A study of categories of algebras and coalgebras

This thesis is intended to help develop the theory of coalgebras by, first, taking classic theorems in the theory of universal algebras and dualizing them and, second, developing an internal logic for categories of coalgebras. We begin with an introduction to the categorical approach to algebras and the dual notion of coalgebras. Following this, we discuss (co)algebras for a (co)monad and develop a theory of regular subcoalgebras which will be used in the internal logic. We also prove that categories of coalgebras are complete, under reasonably weak conditions, and simultaneously prove the well-known dual result for categories of algebras. We close the second chapter with a discussion of bisimulations in which we introduce a weaker notion of bisimulation than is current in the literature, but which is well-behaved and reduces to the standard definition under the assumption of choice. The third chapter is a detailed look at three theorem’s of G. Birkhoff [Bir35, Bir44], presenting categorical proofs of the theorems which generalize the classical results and which can be easily dualized to apply to categories of coalgebras. The theorems of interest are the variety theorem, the equational completeness theorem and the subdirect product representation theorem. The duals of each of these theorems is discussed in detail, and the dual notion of “coequation” is introduced and several examples given. In the final chapter, we show that first order logic can be interpreted in categories of coalgebras and introduce two modal operators to first order logic to allow reasoning about “endomorphism-invariant” coequations and bisimulations internally. We also develop a translation of terms and formulas into the internal language of the base category, which preserves and reflects truth. Lastly, we introduce a Kripke-Joyal style semantics for L(EG ), as well as a pointwise semantics which reflects the intuition of coequation forcing at a point or subset of a coalgebra. Acknowledgments I have been fortunate to have two advisors on this dissertation. I first became interested in the subject thanks to Dana Scott, who helped guide the questions and suggested the Birkhoff’s theorem research in particular. Steve Awodey taught me everything I know about category theory, but I am grateful anyway. Both advisors helped my writing immensely, in addition to guiding my research, and I am thankful for their patience and wisdom. When Dana first suggested I look into coalgebras, he pointed me to Vicious Circles, by Jon Barwise and Larry Moss. Since that book was the start of my study of coalgebras, it seemed only fair that Larry Moss should have to read this dissertation. He graciously agreed to be my outside reader. I am grateful for the advice he and Jeremy Avigad gave as members of my committee. My research has benefited through discussions and correspondence with many people, including Peter Aczel, Jǐŕı Adámek, Andrej Bauer, Lars Birkedal, Steve Brookes, Corina Ĉırstea, Federico do Marchi, Neil Ghani, Jeremy Gibbons, Peter Gumm, Bart Jacobs, Alexander Kurz, Bill Lawvere, John Reynolds, Tobias Schröder James Worrell and Jaap van Oosten and others I’m sure to have missed here. I also want to thank the organizers of the Coalgebraic Methods for Computer Science workshop for providing a great opportunity to meet and discuss our research. On a more personal note, I could not have completed this work without the extraordinary patience and generosity of my wife, Ling Cheung. In fact, I am the rare husband who’s also grateful for the extended visits of his mother-in-law, Siu Kai Lam. She helped out considerably when two graduate students were overwhelmed with a newborn, Quincy Prescott Hughes. I also enjoyed the distractions from my work, including Penguins hockey, regular fishing trips with Dirk Schlimm, exciting demolition derbies at New Alexandria and captivating and suspenseful games of Peeka-boo with Quincy.