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\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

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