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The Differential -Calculus UNIVERSITY OF CALGARY The differential λ-calculus: syntax and semantics for differential geometry by Jonathan Gallagher A THESIS SUBMITTED TO THE FACULTY OF GRADUATE STUDIES IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY GRADUATE PROGRAM IN COMPUTER SCIENCE CALGARY, ALBERTA May, 2018 c Jonathan Gallagher 2018 Abstract The differential λ-calculus was introduced to study the resource usage of programs. This thesis marks a change in that belief system; our thesis can be summarized by the analogy λ-calculus : functions :: @ λ-calculus : smooth functions To accomplish this, we will describe a precise categorical semantics for the dif- ferential λ-calculus using categories with a differential operator. We will then describe explicit models that are relevant to differential geometry, using categories like Sikorski spaces and diffeological spaces. ii Preface This thesis is the original work of the author. This thesis represents my attempt to understand the differential λ-calculus, its models and closed structures in differential geometry. I motivated to show that the differential λ-calculus should have models in all the geometrically accepted settings that combine differential calculus and function spaces. To accomplish this, I got the opportunity to learn chunks of differential ge- ometry. I became fascinated by the fact that differential geometers make use of curves and (higher order) functionals everywhere, but use these spaces of func- tions without working generally in a closed category. It is in a time like this, that one can convince oneself, that differential geometers secretly wish that there is a closed category of manifolds. There are concrete problems that become impossible to solve without closed structure. One of the first geometers to propose a generalization of manifolds was Chen, who wanted to iterate a path space construction to formalize how iterated integrals can be computed. These spaces, now called Chen spaces, and their closely related counterparts, diffeological spaces and Sikorski spaces have many properties that make them categorically more well behaved (they are Cartesian closed categories with all limits and colimits). Yet, differential geometers do not seem to have simply flocked to these generalized smooth spaces; the reason is that they have badly behaved limits – that is, they admit non-transverse intersections. Examples of badly behaved limits come from catastrophe theory: one has two submanifolds M ,N of some manifold Z , and in Z there is an intersection of M ,N where the tangent vectors at those intersections do not span Z . When taking the pullback of M ,N one obtains a space B with degenerate behaviour: there can be iii places where the dimension of B collapses leading to catastrophic behaviour. I find it exciting that these sorts of issues come up when trying to construct models of the differential λ-calculus. It seems perhaps important that a simple coherence required to model the differential λ-calculus in settings for differential geometry brings up the kind of issues with “bad” limits in differential geometry, which is the subject of part of chapter6. Chapter2 revisits the term logic developed in (Blute, Cockett, and Seely, 2009) from the point of view of a representable multicategory. The view of representable multicategory I took is based on conversations and an unpublished paper with Robin Cockett on Monoidal Turing Categories. The separation of the product from the rules of differentiation is new. The equivalence of categories between the category of differential theories and the category of Cartesian differential categories is new. The approach to type theories I take in regarding sytax as a presentation of a Cartesian multicategory is heavily influenced and shaped by the opportunity to serve as a teaching assistant for the AARMS summer school “Higher category theory and categorical logic” which was taught by Mike Shulman and Peter LeFanu Lumsdaine. Chapter3 is based on reading (Bucciarelli, Ehrhard, and Manzoneto, 2010). I discussed a bit of their paper with the third named author, Giulio Manzonetto. Their paper has no completeness theorem, and Robin Cockett suggested that the chain rule might not hold for theories with function symbols. I wrote this chap- ter to complete the story. (Bucciarelli, Ehrhard, and Manzoneto, 2010) showed soundness for the differential λ-calculus in Cartesian closed categories for a free theory with no function symbols. The soundness theorem for the differential λ-calculus with function symbols and equations is new, and the completeness theorem is new. Chapter4 is based on a review for a journal version of a paper Robin Cockett and I wrote, and have submitted to MSCS on the untyped differential λ-calculus. The reviewer wrote that it was not obvious that the type theory I used was related to the differential λ-calculus. I constructed the interpretations to ensure that they were indeed equivalent. However, this review prompted me to learn about some of the more delicate aspects of constructing categorical models for intensional cal- culi and untyped calculi. Reading papers by Scott, Koymans, Lambek, Barendregt, iv Meyers, and others leads to a much richer and nuanced history than one might expect to find. In the end, I think the objection was based on a misunderstanding, but I think this detour was important because it lead to a bit more discernment in constructing models. Chapter5 is based on a talk I gave in the Peripatetic Seminar at the University of Calgary, in which I proved that synthetic differential geometry gives a model of the differential λ-calculus. Robin Cockett verbally suggested at the end of the talk, that my proof did not involve the particulars of synthetic differential geometry, and seemed to revolve around the strength. I then spent some time analyzing the strength, and proved that indeed, a property of the strength is crucial for modelling the differential λ-calculus. Chapter6 arose from trying to understand two seemingly separate things. First was the work of Nishimura, and second was as to whether diffeological spaces have any non-trivial tangent structure. I had never managed to adequately figure out either, until I read Leung’s paper (Leung, 2017). Leung proved that tangent structures on a category X are equivalent to monoidal functors from Weil algebras (with tensor given by coproduct) into the functor category [X,X] (with tensor given by functor composition). Nishimura had been asking for what he called Weil prolongation, and I realized that Leung’s description of tangent structure could be immediately reformulated as Weil prolongation using actegories which give a categorically coherent version of Nishimura’s notion of prolongation. (Kock, 1986) also used Weil prolongation to obtain tangent structure under the name semidirect product of categories instead of actegories. I met the notion of actegory while working with an MSc student of Robin Cockett, named Masuka Yeasin, on linear protocols for message passing types. Garner also used actegories in characterizing tangent categories as enriched categories (Garner, 2018). In this thesis, I try to attribute the theorems Garner proved, to Garner. The intended audience for this thesis is people interested in the differential λ-calculus and people interested in how to obtain closed categories for differential geometry. I try to take little for granted at the level of theories, but at the level of category theory, I do take some basic material for granted. v Acknowledgments I would like to thank the PIMS organization for funding the peripatetic seminar which allowed visitors to interact with our group on differential and tangent structure. I would like to thank NSERC who, through my supervisor Robin Cockett, allowed me to travel and disseminate my research. I would like to thank my supervisor Robin Cockett for directing my thesis. I would also like to thank Kristine Bauer for years of open minded conversations and support. I would like to thank the whole peripatetic seminar with special thanks to Ben MacAdam, Chad Nester, Prashant Kumar, JS Lemay, Brett Giles, Pavel Hrubes, Geoff Cruttwell, Matthew Burke, Ximo Diaz-Boils, Masuka Yeasin, Priyaa Srinivasan, Daniel Satanove, and Cole Comfort as well as Pieter Hofstra, Rick Blute, and Robert Seely, summer visitor Matvey Soloviev, and labmates James King, Chris Jarabek, Eric Lin, and Sutapa Dey for useful conversations and great times during my stay in Calgary. vi For Mom, Dad, Grandma, Poppy, Grandma, Andrew, Jakob, BJ, Josh, and Veronika. In other words, that the so called involuntary circulation of your blood is one continuous process with the stars shining. If you find out that it’s you who circulates your blood you will at the same moment find out that you are shining the sun. Because your physical organism is one continuous process with ev- erything else that is going on. Just as the waves are continous with the ocean. Your body is continous with the total energy system of the cosmos and its all you. Only you’re playing the game that you’re only this bit of it. – Alan Watts vii Table of Contents Abstract ................................................. ii Preface .................................................. iii Acknowledgments ......................................... vi Dedication ............................................... vii Table of Contents .......................................... viii List of Tables .............................................. x 1 Introduction ..........................................
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