Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich Dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 ABSTRACT Diagrammatics in Categorification and Compositionality by Dmitry Vagner Department of Mathematics Duke University Date: Approved: Ezra Miller, Supervisor Lenhard Ng Sayan Mukherjee Paul Bendich An abstract of a dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy in the Department of Mathematics in the Graduate School of Duke University 2019 Copyright c 2019 by Dmitry Vagner All rights reserved Abstract In the present work, I explore the theme of diagrammatics and their capacity to shed insight on two trends|categorification and compositionality|in and around contemporary category theory. The work begins with an introduction of these meta- phenomena in the context of elementary sets and maps. Towards generalizing their study to more complicated domains, we provide a self-contained treatment|from a pedagogically novel perspective that introduces almost all concepts via diagrammatic language|of the categorical machinery with which we may express the broader no- tions found in the sequel. The work then branches into two seemingly unrelated disciplines: dynamical systems and knot theory. In particular, the former research defines what it means to compose dynamical systems in a manner analogous to how one composes simple maps. The latter work concerns the categorification of the slN link invariant. In particular, we use a virtual filtration to give a more diagrammatic reconstruction of Khovanov-Rozansky homology via a smooth TQFT. Finally, the work culminates in a manifesto on the philosophical place of category theory. iv Acknowledgements I am grateful to the following institutions • The city of Durham: for being a both cozy and exciting place to live. • Duke University: for being my intellectual incubator and patron. • Cocoa Cinnamon: for being my second home during graduate school. I am grateful to the following academic relationships • My meta-advisor Ezra Miller: for providing me with both guidance and free- dom, for welcoming me into your home and family, and for being simultaneously a mentor and friend. • My research mentor and kindred spirit David Spivak: for opening my mind to new dimensions of thought, shepherding me into the world of category theory, and taking me on some of the most exciting conversational joy rides. • My collaborator and friend Michael Abel: for thoroughly answering any ques- tion, be it silly or far afield, making doctoral research an experience equal parts exciting and warm. • Lenny Ng: for helping me become a more sophisticated mathematician and for introducing me to the world of knot theory as a context for my categorical ambitions. • Sayan Mukherjee: for facilitating my becoming-computer-scientist and open- ing my eyes to the (slightly) more concrete applications of my extravagantly abstract interests. v • Paul Bendich: for providing a spark of cultural vibrancy in the Duke mathe- matics culture, helping me get deeper exposure into data science, and serving on my defense committee. • Andrew Janiak: for giving me the opportunity to formally study philosophy while allowing me to roam freely in its deep end. • Michael Hardt and Frederic Jameson: for introducing me to mind bending ideas that a mathematics graduate student rarely has the opportunity to encounter. • John Baez: for sharing ideas that furnished my mind with new architectural possibility and sparked a rennaisance in my interest for pursuing mathematics • Mike Shulman and Aleks Kissinger: for respectively mentoring me during the deeply influential homotopy type theory and applied category research schools. • Aaron Mazel-Gee, Saul Glasman, and Patrick Schultz: for teaching me various notions of higher category theory. • Radmila Sazdanovic, Tye Lidman, Lev Rozansky, Dave Rose and the broader Triangle Topology community: for providing valuable mentorship. • Bjorn Sandstede: for being my mathematical guide throughout college and maintianing my interest in the field. • Tom Goodwillie: for introducing me to the kind of psychedelic math to which I would dedicate my graduate career. • Richard Piccirillo: for supporting my mathematical ambition and for living up to your reputation as Ranney's most respected faculty member. vi I am grateful to my loved ones • my friends: for enriching my life with both companionship and adventure • my non-human companions: Chansey, Umka, Loloz, and Theory: for bringing me so much joy and warmth. • my darling Thu. c Anh: for creating a home together with me and making my life far more joyous, meaningful, and cozy • my parents, Anna and Leon: for their unconditional love, deep connection, and unwavering commitment, both spiritual and material, to my upbringing and education • my grandpa Boris: for instilling in me an early love of doing mathematics and taking long walks • my late grandmother Yelena, to whom I dedicate this thesis: for an influence on my life that is so profound I haven't the language to articulate it. vii Contents Abstract iv Acknowledgementsv 1 Introduction: Aesthetics of Abstraction1 2 Compositionality, Categorification in Sets4 2.1 Compositionality of Maps........................6 2.1.1 Composites............................6 2.1.2 Compositions...........................6 2.1.3 Inverses.............................. 10 2.1.4 Products.............................. 12 2.1.5 Swaps............................... 13 2.1.6 Flows............................... 15 2.1.7 Forks................................ 18 2.1.8 Algebras.............................. 24 2.2 Categorification of Counting....................... 27 2.2.1 Equality.............................. 28 2.2.2 Multiplication........................... 28 viii 2.2.3 Exponentiation.......................... 30 2.2.4 Addition.............................. 31 2.2.5 Combinatorics........................... 36 3 Categorical Technology 38 3.1 Classical Category Theory........................ 38 3.1.1 Categories............................. 38 3.1.2 Functors.............................. 41 3.1.3 Naturality............................. 43 3.1.4 Universality............................ 44 3.1.5 Exponentials........................... 48 3.1.6 Theories.............................. 49 3.2 Monoidal Category Theory........................ 53 3.2.1 Tensors.............................. 53 3.2.2 Braids............................... 59 3.2.3 Strength.............................. 62 3.2.4 Monoids.............................. 65 3.2.5 Scarcity.............................. 69 3.2.6 Bimonoids............................. 78 ix 3.2.7 Closure.............................. 79 3.2.8 Duals............................... 87 4 Compositional Systems 90 4.1 Open Dynamical Systems Preliminaries................. 90 4.1.1 Motivation............................. 90 4.1.2 Motivating example........................ 92 4.1.3 Preliminary Notions....................... 93 4.1.4 Monoidal categories and operads................ 93 4.1.5 Typed sets............................. 95 4.1.6 Open systems........................... 98 4.2 Compositionality with Wiring Diagrams................ 102 4.2.1 Chains............................... 102 4.2.2 Flows............................... 105 4.2.3 The Operad of Wiring Diagrams................ 110 4.3 Compositional Dynamical Systems................... 122 4.3.1 The Algebra of Open Dynamical Systems............ 122 4.3.2 The Subalgebra of Open Linear Systems............ 126 5 Categorified Invariants 134 x 5.1 Quantum Link Invariants Preliminaries................. 134 5.1.1 Isotopy Types........................... 134 5.1.2 Resolution Cubes......................... 136 5.2 Categorification with Linear Structures................. 138 5.2.1 Categorifying Natural Numbers................. 138 5.2.2 Categorifying Polynomials.................... 140 5.2.3 DG Categories and Twisted Complexes............. 143 5.3 Categorified Link Invariants....................... 153 5.3.1 Khovanov Homology....................... 153 5.3.2 Khovanov-Rozansky Homology................. 155 5.3.3 Matrix factorizations....................... 157 5.3.4 Smooth Resolutions for MOY Graphs.............. 168 5.3.5 Smooth Resolution Meta-Cube................. 186 5.3.6 Recovering Khovanov-Rozansky Homology........... 204 6 Conclusion: Broader Cosmologies of Structure 212 6.1 The Notionographic Perspective..................... 212 6.2 The Role of Mathematics in Notionography.............. 214 6.3 The Prismatic Notionography of Categories.............. 217 xi Bibliography 228 Biography 231 xii Chapter 1 Introduction: Aesthetics of Abstraction The present work is obsessed with the visual language used to to render and manip- ulate mathematical notions. This yoga, which we call diagrammatics, was formally introduced in [JS91] by Joyal and Street to mechanize reasoning within monoidal cat- egories. In departure from tradition, we develop from the ground up the entire theory of categories with diagrammatics as first principles. Diagrammatics are in some sense a higher-dimensional|typically, given the dimensionality of our communication me- dia, planar|notation for formal symbolics in place of the traditional one-dimensional notation for mathematical expressions. Such