arXiv subjects

Eduardo Ochs

Publications and source records attributed to Eduardo Ochs.

6 recordsLinked to original sources

arXiv2026

Bad Foundations and Manipulable Objects

Imagine a student -- let's call him `E', and make him a "he" -- that is enrolled in Calculus 2, and who believes that to pass in maths courses he only needs to memorize methods and apply them quickly and without errors. Let's imagine that `E' is an `E'xtreme case of "bad foundations" and that he knows how to solve $x+2=5$ by doing $x=5-2=3$, but he doesn't know how to substitute the $x$ in $x+2=5$ by 3, and the only way that he knows of "testing the solution" is to apply the same method again and check that he got the same result. When we are teaching Calculus to classes that have many students that are extreme cases of bad foundations we need new strategies and tools; for example, we can't pretend that "taking a particular case" is an obvious operation anymore -- instead we need ways to make these operations easy to visualize. This article shows a way to do that using Maxima.

math.HO↗

arXiv2022

On the the missing diagrams in Category Theory (first-person version)

Most texts on Category Theory are written in a very terse style, in which people pretend a) that all concepts are visualizable, and b) that the readers can reconstruct the diagrams that the authors had in mind based on only the most essential cues. As an outsider I spent years believing that the techniques for drawing diagrams were part of the oral culture of the field, and that the insiders could read texts on CT reconstructing the "missing diagrams" in them line by line and paragraph by paragraph, and drawing for each page of text a page of diagrams with all the diagrams that the authors had omitted. My belief was wrong: there are lots of conventions for drawing diagrams scattered through the literature, but that unified diagrammatic language did not exist. In this chapter I will show an attempt to reconstruct that (imaginary) language for missing diagrams: we will see an extensible diagrammatic language, called DL, that follows the conventions of the diagrams in the literature of CT whenever possible and that seems to be adequate for drawing "missing diagrams" for Category Theory. Our examples include the "missing diagrams" for adjunctions, for the Yoneda Lemma, for Kan extensions, and for geometric morphisms, and how to formalize them in Agda.

math.CT↗

arXiv2021

Each closure operator induces a topology and vice-versa ("version for children")

One of the main prerequisites for understanding sheaves on elementary toposes is the proof that a (Lawvere-Tierney) topology on a topos induces a closure operator on it, and vice-versa. That standard theorem is usually presented in a relatively brief way, with most details being left to the reader and with no hints on how to visualize some of the hardest axioms and proofs. These notes are, on a first level, an attempt to present that standard theorem in all details and in a visual way, following the conventions in "On my favorite conventions for drawing the missing diagrams in Category Theory" [Ochs2020]; in particular, some properties, like stability by pullbacks, are always drawn in the same "shape". On a second level these notes are also an experiment on doing these proofs on "archetypal cases" in ways that makes all the proofs easy to lift to the "general case". Our first archetypal case is a "topos with inclusions". This is a variant of the "toposes with canonical subobjects" from [Lambek/Scott 1986]; all toposes of the form $\mathbf{Set}^\mathbf{C}$, where $\mathbf{C}$ is a small category, are toposes with inclusions, and when we work with toposes with inclusions concepts like subsets and intersections are very easy to formalize. We do all our proofs on the correspondence between closure operators and topologies in toposes with inclusions, and then we show how to lift all our proofs to proofs that work on any topos. Our second archetypal case is toposes of the form $\mathbf{Set}^\mathbf{D}$, where $\mathbf{D}$ is a finite two-column graph. We show a way to visualize all the LT-topologies on toposes of the form $\mathbf{Set}^\mathbf{D}$, and we define formally a way to "add visual intuition to a proof about toposes"; this is related to the several techniques for doing "Category Theory for children" that are explained in "On my favorite conventions...".

math.CT↗

arXiv2021

On a formula that is not in "Grothendieck Topologies in Posets"

The paper "Grothendieck Topologies on Posets" by A.J. Lindenhovius shows that when $\mathbf{P}$ is an Artinian poset and $\mathbf{E}$ is the topos $\mathbf{Set}^\mathbf{P}$ then there are bijections between the set of subsets of $\mathbf{P}$, the set of Grothendieck topologies on $\mathbf{E}$, and the set of nuclei on the Heyting Algebra $\mathrm{Sub}(1_\mathbf{E})$. It also shows that there are nice formulas for converting between subsets, Grothendieck topologies, and nuclei, but the formula for converting a nucleus to a subset is not spelled out explicitly. These notes fix that gap.

math.CO↗

arXiv2020

On my favorite conventions for drawing the missing diagrams in Category Theory

I used to believe that my conventions for drawing diagrams for categorical statements could be written down in one page or less, and that the only tricky part was the technique for reconstructing objects "from their names"... but then I found out that this is not so. This is an attempt to explain, with motivations and examples, all the conventions behind a certain diagram, called the "Basic Example" in the text. Once the conventions are understood that diagram becomes a "skeleton" for a certain lemma related to the Yoneda Lemma, in the sense that both the statement and the proof of that lemma can be reconstructed from the diagram. The last sections discuss some simple ways to extend the conventions; we see how to express in diagrams the ("real") Yoneda Lemma and a corollary of it, how to define comma categories, and how to formalize the diagram for "geometric morphism for children". People in CT usually only share their ways of visualizing things when their diagrams cross some threshold of of mathematical relevance - and this usually happens when they prove new theorems with their diagrams, or when they can show that their diagrams can translate calculations that used to be huge into things that are much easier to visualize. The diagrammatic language that I present here lies below that threshold - and so it is a "private" diagrammatic language, that I am making public as an attempt to establish a dialogue with other people who have also created their own private diagrammatic languages.

math.CT↗

arXiv2020

Planar Heyting Algebras for Children 2: Local Operators, J-Operators, and Slashings

Choose a topos $E$. There are several different "notions of sheafness" on $E$. How do we visualize them? Let's refer to the classifier object of $E$ as $Ω$, and to its Heyting Algebra of truth-values, $Sub(1_E)$, as $H$; we will sometimes call $H$ the "logic" of the topos. There is a well-known way of representing notions of sheafness as morphisms $j:Ω\to Ω$, but these `$j$'s yield big diagrams when we draw them explicitly; here we will see a way to represent these `$j$'s as maps $J:H\to H$ in a way that is much more manageable. In the previous paper of this series we showed how certain toy models of Heyting Algebras, called "ZHAs", can be used to develop visual intuition for how Heyting Algebras and Intuitionistic Propositional Logic work; here we will extend that to sheaves. The full idea is this: notions of sheafness correspond to local operators and vice-versa; local operators correspond to J-operators and vice-versa; if our Heyting Algebra $H$ is a ZHA then J-operators correspond to slashings on $H$, and vice-versa; slashings on $H$ correspond to "sets of question marks" and vice-versa, and each set of question marks induces a notion of erasing and reconstructing, which induces a sheaf. Also, every ZHA $H$ corresponds to an (acyclic) 2-column graph, and vice-versa, and for any two-column graph $(P,A)$ the logic of the topos $\mathbf{Set}^{(P,A)}$ is exactly the ZHA $H$ associated to $(P,A)$.

math.CT↗