generated from nerf/texTemplate
51 lines
2.7 KiB
TeX
51 lines
2.7 KiB
TeX
\documentclass[../Main.tex]{subfiles}
|
||
\begin{document}
|
||
\section{Leibniz Construction}
|
||
We will see use a well know construction in homotopy theory, to
|
||
elegantly construct a lot of interesting objects, the Leibniz construction.
|
||
This section will recall the definition establish some facts and give some examples.
|
||
There are multiple
|
||
|
||
We start be giving the definition
|
||
\begin{definition}[Leibniz Construction]
|
||
Let \(\A\), \(\B\) and \(\C\) be categories and \(\C\) has finite pushouts. Let \(⊗ : \A × \B → \C \) be a bifunctor.
|
||
Then we define \(\hat{⊗} : \arr{\A} × \arr{\B} → \arr{\C}\) to defined as follows. Given \(f : A → A' \) in \(\A\) and
|
||
\( g : B → B' \) in \(\B\), then \( f \hat{⊗} g \) is defined as in the following diagram.\\
|
||
\begin{equation*}
|
||
\begin{tikzcd}
|
||
A ⊗ B \arrow[r, "f ⊗ \id"] \arrow[d, "\id ⊗ g"] \arrow[dr, phantom, "\ulcorner", very near end] & A' ⊗ \mathrlap{B} \phantom{A'} \arrow[ddr, bend left, "\id ⊗ g"] \arrow[d] & \\
|
||
\phantom{B'} \mathllap{A} ⊗ B' \arrow[drr, bend right, "f ⊗ \id"] \arrow[r] & A ⊗ B' ∐\limits_{A ⊗ B} A' ⊗ B \arrow[dr, dashed ,"f \hat{⊗} g"] & \\
|
||
& & A' ⊗ B'
|
||
\end{tikzcd}
|
||
\end{equation*}
|
||
\end{definition}
|
||
When the bifunctor is the cartesian product functor, this construction is also known as the \emph{pushout-product}.
|
||
Now we will give some examples, some of them instructive and some that will appear later again.
|
||
|
||
\begin{example}
|
||
Let \(\C\) be your favourite category of spaces, \( × \) the cartesian product and \(f = g\) be the
|
||
inclusion of the point into the start of the interval. Then the pushout-product \(f \hat{×} g \) is the inclusion of
|
||
two adjacents sides of a square, into that square.
|
||
\end{example}
|
||
Or a bit more general
|
||
\begin{example}\label{Leibniz:Ex:OpenBox}
|
||
Let \(\C\) be your favourite category of spaces, \( × \) the cartesian poduct. Let \(f\) be the
|
||
inclusion of one endpoint into the interval. Let \(g : ∂\I^n → \I^n \) be the boundary inclusion into
|
||
the \(n\)-cube. Then \(f \hat{×} \) is the filling of an \(n\) dimensional open box.
|
||
\end{example}
|
||
\begin{example}
|
||
Let \(\C\) be your favourite category of spaces, and \( @ : \C^{\C} × \C → \C \) defined as
|
||
\( F @ x ≔ F(x)\) the functor application functor. Let \(\I ⊗ (−)\) be a functorial cylinder and \(i_0\).
|
||
be one of the inclusions \( i_0 : \Id → \I ⊗ (−) \) and \( ∂X → X \) be the boundery inclusion of \(X\).
|
||
Then \( i_0 \hat{@} (∂X → X) \) is the filling of a cylinder with one base surface of shape \(X\).
|
||
\end{example}
|
||
|
||
As we will come across this example multiple times it will get some special syntax.
|
||
|
||
\begin{definition}
|
||
|
||
\end{definition}
|
||
|
||
% TODO Left adjoint
|
||
|
||
\end{document}
|