I have been writing and thinking and writing and thinking. Here are some thoughts.
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}}}
One thing I was dissatisfied with was the presentation of the duploids-from-an-adjunction chapter. In particular:
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.
]]>In the past few weeks I prodded a little more at bicategorical proofs about left universal arrows. However, in the interest of time, I’m turning my attention more to (re)writing.
The abridged draft of my dissertation, to be suitable for publication in a conference, will be available at https://notes.eutro.dev/cs/diss/output/abridged.pdf shortly. Currently, the Preliminary Type Theory section has been considerably reduced, but I intend to reintroduce pointers to relevant material. I am thinking about how best to treat the introduction and Semantics and Evaluation Order sections. I have also cut various proofs and explicit definitions, where I’ll be reading through to make sure things are still coherent. The body is currently at 36 pages, down from 49 for the dissertation, but I’ll have to make sure there is enough information for things to be clear.
I am wondering if the introduction I wrote for my dissertation is the best introduction to the actual content, viewed as a mathematics paper. (The introduction walks through call-by-name and call-by-value giving rise to (an example of) non-associative composition.) From my perspective, thinking about programming language semantics is entirely irrelevant to the mathematics I did, even though it is a substantial motivator for duploids. On the other hand, the context is important, especially in referencing the related work like Levy’s contextual isomorphisms.
Comparing to other papers, the introduction of Mangel et al. [1] conveys somewhat similar information to my dissertation. This makes sense, because their paper is actually about duploids and semantics, which mine is not. By contrast, the work of van der Weide [2] on monads in Univalent Foundations does little more than mention the uses of monads, without belabouring the details. This also makes sense, in part because the use of monads is well-established.
In my paper I believe it will be sufficient to mention programming language semantics only in passing, so I can focus more on the specific motivation of my own work instead. I’m not sure how much I need to motivate duploids themselves, though. I will see how it goes.
This week I spent more time bashing my head against UniMath’s bicategories. This has resulted in me proving 2.5 more coherence laws for the pseudofunctor-from-left-universal-arrows, leaving only 1.5 more to go. Mind, the computationally relevant data of the pseudofunctor is already well-defined, so this grunt work is unnecessary if my interest is in using Rocq as a blackboard1.
A 2-categorical way to define the equalizing requirement is helpful for studying adjunctions and their relation to duploids. Every adjunction \(L \dashv R : \NN \to \PP\) gives rise to a comparison functor \(K^{RL} \colon \NN \to \PP^{RL}\) to the Eilenberg-Moore category of the monad \(RL\). It is known that an adjunction is negative equalizing if and only if \(K^{RL}\) is fully-faithful, and I have now proven this in UniMath. Likewise, this can be dualized to comonads for the positive side. This functor and its fully-faithfulness can both be expressed 2-categorically, which (I hope) will be easier to work with for the 2-categorically-defined image-of-an-adjunction-in-a-pseudofunctor.
It is clear that the 2-categorical Rezk completion via adjunctions and the envelope duploid still has lots of work to be done. Given this, I may take a fresh stab at defining a Rezk completion of duploids directly. Such a construction may provide more insight anyhow, and it will force me to refactor some existing code too (e.g. the cursed Yoneda stuff).
1 Someone else proved these theorems on paper for
2-categories so it must be true! I can safely use admit, right?
These past few weeks I have spent some time working on the Rezk completion for duploids. Otherwise, I would like to collect my thoughts on what I wish to do, now that I have submitted the final version of the part III project and graduated.
I intend to write a post here presenting the final version of my dissertation,1 but that is not yet written. In the meantime, please find it at https://notes.eutro.dev/cs/diss/output/writeup.pdf2
These are some ideas for what I could do going forward. This also serves as a reminder of what my unfinished/future work was, and the progress I made on them nevertheless.
I have made some progress formalizing the pseudofunctor from a left universal arrow of 2-categories.
Definition 1. Let \(R : \C \to \D\) be a pseudofunctor. A left universal arrow \((L_0x, \eta_x)\) from \(x : \D\) to \(R\) consists of:
Proposition 2. A family \((L_0x, \eta_x)_{x : \D}\) of left universal arrows to \(R : \C \to \D\) assembles into a pseudofunctor \(L : \D \to \C\).
The proof of Thomas Fiore’s Theorem 9.17 [1] should somewhat be usable. However, it is complicated by UniMath using weak 2-categories (i.e. bicategories) rather than strict 2-categories as Fiore does.
I am… getting through it. I have got the data of the pseudofunctor, have proven that its identitor and compositor are invertible, and have proven a couple easier laws. It remains to prove the more difficult laws, for which I’m better served by some pen-on-paper diagram chasing than by the proof assistant.
1 Sorry, “part III project”.
2 The newest
version of writeup.pdf is a different rendering to that submitted
for marking. It links directly to GitHub for source links. The links
from the submitted rendering claim to redirect, but they do not.
TODO-s. Then I intend
to enter an editing phase in time for submission by June 1st.]]>The most recent draft of my dissertation is now uploaded at https://notes.eutro.dev/cs/diss/output/writeup-2026-05-15.pdf. In the past couple of days I have written some more of the final theorems about univalence of the envelope and oblique duploids, and made minor revisions to the unital magmoids chapter. I have revised some of the background chapter and introduction, but more revision is still due.
I am likely still missing some theorems here and there, and I am quite a few pages over the limit already, even without some necessary prose that still needs to be (re)written. The pruning and revision necessary to bring me under the limit, and of course to a more coherent paper, will require a thorough read-through and of the whole paper, that I intend to make a start on this weekend. There are also definitely grammar mistakes and typos that I feel the need to apologise for.
]]>This past week I made a great deal of progress on writing. The new version is up on my notes site. Notably, I reorganized the Unital Magmoids chapter to include the definition of categories (and hence its generalizations more closely), added most definitions to the Duploids chapter, and wrote most of the definitions for the “Adjunctions and Duploids” chapter, which includes definitions of the oblique and envelope duploids. I have also made some minor stylistic and notational changes.
I still have to write about the equalizing requirement, and how it
makes the envelope duploid univalent; and the oblique duploid, when
univalence is defined to include the polarization choice structure
(see is_univalent_split_oblique_duploid). My remaining results are
then the weak equivalence of each duploid to the envelope duploid on
its shifts (weak_dupoid_equivalence_duploid_to_envelope_on_shifts),
my beautiful 2-categorical UniMath formalization that
adjunctions are preserved by pseudofunctors
(map_left_adjoint_axioms), and then a handwave hint at the Rezk
completion which is ever too painful to actually formalize. I also
might have a handful of theorems missing here and there, and some of
the existing ones are just tedious computations that have been evading
nicer proofs than “Proof. This computation is an exercise for the
reader. \(\square\)” It is the primary benefit of Rocq that I can do
like induction x, y, z; reflexivity. and eat through a bunch of
tedious cases…
Outside of technical results to write down, I have a great deal of prose (outline, conclusions, related work, future work, etc.) and exposition to write, and yet I am running out of space. In the next week or so I shall have to tackle these more seriously, now that most of the technical content is written up.
]]>I have written the bulk of the background chapter of my dissertation, these past few days spent writing the section titled “Mixed Evaluation Order.” I went back-and-forth on whether to put this section before or after the other two sections (currently titled “Preliminary Type Theory” and “Univalent Categories and Reflexive Graphs,” despite a stark lack of Reflexive Graphs in the latter…). I have decided to put it first, because it flows on nicer from the introduction and motivation, and means we can forget about the syntax of these non-associative languages as soon as we have mentioned them.
I have written all the sections of the background and introduction, but have not yet had the opportunity to rewrite them to fit a completely coherent narrative. I believe they are legible, mostly stand on their own (except the new section which I feel is a bit weak), and cover most of the background I think I should need. The version at time of writing may be read here. I still include many notes and TODOs, if you can forgive me.
I have finally found some use of the barely salvageable Yoneda lemma which so horribly fails for unital magmoids. In particular, I can use this to show that linear-and-thunkable-and-intermediate isomorphisms correspond to the indiscernibilities derived mechanically by Ahrens et al. [1]. I did this because I was dreading writing down the univalence principle for unital magmoids without being sufficiently reassured that it is the correct notion.
Part of the Yoneda lemma for categories is that the Yoneda/post-composition functor \(\yo(a : \C) \defequiv \morsof\C{-}{a} : [\C^\op, \Set]\) is fully-faithful. As a consequence, morphisms \(a \to b\) are in correspondence with natural transformations \(\morsof\C{-}{a} \Rightarrow \morsof\C{-}{b}\), and isomorphisms \(a \cong b\) are in correspondence with natural isomorphisms \(\morsof\C{-}{a} \cong \morsof\C{-}{b}\).
Unfortunately, in a unital magmoid most of this fails, but we can scrape back some useful corollaries even in the absence of a full (i.e. nicely stated) Yoneda lemma.
Lemma 1. In a unital magmoid \(\M\), the morphism \(f : \morsof\M{a}{b}\) is linear if and only if the post-composition transformation \((g \mapsto g \dcomp f) : \morsof\M{-}{a} \to \morsof\M{-}{b}\) between reflexive graph functors \(\morsof\M{-}{a} : \M^\op \to \Set\) is natural.
Proof. This is amounts to unfolding the definition of naturality.
Lemma 2. The Yoneda/post-composition functor restricted to linear morphisms \(\Cl\yo(a : \Cl\M) \defequiv \morsof\M{-}{a} : [\M^\op, \Set]\):
Proof. Following your nose (um_yoneda_linear_fully_faithful).
Remark 3. This is a little bit stronger than just applying Yoneda’s lemma to the linear subcategory \(\Cl\M\), because only one of the hom-sets in the entire equation is forced to be linear.
Corollary 4. Linear isomorphisms \(a \Cl\cong b\) in \(\M\) are equivalently natural isomorphisms \(\morsof\M{-}{a} \cong \morsof\M{-}{b}\).
Proof. This follows from usual properties of fully faithful functors.
Corollary 5. Thunkable isomorphisms \(a \Ct\cong b\) in \(\M\) are equivalently natural isomorphisms \(\morsof\M{a}{-} \cong \morsof\M{b}{-}\).
Proof. This is formally dual to the previous statement.
Theorem 6. Linear-and-thunkable-and-intermediate isomorphisms \(a \Clti\cong b\) in a unital magmoid \(\M\) correspond to indistinguishabilities \(a \asymp b\) obtained by following the procedure of Ahrens et al. [1].
Proof. Our diagram signature is exactly the same as categories, but we do not impose the associativity axiom. Therefore most of the extended example in chapter 3 [1] applies. One can verify that the data and properties (3.13–3.26) of indiscernibilities in section 3.4 correspond to a linear-and-thunkable-and-intermediate isomorphism. The data of an indiscernibility \(a \asymp b\) includes a natural isomorphism in the codomain (3.13, 3.16) \(\morsof\M{-}{a} \cong \morsof\M{-}{b}\), hence a linear isomorphism \(p : a \Cl\cong b\); and a natural isomorphism in the domain (3.14, 3.18) \(\morsof\M{a}{-} \cong \morsof\M{b}{-}\), hence a thunkable isomorphism \(q : a \Ct\cong b\). These are forced to be the same by the equation \((f \dcomp p) \dcomp (q^{-1} \dcomp g) = f \dcomp g\) for all \(x \xrightarrow{f} a \xrightarrow{g} z\) (3.17), and the same property also forces the isomorphism to be intermediate. Thus we have \(p = q : a \Clti\cong b\). The last bit of data is an identity-preserving isomorphism (3.15, 3.23) \(\morsof\M{a}{a} \cong \morsof\M{b}{b}\) which is forced to be \(f \mapsto p^{-1} \dcomp f \dcomp p\) (3.19–3.21). The remaining properties (3.22–3.23, 3.24–3.26) are automatic.
This justifies the following definition.
Definition 7. A unital magmoid \(\M\) is univalent when the linear-and-thunkable-and-intermediate subcategory \(\Clti\M\) is univalent.
We already know that this coincides with univalence for preduploids 🙂.
I have continued to write this week. I do not have much to say on this, except that I have now made it so that I can keep multiple dated versions of the rendered writeup available at https://notes.eutro.dev/cs/diss/output/.1 As well as writing, it seems inevitable that I will continue to ponder the outstanding mathematical problems. Fortunately, working with bicategories in UniMath is incredibly painful, so that keeps me away for the most part. My most significant achievement this week is that I have constructed a unital magmoid with a linear-and-thunkable isomorphism which is not intermediate.
One of my early concerns was this: what is a suitable notion of an “indistinguishability” in a unital magmoid? This would be a type of morphism \(p : a \cong_* b\) for which “indistinguishability induction” (handling only \(b \equiv a\) and \(p \equiv \id_a\)) is valid. In other words, a type for which an equivalence \((a = b) \cong (a \cong_* b)\) (taking \(\refl_a\) to \(\id_a\)) could feasibly exist. This is the same as asking, “what morphisms in a unital magmoid correspond to identifications when we consider a univalent unital magmoid?”
As is the case for isomorphisms in categories, an indistinguishability \(p : a \cong_* b\) in a unital magmoid must have an inverse \(p^{-1} : b \cong_* a\). However, this is not sufficient in a unital magmoid: we need to place further associativity conditions on \(p\) for it to be able to behave like an indistinguishability. That is, to have all the properties of \(\id_a\).
In a duploid, we have seen that being a linear-and-thunkable isomorphism seems to be the correct notion of an indistinguishability. We have also seen that all linear-and-thunkable isomorphisms in a preduploid are intermediate (see the definition below). However, this does not appear to be true in a unital magmoid in general, and I have been trying to pin down a counterexample. I present one today.
The terms “linear” and “thunkable” are the names Guillaume gave them, and “intermediate” is a name I made up. The following is a definition from my writeup. Note that I have taken to using \(f \dcomp g\) to denote composition in diagrammatic order, rather than taking \(f \cdot g\) from UniMath .
Definition 1. We say that a path of morphisms
\begin{equation} a_0 \xrightarrow{f_1} a_1 \xrightarrow{f_2} a_2 \rightarrow \cdots \rightarrow a_{n-1} \xrightarrow{f_n} a_n \end{equation}
in a unital magmoid associates when all ways of parenthesising their composite \(f_1 \dcomp f_2 \dcomp \cdots \dcomp f_n\) are equal. In the simplest case, a triple of morphisms \(a \xrightarrow{f} b \xrightarrow{g} c \xrightarrow{h} d\) associates if and only if we have \(f \dcomp (g \dcomp h) = (f \dcomp g) \dcomp h\).
We say that:
Notation 2. Let \(a, b : \M\) be objects in a unital magmoid \(\M\). We may write the subscripts from the Subcategories of a Unital Magmoid onto the binary operators \(a \rightarrow b\) (morphisms) or \(a \cong b\) (isomorphisms) to denote that the morphism or isomorphism resides in the corresponding subcategory of the unital magmoid. For example, \(a \Cl\rightarrow b\) denotes a linear morphism; \(a \Clt\cong b\) denotes a linear-and-thunkable isomorphism: an isomorphism which is linear-and-thunkable, and has a linear-and-thunkable inverse.
The question I wanted to answer is this:
Proposition 3. Let \(\M\) be a unital magmoid. Could there be a linear-and-thunkable isomorphism \(a \Clt\cong b\) in \(\M\) which is not intermediate?
I can answer this in the affirmative today with a small counterexample.
Definition 4. There is a unital magmoid \(\wkcat{Lini}\)2 generated by four objects: \(a,b,c,d:\wkcat{Lini}\) and the morphisms \[\begin{gathered} a \xrightarrow{f} b \xrightarrow{g} c \\ b \Clt{\xrightarrow{p}} d \Clt{\xrightarrow{q}} b \end{gathered}\] such that \(p \dcomp q = \id_b\) and \(q \dcomp p = \id_a\). In other words, \(p\) and \(q\) form a linear-and-thunkable isomorphism \(p : b \Clt\cong d\) such that \(p^{-1} = q\). This is depicted in the diagram below.
Theorem 5. The unital magmoid \(\wkcat{Lini}\) provides a counterexample to the above proposition. In particular, \(p : b \Clt\cong d\) is not intermediate.
Proof. The unital magmoid \(\wkcat{Lini}\) can be described explicitly in the following way. The type of objects \(\ob\wkcat{Lini}\) is, of course, the 4-element finite type \(\llbracket4\rrbracket\), whose elements I will write \(a,b,c,d:\wkcat{Lini}\). The hom-sets \(\morsof{\wkcat{Lini}}{x}{y}\) are given by the following table:
| \(\morsof{\wkcat{Lini}}{x}{y}\) | \(y=\) | \(a\) | \(b\) | \(c\) | \(d\) |
|---|---|---|---|---|---|
| \(x=\) | |||||
| \(a\) | \(\unit\) | \(\unit\) | \(\bool\) | \(\unit\) | |
| \(b\) | \(\ttempty\) | \(\unit\) | \(\unit\) | \(\unit\) | |
| \(c\) | \(\ttempty\) | \(\ttempty\) | \(\unit\) | \(\ttempty\) | |
| \(d\) | \(\ttempty\) | \(\unit\) | \(\unit\) | \(\unit\) |
This is rather opaque. We may instead write the elements more informatively as formal composites generated by \(f,g,p\) and \(q\).
| \(\morsof{\wkcat{Lini}}{x}{y}\) | \(y=\) | \(a\) | \(b\) | \(c\) | \(d\) |
|---|---|---|---|---|---|
| \(x=\) | |||||
| \(a\) | \(\{\id_a\}\) | \(\{f\}\) | \(\{[f\dcomp g], [[f\dcomp p]\dcomp[q\dcomp g]]\}\) | \(\{[f\dcomp p]\}\) | |
| \(b\) | \(\emptyset\) | \(\{\id_b\}\) | \(\{g\}\) | \(\{p\}\) | |
| \(c\) | \(\emptyset\) | \(\emptyset\) | \(\{\id_c\}\) | \(\emptyset\) | |
| \(d\) | \(\emptyset\) | \(\{q\}\) | \(\{[q \dcomp g]\}\) | \(\{\id_d\}\) |
The element \([f\dcomp g] : \morsof{\wkcat{Lini}}{a}{c}\) is just notation for, say \(\orgcode{false} : \bool\). Composition in \(\wkcat{Lini}\) is defined so that these formal composites do in fact denote the composition of the respective morphisms. In particular, we set \[\begin{aligned} f \dcomp g &\defequiv [f \dcomp g] : \morsof{\wkcat{Lini}}{a}{c} \end{aligned}\] and \[[f \dcomp p] \dcomp [q\dcomp g] \defequiv [[f \dcomp p] \dcomp [q\dcomp g]] : \morsof{\wkcat{Lini}}{a}{c}\] for the two non-\(\id\) cases which compose into \(\morsof{\wkcat{Lini}}{a}{c}\). All other composites are either vacuous or defined to be the single element \(\orgcode{tt} : \unit\), by whichever name it is given.
We may reassure ourselves that this does in fact construct \(\wkcat{Lini}\) as described initially, in that the morphisms of this construction are no more and no less than those generated by \(f\), \(g\) and linear-and-thunkable inverses \(p\) and \(q\). In most cases, it is easy to identify the composite in this construction with the corresponding formal-composite-of-morphisms-modulo-laws-and-assumptions. For example, we hypothesized that \(p\) and \(q\) are inverses in \(\wkcat{Lin}\), and we can compute that we have \[\begin{aligned} p \dcomp q &\equiv \id_b \\ q \dcomp p &\equiv \id_d\,. \end{aligned}\] More interestingly, we have \[\begin{aligned} [f \dcomp p] \dcomp q &\equiv f \\ p \dcomp [q \dcomp g] &\equiv g\,. \end{aligned}\] This too is as it should be for a construction of \(\wkcat{Lini}\). We must have \[(f \dcomp p) \dcomp q = f \dcomp (p \dcomp q)\] for \(q\) to be linear, and we must have \[f \dcomp (p \dcomp q) = f\] for \(p\) and \(q\) to be inverses. The case with \(g\) is analogous.
It is easy to verify that \(p\) and \(q\) are linear-and-thunkable. To do so,3 observe that for all \(x : \wkcat{Lini}\) we have that \(\morsof{\wkcat{Lini}}{b}{x}\) and \(\morsof{\wkcat{Lini}}{x}{d}\) are propositional (they are each either \(\ttempty\) or \(\unit\)). Thus all composites starting with \(p\) are equal, all composites ending with \(p\) are equal, and likewise for \(q\). Given this, we can see that \(p\) is a linear-and-thunkable isomorphism \(p : b \Clt\cong d\). Again, this is as it should be for a construction of \(\wkcat{Lini}\).
The linear-and-thunkable isomorphism \(p : b \Clt\cong d\) is not intermediate, however. This may be verified by the counterexample of the triple \[a \xrightarrow{f} b \xrightarrow{p} d \xrightarrow{[q \dcomp g]} c\] which does not associate. We can compute \[(f \dcomp p) \dcomp [q \dcomp g] \equiv [f \dcomp p] \dcomp [q \dcomp g] \equiv [[f \dcomp p] \dcomp [q \dcomp g]]\,,\] but \[f \dcomp (p \dcomp [q \dcomp g]) \equiv f \dcomp g \equiv [f \dcomp g]\,.\] These are the two distinct elements of \(\morsof{\wkcat{Lini}}{a}{c} \equiv \bool\). Thus the triple does not associate, and so \(p : b \Clt\cong d\) is not intermediate.
1 The name I use for myself on https://notes.eutro.dev is pronounced like the final syllable of my given name, but stressed.
2 An initialism of “Linear-and-thunkable Isomorphisms are Not (in general) Intermediate”
3 Sixty-four cases of computation are the easier solution on a computer.
]]>I have been writing the text of my dissertation and made the rendered draft available at https://notes.eutro.dev/cs/diss/output/writeup.pdf.1 I have taken the liberty of sprinkling it with all kinds of notes intended for the reader, about omissions and planned direction. The reader most usually being me, but it could potentially even be you! This is of course a fairly early draft so I am not expecting detailed feedback at this time.
At the moment, the “best-written” part is the introduction. This let me motivate a little bit the project and set out what I intend to include. It is also something I can point to for anyone who asks “what is your dissertation on?” with the hope that it might take at least two sentences before they get completely lost.
Other than that, I am still piecing together the work I have done and text I have already written, trying to fit them neatly into a coherent linear narrative, at points where they will make sense to the reader. I have made some progress so far, but not quite as much as I would have liked. For instance, I am struggling with the placement and explanation of the envelope duploid’s construction. I think I am getting into a workable rhythm with writing though, and my plan and prose are developing healthily in tandem.
I have done some thinking and proving towards the Rezk completion of duploids. In summary: on paper, everything just works. I won’t write about it today though, since I’m just trying to get the draft text up.
1 This link always points to the most up-to-date draft that I have committed and pushed.
]]>