} C$ in $S$, then $T\in J(C)$.
- The subcategory of $\cat{C}\slice{C}$ generated by $\id_C$ is in $J(C)$.
:::

:::{.definition title="Assemblers"}
A **(closed) assembler** is a category $\cC$ with a Grothendieck topology such that 

- $\cC$ has pullbacks (the "closed" condition),
- $\cC$ has an initial object $\emptyset$,
- The empty sieve covers $\emptyset$,
- All morphisms in $\cC$ are monic.
:::

:::{.example title="?"}
Let $\mcg_n$ be the assembler whose objects are $n\dash$dimensional closed polytopes in $\RR^\infty$ and morphisms are "inclusions-after-isometries":

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![](figures/2022-06-14_11-56-14.png)

Take covers to be $\ts{A_i\to A}_{i\in I}$ where $A = \union_i A_i$.
:::

:::{.definition title="Disjoint covers"}
A cover is **disjoint** if it's generated by $\ts{A_i \to A}_{i\in I}$ where all pullbacks satisfy $\lim(A_i \to A \from A_j) = \emptyset$.
:::

:::{.example title="?"}
This captures the notion of intersections being lower-dimensional.
Let $\OO$ be the assembler of all polytopes (not necessarily closed).
:::

## \(\K\dash\)theory 

:::{.remark}
We'll want a spectrum $K(\cC)$ where we'll define $\K_i(\cC) \da \pi_i \K(\cC)$.
:::

:::{.definition title="Formal sums"}
A **formal sum** $X$ of elements in $\cC$ is a finite set $I$ and for all $i\in I$ an object $C_i \in \cat{C}$.
We'll write this as the pair $(I, (A_i))$.
A *morphism* of formal sums is an assignment $(I, (A_i)) \to (J, (A_j))$ where $I \injectsvia{f} J$ is an inclusion of sets and $A_i \cong B_{f(i)}$ for all $i\in I$.
An equivalence (which we'll denote a *weak equivalence*) is a surjection $I\surjects J$ and for all $j\in J$ a disjoint cover $\ts{A_i\to B_j}_{i\in f\inv(j)}$ 
Note that if $J=\emptyset$, this is a disjoint cover of $B$.
:::

:::{.definition title="Scissors congruence"}
Two objects $A, B\in \cC$ are **scissors congruent** iff there exists a set $I$ and weak equivalences
\[
(A, (A_i)) \iso (\emptyset, A) \\
(B, (B_i)) \iso (\emptyset, B) 
.\]
In particular, this induces isomorphisms $A_i \cong B_i$.
:::

:::{.definition title="K theory"}
For $\cat{C}$ a closed assembler,
\[
\K(\cC) = \Loop \realize{ wS_\cdot (\cC)}
\]
where $wS_\cdot$ is a simplicial category and this is the realization of the diagonal bisimplicial set $[n] \mapsto \nerve{wS_\cdot(\cC)}$.
:::

:::{.remark}
On the Waldhausen construction: $wS_n(\cC)$ is the category whose objects are sequences of formal sums $X_i$:

\begin{tikzcd}
	& {X_2} &&& {X_{n-1}} \\
	{X_1} && {X_3} & \cdots && {X_n}
	\arrow["\sim"', from=1-2, to=2-1]
	\arrow["\sim"', from=1-5, to=2-4]
	\arrow[tail, from=1-5, to=2-6]
	\arrow[tail, from=1-2, to=2-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNixbMCwxLCJYXzEiXSxbMSwwLCJYXzIiXSxbMiwxLCJYXzMiXSxbNSwxLCJYX24iXSxbNCwwLCJYX3tuLTF9Il0sWzMsMSwiXFxjZG90cyJdLFsxLDAsIlxcc2ltIiwyXSxbNCw1LCJcXHNpbSIsMl0sWzQsMywiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoibW9ubyJ9fX1dLFsxLDIsIiIsMix7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Im1vbm8ifX19XV0=)

Morphisms are commuting ladders with levelwise weak equivalences where squares between weak equivalences are pullbacks and the remaining squares commute.
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:::{.remark}
Waldhausen showed that this is gives an infinite loop space, and thus there is a naturally associated spectrum.
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:::{.proposition title="?"}
\[
\K_0(\cC) = \gens{[A] \in \cC \st [A] = \sum_j [A_j] \text{ if } \ts{A_i \to A}_{i\in I} \text{ is a cover}}
.\]
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:::{.example title="?"}
If $\cat{C} = \emptyset \to \pt$ and $\pt\to \pt$ is the only cover, $\K(\cat C) = \SS$.
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:::{.example title="?"}
$\K_0(\mcg_n)$ is the scissors congruence group of $n\dash$dimensional polytopes, and $\K(\OO)$ is the group for *all* polytopes.
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:::{.remark}
For suitably nice assemblers with $\cat D \injects \cat C$,
there is a cofiber sequence 
\[
\K(\cat D)\to \K(\cat C) \to \K(\cat C \sm \cat D)
.\]
Since $\OO$ has a dimension filtration $\OO_0 \injects \OO_1 \injects \cdots \injects \OO$ where $\OO_n$ are the polytopes of dimension at most $n$, and there is an induced cofiber sequence 
\[
\K(\OO_{n-1}) \to \K(\OO_n) \to \K(\mcg_n)
.\]

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