λ, etc)
(Chapa 1)


Eduardo Ochs - Academic Research - Categorical Semantics, Downcasing Types, Skeletons of Proofs, and a bit of Non-Standard Analysis

Quick index:

Note: I don't use this page very much myself, and so I don't usually notice when the links to papers stop working... 8-\ I always access (my local copies of) online papers with the functions defined in my .emacs.papers file.

IPL For Children and Meta-Children, or: How Archetypal Are ZHAs?

This is a 20-minute talk that I will give in the EBL2017.
The full title is: "Intuitionistic Propositional Logic For Children and Meta-Children, or: How Archetypal Are Finite Planar Heyting Algebras?"
Abstract: PDF.
Slides: PDF (work in progress).
Handouts: not yet!

Intuitionistic Logic for Children, or: 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.

These notes cover roughly the same material as the ones below, that have the same title but a "2015" at the end, but in a self-contained way. Everything has been rewritten, and some of the sections have been tested on real "children" (sophomore CS students); they found the text of those sections readable enough.

This is a work in progress - it still has some sections missing in the end.

The PDF (current version: 2017mar07).
An older PDF (2016dec18).

Intuitionistic Logic for Children, or: Planar Heyting Algebras for Children (2015)

Seminar notes, with lots of figures (all drawn with Dednat6).
Not self-contained. Superseded by the 2017 version above.
I never wrote a textual abstract for this, but the page 2 of the PDF is a nice "One page intro" with text and diagrams.

The PDF is here (and here).
Current version: 2015oct19b.
(First version: 2015sep24.)

Logic and Categories, or: Heyting Algebras for Children (2015)

A tutorial presented at the UniLog 2015 conference (Istanbul), 20-22/jun/2015.
Abstract: PDF, HTML.
Slides: PDF.
Handouts: PDF.

Sheaves for children (2014)

A 20-minute talk - here are its slides - presented at the XIV EBL, on 2014apr09. Its abstract:

First-year university students - the ``children'' of the title - often prefer to start from an interesting particular case, and only then proceed to general statements. How can we make intuitionistic logic, toposes, and sheaves accessible to them?

Let D be a finite subset of N2. Draw arrows for all the ``black pawns moves'' between points of D, and let D be the poset generated by that graph; D is what we call a ``ZDAG'', and SetD is a ``ZDAG-topos''. It turns out that the truth-values of a SetD can be represented in a very nice way as two-dimensional ASCII diagrams, and that all the operations leading to sheaves and geometric morphisms can be understood via algorithms on diagrams.

In this talk we will present a computer library for performing computations interactively on the truth-values of ZDAG-toposes. The diagrams are rendered in ASCII by default, but there is a module that typesets them in LaTeX.

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)

A paper submitted to Logica Universalis, and published in their special issue on Categorical Logic in 2013, with a longer name: "Internal Diagrams and Archetypal Reasoning in Category Theory". Its abstract:

We can regard operations that discard information, like specializing to a particular case or dropping the intermediate steps of a proof, as projections, and operations that reconstruct information as liftings. By working with several projections in parallel we can make sense of statements like "Set is the archetypal Cartesian Closed Category", which means that proofs about CCCs can be done in the "archetypal language" and then lifted to proofs in the general setting. The method works even when our archetypal language is diagrammatical, has potential ambiguities, is not completely formalized, and does not have semantics for all terms. We illustrate the method with an example from hyperdoctrines and another from synthetic differential geometry.

The paper was produced from the material that I created for the talk below. The slides covers much more ground, but the paper explains the ideas much more clearly. The revised version (2013), that has more sections and more words in the title: "Internal Diagrams and Archetypal Reasoning in Category Theory" (if you prefer a .dvi, it needs this .eps). The original version (just "Internal Diagrams in Category Theory", 2010) is here.

(2013) Internal Diagrams and Archetypal Reasoning in Category Theory
(2010) Internal Diagrams 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 the volcanic ashes many people could not fly to Portugal, and from all these programmed talks only these ended up happening. The abstract was:

When we represent a category C in a type system it becomes a 7-uple, whose first four components - class of objects, Hom, id, composition - are "structure"; the other three components are "properties", and only these last three involve equalities of morphisms.

We can define a projection that keeps the "structure" and drops the "properties" part; it takes a category and returns a "proto-category", and it also acts on functors, isos, adjunctions, proofs, etc, producing proto-functors, proto-proofs, and so on.

We say that this projection goes from the "real world" to the "syntactical world"; and that it takes a "real proof", P, of some categorical fact, and returns its "syntactical skeleton", P-. This P- is especially amenable to diagrammatic representations, because it has only the constructions from the original P --- the diagram chasings have been dropped.

We will show how to "lift" the proto-proofs of the Yoneda Lemma and of some facts about monads and about hyperdoctrines from the syntactical world to the real world. Also, we will show how each arrow in our diagrams is a term in a precise diagrammatic language, and how these diagrams can be read out as definitions. The "downcased" diagrams for hyperdoctrines, in particular, look as diagrams about Set (the archetypical hyperdoctrine), yet they state the definition of an arbitrary hyperdoctrine, plus (proto-)theorems.

(A longer version of the abstract: pdf.)

First official release of the slides (2010jun21, 100 pages): pdf.

Latest version of the slides (109 pages): dvi, pdf.

Some related posts at cat-dist:

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.

Preliminary version (2008jul13), including the message to the mailing list: pdf, dvi, source.

A (long-ish) abstract for a presentation intended for undergrads: pdf, source.
I presented that at the students' colloquium at PUC-Rio, on 2008aug20, 17:00-18:00hs.
I will talk about it again at IMPA, on 2008sep17, 15:30 (pdf).

(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:

Take a set of "worlds", W, and a directed acyclical graph on W, given by a relation R ⊂ W × W. Let's call the functions W → {0,1} "modal truth-values", and the R-non-decreasing functions W → {0,1} "intuitionistic truth-values". If we see W as a topological space with the order topology induced by R, the intuitionistic truth-values correspond to open sets.

The pair (O(W), ⊆) is a Heyting algebra --- meaning that we can interpret intuitionistic propositional logic on it --- and it is a (bigger) DAG, and so we can repeat the above process with it, to generate a (bigger) topological space (O(W), O(O(W))), which is the natural setting for talking about "covers", "saturated covers", and "unions of covers".

This presentation will be focused on understanding all these ideas (and more!), mainly in the case where W has three worlds forming a "V", and R has two arrows pointing downwards. The operation of "taking the union of a cover" turns out to be a particular case of a "Lawvere-Tierney modality"; the double negation is another LT-modality.

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...):

General links

Some articles and books that I'm reading:

(The next sections are links to online texts.)

Category Theory (in general):




Type Theory (mainly the Calculus of Constructions):


Non-standard Analysis, SDG and friends:

Natural Deduction:


Etc (unclassified):

Links to the home pages of some category theorists (and a few type theorists): Abadi, Abramsky, Aczel, Adamek, Altenkirch, Aspinall, Atanassow, Avigad, Awodey, Baez, Barendregt, Barr, Bauer, Beeson, Bell, Berardi, Berg, Birkedal, Blass, Blute, Brown, Bunge, Butz, Caccamo, Cockett, Coquand, Crosilla, Dawson, DePaiva, Diaconescu, Dosen, Dybjer, Egger, Ehrhard, Escardo, Fiore, Funk, Gaucher, Geuvers, Girard, Grandis, Gurevich, Hasegawa, Hermida, Hofmann, Hofstra, Honsell, Hyland, Jacobs, Jardine, Jibladze, Joyal, Kock, Koslowski, Lack, Lamarche, Lambek, Laurent, Lawvere, Leinster, Levy, Longo, Luo, MPJones, Maietti, Mairson, Makkai, Marcos, McLarty, Maltsiniotis, Milner, Mislove; Moerdijk, Moggi, Nelson, Negri, Niefield, Palmgren, Pare, Pastro, Pavlovic, Petric, Phoa, Pierce, Pitts, Plotkin, Porter, Pratt, Pronk, Queiroz, Regnier, Reynolds, Rosebrugh, Rosolini, Sambin, Scedrov, Schalk, Schuster, Scott, ScottD, Seely, Seldin, Selinger, Simmons, Simpson, Spitters, Street, Streicher, Taylor, Tholen, VanOosten, Vickers, Wadler, Weirich, Wells, Wiedijk, Winskel, Wood, Wraith, Yanofsky.

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 a few holes) in August, 2003 and presented the final version - filling out most of the gaps - in February, 2004. Then I spent most of 2004 teaching part-time in an university at the outskirts of Rio (the more realworldish job that I've ever had!), and also finishing 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 - you want to see the slides that I'm working on (it's 2005mar12 as I write this), in which the method for interpreting diagrams and "lifting" them from Set to an arbitrary category with the adequate structure in explained in a really nice way. But if you are really anxious you can get in touch with me.

News (October 2005): I'm giving a series of talks about my PhD thesis at UFF (see http://www.uff.br/grupodelogica/). Expect slides soon and articles not so soon, but as soon as possible.

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 =).

This is the abstract for a talk that I gave at the FMCS2002 in June 8, 2002.

Title: A System of Natural Deduction for Categories

We will present a logic (system DNC) whose terms represent categories, objects, morphisms, functors, natural transformations, sets, points, and functions, and whose rules of deduction represent certain constructive operations involving those entities. Derivation trees in this system only represent the "T-part" (for "terms" and "types") of the constructions, not the "P-part" ("proofs" and "propositions"): the rules that generate functors and natural transformations do not check that they obey the necessary equations. So, we can see derivations in this system either as constructions happening in a "syntactical world", that should be related to the "real world" in some way (maybe through meta-theorems that are yet to be found), or as being just "skeletons" of the real constructions, with the P-parts having been omitted for briefness.

Even though derivations in DNC tell only half of the story, they still have a certain charm: DNC terms represent "types", but a tree represents a construction of a lambda-calculus term; there's a Curry-Howard isomorphism going on, and a tree can be a visual help for understanding how the lambda-calculus term works -- how the data flows inside a given categorical construction. Also, if we are looking for a categorical entity of a certain "type" we can try to express it as a DNC term, and then look for a DNC "deduction" having it as its "conclusion"; the deduction will give us a T-part, and we will still have to go back to the standard language to supply a P-part, but at least the search has been broken in two parts...

The way to formalize DNC, and to provide a translation between terms in its "logic" and the usual notations for Category Theory, is based on the following idea. Take a derivation tree D in the Calculus of Constructions, and erase all the contexts and all the typings that appear in it; also erase all the deduction steps that now look redundant. Call the new tree D'. If the original derivation, D, obeys some mild conditions, then it is possible to reconstruct it -- modulo exchanges and unessential weakenings in the contexts -- from D', that is much shorter. The algorithm that does the reconstruction generates as a by-product a "dictionary" that tells the type and the "minimal context" for each term that appears in D'; by extending the language that the dictionary can deal with we get a way to translate DNC terms and trees -- and also, curiously, with a few tricks more, and with some minimal information to "bootstrap" the dictionary, categorical diagrams written in a DNC-like language.

I also gave a shorter version of that talk at the CMS Summer 2002 Meeting, in June 17.

Slides for the longer talk (45 minutes, 26+3 pages): pdf, ps, dvi, source.

Slides for the shorter talk (one week later, 15 minutes, 16+2 pages: pdf, ps, dvi, source.

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.

Abstract (3 pages and a bit): pdf, ps, dvi, source.

Slides (16 slides): pdf, ps, dvi+eps's, source.

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".

The notes from which the slides were made: ps, dvi+eps's, source.

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.

Thesis (7+116 pages): ps, pdf, dvi+eps's, source.

Slides (15 slides): ps, pdf, dvi+eps's, source.

The diagrams were made with diaglib and the deduction trees with dednat.icn.

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 to help me typeset them. Currently (March 2005) I'm trying to pack that preprocessor and document it; its README is still horribly incomplete. The source code for the examples below is here.

if a functor R has a left adjont
then it preserves limits
The Beck-Chevalley map
in an LCCC
The Frobenius map
in an LCCC
Downcasing A×
(three views)

Technical information: this page was made with blogme; the source is here. I access local copies of papers with the functions defined in my .emacs.papers. The diagrams were made by processing this file (oops, which?) with dednat4, then viewing the resulting dvi file with xdvi and taking screenshots with Xscreenshot-rect.