Sandbox

Introduction

This is a sandbox document to test typesetting and layout. This website and notes are built using Quarto¹.

¹ Quarto has a website and a Github repository.

Math

\[ \newcommand{\blank}{{-}} \newcommand{\ladj}{\dashv} \newcommand{\dual}{^{\perp}} \newcommand{\parr}{ \require{html} \mathop{\style{display: inline-block; transform: rotate(180deg)}{\&}} } \]

This is a math character \(\alpha\), and this is some inline math \((\blank) \otimes A \ladj A\dual \parr (\blank)\).

This is a prooftree in display math mode². \[ \require{bussproofs} \begin{prooftree} \AxiomC{ $\Gamma, A \vdash B$ } \RightLabel{ $\to$I } \UnaryInfC{ $\Gamma \vdash A \to B$ } \AxiomC{ $\Gamma \vdash A$ } \RightLabel{ $\to$E } \BinaryInfC{ $\Gamma \vdash B$ } \end{prooftree} \]

² This is another prooftree but in the margin. \[ \begin{prooftree} \AxiomC{ $P$ } \AxiomC{ $P \to Q$ } \BinaryInfC { $Q$ } \end{prooftree} \]

Theorems

This is a theorem with proof, labeled and cross-referenced Theorem 1.

Theorem 1 (Yoneda lemma). For a locally small category \(\mathcal{C}\), a functor \(F : \mathcal{C} \to \mathbf{Set}\), and an object \(A \in \mathcal{C}\), there is a natural isomorphism \[ \mathrm{Nat}(\mathrm{Hom}_{\mathcal{C}}(-, A), F) \cong F(A). \]

Proof. Obvious.

Diagrams

This renders diagrams in markdown Figure 1.

This is some imagified LaTeX math.

A TikZ picture Figure 1 (a).

(a)

A commutative diagram Figure 1 (b).

(b)

Figure 1

This is a quiver iframe.