Multiple hierarchical structures can be imposed on the same associative network of nodes; a hierarchical structure amounts to a “narrative” that contextualizes a given subgraph of the network. One example could be the construction of lecture notes; another example could be a homework sheet; a further example could be a book chapter or scientific article. Although these may draw from the same body of definitions, theorems, examples, and exercises, these objects are contextualized within a different narrative, often toward fundamentally different ends.

As a result, any interface for navigating the neighbor-relation in hierarchically organized notes would need to take account of the multiplicity of parent nodes. Most hypertext tools assume that the position of a node in the hierarchy is unique, and therefore have a single “next/previous” navigation interface; we must investigate the design of interfaces that surface all parent/neighbor relations.

Both HTML and \LaTeX support a form of hierarchical organization with “absolute” heading levels, i.e. levels that count upward from a single root. In HTML, there is <h1>, <h2>, <h3>…, and in \LaTeX there is \part, \chapter, \section, \subsection,\subsubsection, …, \paragraph depending on the document class. This is in contrast to a relative model of hierarchy, in which there is a single command to introduce a section heading at the “current” level, and there are other commands to introduce hierarchical nesting.

The absolute sectioning model is completely inadequate for the hierarchical organization of ideas, for the simple reason that it is the context of a node that determines what its level in the hierarchy is, not the node itself. When this is mixed up, it makes re-contextualization an extremely painful and time-consuming process: you must recursively increment or decrement all section levels that occur underneath a given node, as anyone who has used \LaTeX for any significant writing project can attest.

In traditional texts, re-contextualization occurs when you want to move a section from one place in the hierarchy to another; in the more fluid media I am pursuing, there may be many orthogonal hierarchical structures imposed on the network, so re-contextualization ceases to be a refactoring task and is elevated as a basic unit of mathematical activity. In either case, we are drawn to prefer relative hierarchy over absolute hierarchy. See relative hierarchy in existing tools for existing implementations of this idea.

This is similar to the relationship between De Bruijn levels (global levels) and De Bruijn indices (local levels) in type theory: conventional section headings are like De Bruijn indices in that they count from the root node, whereas what we would want are section headings that count from the present node.

Many document markup languages, including \LaTeX and HTML, use sectioning commands that evince an implicit hierarchical structure: for instance, consider the following HTML code:

<h1>Foo</h1>
<h2>Bar</h2>
<h3>Baz</h3>
<h3>Qux</h4>
<h1>Boo</h1>

The above corresponds to the tree [Foo > [Bar > [Baz, Qux]], Boo]. On the other hand, it is also possible to consider a model in which the hierarchy is made explicit through the syntactical tree structure of the markup language. This style is also supported (but not forced) in HTML:

<section>
  <h1>Foo</h1>
  <section>
    <h2>Bar</h2>
    <section>
      <h3>Baz</h3>
    </section>
    <section>
      <h3>Qux</h3>
    </section>
  </section>
</section>

<section>
  <h1>Boo</h1>
</section>

Atomicity in evergreen mathematical notes is enhanced by adhering to the following principles:

1. no free variables: do not rely on one-off objects that are defined incidentally upwards in the hierarchy; turn them into atomic nodes that can be linked;
2. favor explicit dependency: whenever using a terminology or construction that has been defined elsewhere, link it;
3. notation should be decodable: all notations (except the most very basic) should be recalled via a link.

It can be a bit excessive to link every word: but the pertinent links could be added to a “related pages” section.

Mathematical writing tends to involve a variety of notations which (1) can be difficult to typeset by hand, and (2) will likely change over time. The difficulty of hand-typesetting is somewhat less important than the propensity of notation to change over time: when we change notations within a given mathematical work, we must update every occurrence of the notation: but when the representation of the notation is unstructured, it is not in fact possible for a tool (e.g. find-and-replace) to detect every instance that needs to be updated. Therefore, it is mandatory that the representation of mathematical notations be structured.

\LaTeX allows authors to structure their notations very simply using macros, which can be introduced using \def or \newcommand or \NewDocumentCommand. It is trivial to update all occurences of a notation by simply changing the definition of the corresponding macro.

Unfortunately, most tools that purport to support the inclusion of mathematical expressions do not have adequate support for macros. Both \KaTeX and MathJax have excellent support for configuring macros, but these configuration options are not available in most of the tools that build on \KaTeX and MathJax: for instance, Logseq and Obsidian and Notion all support embedding mathematics, but they do not support configuring macros. In fact there is a community plugin for Obsidian that adds this functionality, but it only supports imposing a global macro library on the entire “vault”, which is inadequate.

In \LaTeX , macros are organized into packages that are then globally imported into a single document. Because a \LaTeX document comprises just one project and thus any transclusions (via \input or \include) are of components local to that one project, this model is adequate — although experienced users of \LaTeX are nonetheless all too aware of the difficulties of namespacing macro commands when interacting with multiple packages or document classes.

The requirements for a tool that aims to bring together multiple projects over a very long period of time are somewhat different: many distinct packages of notation will be used across the body of work, and it is not possible to fix a single global notation package.

Indeed, it is not reasonable to expect that all notes within a person’s mathematical life shall share the same notations, and in many cases, it would moreover be necessary for the names of the macros associated to these notations to clash. This can happen because two projects are orthogonal, or it can happen as the author’s tastes change over time — and it is not reasonable for such a tool to force enormous and weighty refactorings (touching thousands or tens of thousands of notes) every time the author’s taste changes. The need for arduous refactorings of this kind is one of the main ways that large mathematical projects tend to collapse under their own weight.

It follows that any tool for thought whose support for mathematical notations involves a globally-defined macro package is inadequate for mathematical uses. On the other hand, it is also not reasonable to require the author to define all their macros in each note: notes tend to be small, and there will always be large clusters of notes that share the same notations — and for which the small refactoring tasks involved when notations change are a positive feature rather than a negative one, as one of the goals of a cluster is to accumulate cohesion.

Therefore, the precise requirement for macro library support is as follows:

1. The author must be able to define (in their own files) multiple notational macro libraries.

2. A given note must be able to specify which macro library (if any) it employs.

Finally, careful attention must be paid to the interaction between the above requirements and the transclusion of mathematical notes: a transcluded note must be rendered with respect to its own macro library, and not that of the parent note.

A basic requirement of tools for mathematical thought is to support the rendering of mathematical diagrams. What kinds of diagrams are needed depends, of course, on the problem domain: for my own work, the main diagram-forms needed are commutative diagrams and string diagrams.

The situation for hypertext mathematical tools is somewhat less advanced than that of \LaTeX and PFG/TikZ, but there are several options which we discuss below.

Both \KaTeX and MathJax have the benefit that diagrams created using them will respect the ambient macro package with which the tool has been configured; therefore, if one looks past the rudimentary nature of the support for commutative diagrams, our main requirement is indeed satisfied. Another tool worth discussing is quiver.

% https://q.uiver.app/?q=WzAsMixbMCwwLCJBIl0sWzEsMCwiQiJdLFswLDFdXQ==
\[
  \begin{tikzcd}
    A & B
    \arrow[from=1-1, to=1-2]
  \end{tikzcd}
\]

<!-- https://q.uiver.app/?q=WzAsMixbMCwwLCJBIl0sWzEsMCwiQiJdLFswLDFdXQ== -->
<iframe class="quiver-embed" src="https://q.uiver.app/?q=WzAsMixbMCwwLCJBIl0sWzEsMCwiQiJdLFswLDFdXQ==&embed" width="304" height="176" style="border-radius: 8px; border: none;"></iframe>

Because of the currently inadequate support of quiver for embedding diagrams in hypertext settings, we cannot consider it any further. There is a final option that turns out to be the most used in practice: generating SVG images statically from embedded \LaTeX code.

Finally, we comment on more principled approaches using web standards such as SVG and MathML that we hope will take form in the future.

In light of the fundamental self-indexing, the following result for displayed categories generalizes the ordinary “pullback lemma.”

Lemma (generalized pullback lemma) [frct-001H]

Let \DispMor{\bar{f}}{f}{\bar{x}}{\bar{y}} , and suppose that \DispMor{\bar{g}}{g}{\bar{y}}{\bar{z}} is cartesian over g . Then \bar{f};\bar{g} is cartesian over f;g if and only if \bar{f} is cartesian over f .

Proof.

Suppose first that \bar{f} is cartesian. To see that \bar{f};\bar{g} is cartesian, we must construct a unique factorization as follows:

Because \bar{g} is cartesian, we can factor \bar{h} = i;\bar{g} for a unique \DispMor{i}{m;f}{\bar{u}}{\bar{y}} . Then, because \bar{f} is cartesian, we can further factor i = j;\bar{f} for a unique \DispMor{j}{m}{\bar{u}}{\bar{x}} . We conclude that there is a unique \DispMor{j}{m}{\bar{u}}{\bar{x}} for which \bar{h} = j;\bar{f};\bar{g} , as required.

Conversely, suppose that \bar{f};\bar{g} is cartesian. To see that \bar{f} is cartesian, we must construct a unique factorization as follows:

Because \bar{f};\bar{g} is cartesian, we can factor \bar{h};\bar{g} = i;\bar{f};\bar{g} for a unique \DispMor{i}{m}{\bar{u}}{\bar{x}} . On the other hand, because \bar{g} is cartesian, there is a unique \DispMor{j}{m;f}{\bar{u}}{\bar{y}} for which \bar{h};\bar{g} = j;\bar{g} ; as both \bar{h} and i;\bar{f} satisfy this condition, we conclude \bar{h}=i;\bar{f} . Therefore, there is a unique \DispMor{i}{m}{\bar{u}}{\bar{x}} for which \bar{h} = i;\bar{f} , as required.

Warning. Some authors including Grothendieck give an equivalent definition of cartesian fibration that factors through a nonequivalent definition of cartesian morphisms. Such authors refer to our notion of cartesian morphism as hypercartesian (see Streicher).

Definition (hypocartesian morphisms) [frct-002A]

Let E be displayed over B , and let f:x\to y \in B ; a morphism \DispMor{\bar{f}}{f}{\bar{x}}{\bar{y}} in E is called hypocartesian over f when for any \bar{u}\in E\Sub{x} and \DispMor{\bar{h}}{f}{\bar{u}}{\bar{y}} there exists a unique \DispMor{i}{\Idn{x}}{\bar{u}}{\bar{x}} with i;\bar{f} = \bar{h} as follows:

Lemma (hypocartesian = cartesian in a cartesian fibration) [frct-002C]

Let E be a cartesian fibration in the sense of cartesian fibration, and let \DispMor{\bar{f}}{f}{\bar{x}}{\bar{y}} be displayed over \Mor{f}{x}{y} . The displayed morphism \bar{f} is cartesian if and only if it is hypocartesian.

Proof.

Any cartesian map is clearly hypocartesian. To see that a hypocartesian map \DispMor{\bar{f}}{f}{\bar{x}}{\bar{y}} in a cartesian fibration is cartesian, we consider the cartesian lift of \Mor{f}{x}{y} under \bar{y} :

As the cartesian lift \bar{x}\tick\to \bar{y} is also hypocartesian, it follows that there is a unique vertical isomorphism identifying \bar{x} with \bar{x}\tick factoring \DispMor{\bar{f}}{f}{\bar{x}}{\bar{y}} through \DispMor{\bar{f}\tick}{f}{\bar{x}\tick}{\bar{y}} . Being cartesian over f is clearly stable under isomorphism, hence we conclude that \bar{f} is cartesian from the fact that \bar{f}\tick is cartesian.

Grothendieck defines a fibration in terms of (what we refer to as) hypocartesian morphisms rather than (what we refer to as) cartesian morphisms, and therefore imposes the additional constraint that the hypocartesian morphisms be closed under composition. In equivalence with Grothendieck's fibrations below, we verify that these two definitions of cartesian fibration coincide.