Advanced Topics in Category Theory

Part I - Lectures

Part II - Research Seminar

Use the signup sheet to indicate your chosen topic. Access is via your Cambridge google account.

Here is a list of suggested papers:

• Type theory
  • Eric Finster and Samuel Mimram (2017). "A type-theoretical definition of weak ω-categories", Proceedings of LICS 2017.
  • Martin Hoffman and Thomas Streicher (1994). "The groupoid model refutes uniqueness of identity proofs", Proceedings of LICS 1994.
  • Roberto Bruni and Ugo Montanari (1999). "Cartesian closed double categories, their λ-notation, and the π-calculus", Proceedings of LICS 1999.
  • Richard Garner (2009). "Two-dimensional models of type theory", MSCS 19(4).
  • Michael Shulman (2021). "A practical type theory for symmetric monoidal categories".
  • Multiple authors. "Homotopy Type Theory: Univalent Foundations of Mathematics".
• Game theory
  • Joe Bolt, Jules Hedges and Philipp Zahn (2021). "Bayesian open games", Compositionality 5(9).
  • Neil Ghani, Jules Hedges, Viktor Winschel and Philipp Zahn (2016). "Compositional game theory", Proceedings of LICS 2018.
• Machine learning
  • Brendan Fong, David Spivak and Remy Tuyeras (2019). "Backprop as a functor: a compositional perspective on supervised learning", Proceedings of LICS 2019.
  • Geoff Crutwell, Bruno Gavranovic, Neil Ghani, Paul Wilson and Fabio Zanasi (2021). "Categorical foundations of gradient based learning", Proceedings of ESOP 2022.
  • Dan Shiebler, Bruno Gavranovic and Paul Wilson (2021). "Category theory in machine learning".
  • Brendan Fong and Michael Johnson (2019). "Lenses and learners", Proceedings of BX 2019.
  • Matteo Capucci, Bruno Gavranovic, Jules Hedges and Eigil Fjeldgren Rischel (2021). "Towards foundations of categorical cybernetics", Proceedings of ACT 2021.
• Natural language processing and cognition
  • Bob Coecke, Mehrnoosh Sadrzadeh and Stephen Clark (2010). "Mathematical foundations for a compositional distributional model of meaning", Lambek Festschrift (Linguistic Analysis).
  • Mehrnoosh Sadrzadeh, Stephen Clark and Bob Coecke (2013). "The Frobenius anatomy of word meanings", Journal of Logic and Computation 23(6).
  • Joe Bolt, Bob Coecke, Fabrizio Genovese, Martha Lewis, Dan Marsden and Robin Piedeleu (2016). "Interacting conceptual spaces", Chapter in Conceptual Spaces: Elaborations and Applications.
• Database theory
  • David Spivak (2012). "Functorial data migration", Information and Computation 217.
• Programming languages
  • Jules Hedges (2018). "Lenses for philosophers", blog post.
  • Michael Abbott, Thorsten Altenkirch and Neil Ghani (2003). "Categories of containers", Proceedings of FOSSACS 2003.
  • Mario Alvarez-Picallo, Dan Ghica, David sprunger and Fabio Zanasi (2021). "Functorial string diagrams for reverse-mode automatic differentiation",
• Quantum computing
  • Samson Abramsky and Bob Coecke (2004). "A categorical semantics of quantum protocols", Proceedings of LICS 2004.
  • Bob Coecke and Ross Duncan (2011). "Interacting quantum observables: categorical algebra and diagrammatics", New Journal of Physics 13(4).
  • Ross Duncan, Aleks Kissinger, Simon Perdrix and John van de Wetering (2020). "Graph-theoretic simplification of quantum circuits with the ZX-calculus", Quantum 4.
  • Aleks Kissinger and John van der Wetering (2020). "Reducing the number of non-Clifford gates in quantum circuits", Physical Review A 102.
  • Alexandre Clément, Nicolas Heurtel, Shane Mansfield, Simon Perdrix and Benoit Valiron (2023). "A complete equational theory for quantum circuits", Proceedings of LICS 2023.
  • Mathieu Huot and Sam Staton (2019). "Universal properties in quantum theory", EPTCS 287(213-223).
• String diagrams and rewriting
  • Stephen Lack and Pawel Sobocinski (2004). "Adhesive categories", Proceedings of FOSSACS 2004.
  • Peter Selinger, "A survey of graphical languages for monoidal categories" (2009). Springer Lectures Notes in Physics 813(289-255).
  • Antonin Delpeuch and Jamie Vicary (2018). "Normalization for planar string diagrams and a quadratic equivalence algorithm", Logical Methods In Computer Science 18(1).
  • David Reutter and Jamie Vicary (2019). "High-level methods for homotopy construction in associative n-categories", Proceedings of LICS 2019.
  • Filippo Bonchi, Fabio Gadducci, Aleks Kissinger, Pawel Sobocinski and Fabio Zanasi (2016). "Rewriting modulo symmetric monoidal structure", Proceedings of LICS 2006.
  • Samuel Mimram (2014). "Towards 3-dimensional rewriting theory", Logical Methods in Computer Science 10(2).
• Resources and processes
  • Bob Coecke, Tobias Fritz and Robert Spekkens (2016). "A mathematical theory of resources", Information and Computation 250.
  • Filippo Bonchi, Pawel Sobocinski and Fabio Zanasi (2015). "Full abstraction for signal flow graphs", Proceedings of POPL 2015.
  • John Baez, Xiaoyan Li, Sophie Libkind, Nathaniel D. Osgood and Eric Redekopp (2022). "A categorical framework for modeling with stock and flow diagrams", in Mathematics of Public Health, Fields Institute Communications Book Series 88.
  • John Baez, Fabrizio Genovese, Jade Master and Michael Shulman (2021). "Categories of nets", Proceedings of LICS 2021.
• Abstract structures
  • Filippo Bonchi, Pawel Sobocinski and Fabio Zanasi (2014). "Interacting Hopf algebras", Journal of Pure and Applied Algebra 221(1).
  • Brendan Fong and David Spivak (2019). "Hypergraph categories", Journal of Pure and Applied Algebra 223(11).
  • Brendan Fong (2015). "Decorated cospans", Theory and Applications of Categories 30(33).
  • John Baez and Mike Stay (2009). "Physics, topology, logic and computation: a Rosetta Stone", in New Structures for Physics, Lecture Notes in Physics 813.
  • Robin Houston (2007). "Finite products are biproducts in a compact closed category", Journal of Pure and Applied Algebra 212(2).