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I have been writing and thinking and writing and thinking. Here are some thoughts.

Cutting Down

I have reduced the body by one (1) more page since last time. This has involved reducing proofs and removing some definitions. I will continue to do so. Eventually I will have to do fresh read-overs to make sure everything is still coherent.

One definition cut was unnatural transformations, as it was only used in my appendix. I will be sad to see my \Rightharpoonup arrow go. For anyone willing to gaze upon the Tikz incantation used for the arrow, here it is:

\newsavebox{\Rightharpoonupbox}
\begin{lrbox}{\Rightharpoonupbox}%
  \tikz[baseline=-0.53ex, semithick, line cap=round]{
    \draw[arrows={->[width=10pt,length=4pt,harpoon]}] (0,-0.225ex)--(0.4cm,0.-0.225ex);
    \draw(0,0.225ex)--(0.4cm-2.85px,0.225ex);
  }%
\end{lrbox}
\newcommand*\Rightharpoonup{\mathrel{\usebox{\Rightharpoonupbox}}}

Rewriting the Duploids from Adjunctions Section?

One thing I was dissatisfied with was the presentation of the duploids-from-an-adjunction chapter. In particular:

  • There was very little prose.
  • There was a lack of smooth reasoning for how the envelope duploid arose from gazing at the oblique duploid construction for too long.
  • There is a lot of duplication between the definition of the oblique duploid and the envelope duploid. Some duplication arose from me (intentionally!) defining the oblique duploid’s composition effectively as that of the envelope duploid. The rest arises from generalizing Guillaume’s theorems about the oblique duploid to the envelope duploid.

I believe I can kill three birds with one stone by defining and motivating the envelope duploid in a more unbundled fashion (i.e. objects, then composition, then shifts) with more prose and motivation. Then I can simply bundle them together for the envelope duploid, and define the oblique duploid by its forgetful functor into the envelope duploid. This is probably fine, though I am worried about having unbundled definitions too far from their uses. The original oblique duploid construction has appeared in other literature, whereas the envelope duploid and its univalence is my novel contribution, so it stands to reason that I ought to focus on that, rather than effectively duplicating my definitions to include the oblique duploid construction first.

I have already made progress on rewriting the section in this form (chapter link), up to motivating envelope objects, and I believe it is more effective in motivating the definition. I can foresee it being more effective in flowing into the definition of composition too.

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