Eduardo Ochs - Academic Research - Categorical Semantics, Downcasing Types, Skeletons of Proofs, and a bit of Non-Standard Analysis
The best things here are marked like [this].Quick index:
On the missing diagrams in Category Theory (2022) ***
[MD] This is a rewrite of [FavC]. Its abstract is:
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.
Grothendieck Topologies for Children (2021)
The paper "Planar Heyting Algebras for Children" (here) showed how to use Planar Heyting Algebras to visualize the truth-values and the operations of Propositional Calculus in certain toposes; the "...for children" of its title means: "we will start from some motivating examples ('for children') that are easy to visualize, and then go the general case ('for adults') - but there are precise techniques for working on the case 'for children' and on the case 'for adults' in parallel". These techniques are described in detail here. In these notes we will use these techniques to visualize Grothendieck topologies - first in the "archetypal" case of the canonical topology on a certain finite topological space, and then we will generalize that to arbitrary Grothendieck topogies on certain finite posets. that we will treat as "ex-topologies".
I give two informal talks on that, both in Portuguese, here.
Category Theory as An Excuse to Learn Type Theory (2021) **
The organizers of the "Encontro Brasileiro em Teoria das Categorias" invited me to give a 50-minute talk there. My talk was on the thursday, 28 january 2021. The abstract for my talk is here, and the slides are here. The recording is here (in Portuguese).
Each closure operator induces a topology and vice-versa (2020/2021) ***
[Clops&Tops] 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 below; 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 section 2.15 of [Lambek/Scott 86]; all toposes of the form Set^C, where 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 Set^D, where D is a finite two-column graph. We show a way to visualize all the Lawvere-Tierney topologies on toposes of the form Set^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 the first sections of [FavC].
The PDF of the
current version (2021jul26). Arxiv.
On my favorite conventions for drawing the missing diagrams in Category Theory (2020) ***
[FavC] 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" mentioned in section 1.
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.
Notes on Notation (2020)
In 2020 I decided to test my conventions
for drawing missing diagrams by drawing
some of the diagrams that are "missing" in several texts and seeing if
I always get diagrams that are easy to translate into Idris. These are
messy notes for personal use, but if you have your own conventions I'd
love to see them - please get in touch.
What kinds of knowledge do we gain by doing CT in several levels of abstraction in parallel? (2020)
A talk that I submitted to Diagram 2020. Its short abstract is:
Its "real", 4-page abstract is here.
Notes about classifiers and local operators in a Set^(P,A) (2020)
In the last section of Planar Heyting
Algebras for Children 2: Local Operators, J-Operators, and
I wrote that the proofs that my definitions for Ω and j "work as expected" are "routine". Well, they are only routine
if you know some techniques... these notes will
discuss these techniques and show all the calculations, but they
are currently in a very preliminary form.
On two tricks to make Category Theory fit in less mental space (2019)
A talk at the Creativity
2019 in Rio de Janeiro, in the workshop on formal logic and foundations.
Its full title was "On two tricks to make Category Theory fit in
less mental space: missing diagrams and skeletons of proofs", and
the abstract that I submitted is here.
Using Planar Heyting Algebras to develop visual intuition about IPL (2019)
Two presentations with the same title in two small events.
On some missing diagrams in the Elephant (2019) *
My submission was not accepted to become a talk, only to the poster session (on tuesday 16/july).
See the "Planar Heyting Algebras for Children" series.
Notes on the Yoneda Lemma (2019/2020)
My plan was to make a video from this, but I got stuck...
I am trying to implement this in a proof assistant -
An introduction to type systems (and to the "Logic for Children" project) (2019)
This is going to be a series of videos, but I am still
working on the slides... work in progress!
I almost gave a presentation about the part on types at IFCS in 2019jun18, but Jean-Yves Beziau forgot to pick up the projector from the secretary's drawer in the morning... she left everything locked, went out for lunch and only returned many, many hours later.
Five applications of the "Logic for Children" project to Category Theory (2019)
A talk that I gave at the EBL2019 in 2019may09. Its abstract is here. My plans for this talk were very ambitious: I wanted to present the main ideas, motivations and constructions of PHAfC 1, 2 and 3 and Logic for Children in 20 minutes, with lots of figures... but when it was my turn to present all the people who knew a bit of Category Theory were in another room, attending a session with technical talks, and I did not have the 5 or 6 slides that I could have made my talk more accessible to an audience of philosophers. I failed miserably.Slides.
Ensinando Matemática Discreta para calouros com português muito ruim (2019)
I organized a round table about how to teach Logic
to undergraduates in the EBL2019.
How to almost teach Intuitionistic Logic to Discrete Mathematics Students (2019) *
A talk in the World Logic
Day - 2019jan14 - in Rio de
Dednat6: an extensible (semi-)preprocessor for LuaLaTeX *
A talk at TUG2018, in Rio de
Janeiro, in 2018jul20 about
the package that I use to typeset several kinds of
Logic for Children (workshop at UniLog 2018) *
I organized (with Fernando Lucatelli) a workshop called "Logic for Children" that
happened at the UniLog 2018 in
Vichy, France, in june 24, 2018.
Visualizing Geometric Morphisms (talk at UniLog 2018)
A talk given at the workshop "Categories and Logic" in the
UniLog 2018 in june 22, 2018.
Planar Heyting Algebras for Children (2017-) ***
Finite planar Heyting Algebras ("ZHA"s) are very good tools for teaching Heyting Algebras and Intuitionistic Propositional Logic to "children"; "children" here means "people without mathematical maturity", in the sense that they are not able to understand structures that are too abstract straight away, they need particular cases first.
This is going to be a series of three papers.
Notes on notation (2017)
"Different people have different measures for "mental space"; someone with a good algebraic memory may feel that an expression like [...] is easy to remember, while I always think diagramatically, and so what I do is that I remember this diagram [...] and I reconstruct the formula from it." (IDARCT)
These are very informal notes showing my favourite ways to draw the "missing diagrams" in MacLane's CWM, and my favourite choice of letters for them. Work in progress changing often, contributions and chats welcome, etc. My plan is to do something similar for parts of the Elephant next.
IPL For Children and Meta-Children, or: How Archetypal Are ZHAs? (2017)
This is a 20-minute talk that I gave in the
EBL2017 in 2017may09.
Lambda-calculus, logics and translations (course, 2016-)
In 2016 I started giving a very introductory course on lambda-calculus, types, intuitionistic propositional logic, Curry-Howard, Categories, Lisp and Lua in the campus where I work. The course is 2hs/week, has no prerequisites at all, has no homework, and is usually attended by 2nd/3rd semester CompSci students; they spend most of their time in class discussing and doing exercises together in groups on a whiteboard.
Intuitionistic Logic for Children, or: Planar Heyting Algebras for Children (2015)
Seminar notes, with lots of figures (all drawn with Dednat6).
Logic and Categories, or: Heyting Algebras for Children (2015)
Sheaves for children (2014)
The second - and much longer - version of this talk (at the Seminário Carioca de Lógica, 2014may19, 15:00, IFCS) had these slides and these handouts, and was meant for much younger "children". The focus this time was a visual characterization of the subsets of N2 that are Heyting Algebras, and how can we treat their points as truth-values, and so how to interpret intuitionistic logic on them. I call these subsets "ZHAs", the definitions and main theorems for them are in the pages 20 to 27 of the slides, and also at the handouts.
Sheaves on Finite DAGs may be Archetypal (2011)
Can the ideas of my article about "internal diagrams" be used to present the basic concepts of toposes and sheaves starting from simple, "archetypal" examples? I believe so, but this is still a work in progress!
Here are 7 pages of very nice handwritten notes (titled "Sheaves for Children"): pdf, djvu. They were written after discussions with Hugo Luiz Mariano and Claus Akira Matsushige Horodynski in feb/2012, during a minicourse on CT in Brasilia organized by Claus, with me and Hugo as lecturers...
...and here are some slightly older notes - I submitted them, in a admittedly incomplete form, to the XVI EBL, with this abstract - and then I did a bad job at presenting them; here are the slides, they cover only the first ideas =(.
For the sake of completeness, here are some handwritten diagrams describing Kan extensions in an (hopefully) archetypal case, motivated by discussions with G.F. Lima: 1200dpi djvu, 600dpi djvu, 600dpi pdf.
Internal Diagrams in Category Theory (2010/2013) ***
A paper that I published at Logica Universalis in
issue on Categorical Logic in 2013.
This was my first "real" paper.
For my talk at the UniLog 2010 (below) I prepared a HUGE set of slides, and after chatting with several people at the conference I understood that the best way to try to publish those ideas would be to focus on the philosophical side and to leave out most technicalities (e.g., fibrations)... so I wrote "Internal Diagrams in Category Theory" and submitted it to LU. The referees told me to change some things in it, including the title, and to split the paper in two. Instead of splitting it I wrote some new sections to explain how its two "halves" were connected, and this became "Internal Diagrams and Archetypal Reasoning in Category Theory".
Downcasing Types (at UniLog'2010)
I gave a talk about Downcasing Types at the special session on Categorical Logic of UNILOG'2010, on 2010apr22. Very few people attended - because of the volcanic ashes many people could not fly to Portugal, and from all these programmed talks only these ended up happening. The abstract was:
Natural infinitesimals in filter-powers (2008)
"Purely calculational proofs" involving infinitesimals can be "lifted" from the non-standard universe (an ultrapower) to the "semi-standard universe" (a filter-power) through the quotient SetI/F→SetI/U; and after they've been moved to the right filter-power they can be translated very easily to standard proofs. I don't know how much of this idea is new, but I liked it so much that I wrote it down in some detail and asked for feedback in the Categories mailing list.
A (long-ish) abstract for a presentation intended for undergrads:
(News: Reinhard Boerger pointed me to later (post-1958) work by Laugwitz and Schmieden, and I got a copy of the "Reuniting the Antipodes" book; my current impression is that my result is not as trivial as I was afraid it could be. Homework-in-progress: several cleanups on the preliminary version above, and I'm trying - harder - to understand Moerdijk and Palmgren's sheaf models.)
Note (2010): I still don't have the tools for formalizing this idea completely. As what I have is an "incomplete internal language", the ideas in this preprint may help.
Sheaves for Non-Categorists (2008)
This is another presentation that - maybe after some clean-ups - will be accessible to undergrads... The current version of the slides (far from ready, with lots of garbage and gaps!) is here: pdf, source. The presentation will be at the Logic Seminar at UFF, on 2008sep04, 16:00-17:00hs.
Here's an htmlized version of the abstract:
Seminar on downcasing types (nov/2007)
If you are going to attend my seminars at PUC at November/2007 and want to take a peek at my notes (they are very incomplete at the moment, it goes without saying), they have just been split into several parts:
Bad news (?), dec/2007: the seminars will not happen - instead, I got a job at São Paulo, on computer stuff. I'll keep working on maths and on my personal free software projects in my spare time. If you find any of these things interesting, and want to discuss or to encourage me to finish something, get in touch!
2008: I am giving a series of seminars at UFF to try to organize my ideas about downcasing types... here are links to some of TeXed slides (they are very preliminary, too. Should I be embarassed to provide links to these things? Well...):
I moved them to this page.
PhD and post-PhD research *
I did both my MsC and my PhD (and also my graduation, by the way) at the Department of Mathematics at PUC-Rio. The Dept of Mathematics is a fantastic place - tiny, incredibly friendly, well-equipped, lots of research going on -, but (rant mode on) PUC-Rio is a private university, and most of the students from other departments were ultra-competitive rich kids who had never stepped out of the marble towers they live in. I used to find it very hard - very painful, even - to interact with them, and even to stand their looks, like if they were always trying to tag me as either a "winner" or a "loser", as if there weren't any other ways to live. Eeek! But these days are long gone now (rant mode off).
I spent the first eight months of 2002 at McGill University in Montreal, doing research for my PhD thesis there, working with Robert Seely... I was in a "Sandwich PhD" program (thanks CAPES!), which is something that lets us do part of the research abroad and then come back and finish (and defend) the thesis at our university of origin.
I defended my PhD thesis (with lots of holes) in August, 2003 and presented the final version - filling out some of the gaps - in February, 2004. Then I spent most of 2004 teaching part-time in an university at the outskirts of Rio (FEBF/UERJ), and also trying to finish a very important Free Software project that I've been working on since 1999 (GNU eev).
The thesis is in Portuguese and you don't want to see it. Really. =(
News (2010/2013): this paper has all the ideas from my PhD thesis, plus some! It fills all the gaps from the thesis, and it is quite well written =).
News (October 2005): I gave a series of talks about my PhD thesis at UFF.
This is the abstract for a talk that I gave at the FMCS2002 in June 8, 2002.
Fact: all the essential details (i.e., the "T-part", as in the abstract above) of a certain construction of a categorical model of the Calculus of Constructions - and also of categorical models of several fragments of CC - can be expressed in (a few!) categorical diagrams using the DNC language. I'm currently (February/March 2005) preparing talks and articles about that.
An older talk about Natural Deduction for Categories. After using something like the DNC notation for years just because "it looked right", but without any good formal justification for it, in February 2001 I had the key idea: there were rules of both discharge and introduction for the "connectives" for functors and natural transformations. A few months after that (in July 5 2001, to be precise) I gave a talk about it at a meeting called Natural Deduction Rio 2001.
Another talk, even older, about Natural Deduction for Categories. After finding the key idea that I mentioned above I arranged to give a (very informal) talk at the Centro de Lógica e Epistemologia at UNICAMP. It happened in April 25, 2001, and for it I had to assemble my personal notes into something that could be used as slides. The title was ""Set^C is a topos" has a syntactical proof".
MsC Thesis and related things *
My master's thesis: "Categorias, Filtros e Infinitesimais Naturais" (April, 1999). The thesis and the slides used in the defense are in Portuguese.
A few months after the defense (in February 24, 2000, to be precise) I gave a talk at UFF about a kind of "Nonstandard Analysis with Filters", and about skeletons of proofs. Slides (12 slides plus one page), in Portuguese: pdf, ps, dvi+eps's, source.
My advisor at PUC: Nicolau Saldanha
Typesetting categorical diagrams in LaTeX
My PhD thesis included lots of hairy categorical diagrams, and I ended up writing a LaTeX preprocessor in Lua - called "dednat4.lua" - to help me typeset them. Below are some examples of diagrams that I have typeset with dednat4:
Note that dednat4 is obsolete, and that the diagrams above use an obsolete notation - the notation for "downcasings" from IDARCT...