Table Of ContentA Study of Categories of Algebras
and Coalgebras
Jesse Hughes
May, 2001
Department of Philosophy
Carnegie Mellon University
Pittsburgh PA 15213
Thesis Committee
Steve Awodey, Co-Chair
Dana Scott, Co-Chair
Jeremy Avigad
Lawrence Moss, Indiana University
Submitted in partial ful(cid:12)llment of the requirements for the degree of Doctor of
Philosophy
Abstract
This thesis is intended to help develop the theory of coalgebras by, (cid:12)rst, 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 de(cid:12)nition under the assumption
of choice.
The third chapter is a detailed look at three theorem’s of G. Birkho(cid:11) [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
theoremsofinterestarethevarietytheorem,theequationalcompletenesstheoremand
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 (cid:12)nal chapter, we show that (cid:12)rst order logic can be interpreted in categories
ofcoalgebrasandintroduce twomodaloperatorsto(cid:12)rstorderlogictoallowreasoning
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 andre(cid:13)ects truth. Lastly, we introduce aKripke-Joyal style
semantics for L(E ), as well as a pointwise semantics which re(cid:13)ects the intuition of
(cid:0)
coequation forcing at a point or subset of a coalgebra.
Acknowledgments
I have been fortunate to have two advisors on this dissertation. I (cid:12)rst became in-
terested in the subject thanks to Dana Scott, who helped guide the questions and
suggested the Birkho(cid:11)’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 (cid:12)rst suggested I look into coalgebras, he pointed me to Vicious Cir-
cles, 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 bene(cid:12)ted through discussions and correspondence with many
people, including Peter Aczel, Ji(cid:20)r(cid:19)(cid:16) Ada(cid:19)mek, Andrej Bauer, Lars Birkedal, Steve
Brookes, Corina C^(cid:16)rstea, Federico do Marchi, Neil Ghani, Jeremy Gibbons, Peter
Gumm, Bart Jacobs, Alexander Kurz, Bill Lawvere, John Reynolds, Tobias Schro(cid:127)der
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 (cid:12)shing trips with Dirk Schlimm, exciting
demolitionderbiesatNewAlexandriaandcaptivatingandsuspenseful gamesofPeek-
a-boo with Quincy.
iii
Contents
Introduction 1
Chapter synopsis 4
Chapter 1. Algebras and coalgebras 7
1.1. Algebras and coalgebras for an endofunctor 7
1.2. Structural features of E(cid:0) and E 16
(cid:0)
1.3. Subalgebras 22
1.4. Congruences 27
1.5. Initial algebras and (cid:12)nal coalgebras 32
Chapter 2. Constructions arising from a (co)monad 47
2.1. (Co)monads and (co)algebras 47
2.2. Subcoalgebras 61
2.3. Subcoalgebras generated by a subobject 73
2.4. Limits in categories of coalgebras revisited 77
2.5. Bisimulations 89
2.6. Coinduction and bisimulations 105
2.7. n-simulations 109
Chapter 3. Birkho(cid:11)’s variety theorem 113
3.1. The classical theorem 114
3.2. A categorical approach 115
3.3. Categories of algebras 125
3.4. Uniformly Birkho(cid:11) categories 127
3.5. Deductive closure 135
3.6. The coalgebraic dual of Birkho(cid:11)’s variety theorem 140
3.7. Uniformly co-Birkho(cid:11) categories 153
3.8. Invariant coequations 162
3.9. Behavioral covarieties and monochromatic coequations 168
Chapter 4. The internal logic of E 175
(cid:0)
4.1. Preliminary results 175
4.2. Transfer principles 186
v
vi CONTENTS
4.3. A Kripke-Joyal style semantics 198
4.4. Pointwise forcing of coequations 200
Concluding remarks and further research 205
Appendix A. Preliminaries 209
A.1. Notation 209
A.2. Factorization systems 209
A.3. Predicates and Subobjects 212
A.4. Relations 214
A.5. Monads and comonads 216
Appendix. Bibliography 219
Appendix. Index 223
Introduction
The theory of universal algebras has been well-developed in the twentieth cen-
tury. The theory has also proved especially fruitful, with early results (like Birkho(cid:11)’s
variety theorem) providing a basis for model theory and other results providing an
abstract understanding of familiar principles of induction, recursion and freeness.
The theory of coalgebras is considerably younger and less well developed. Coalgebras
arise naturally, as Kripke models for modal logic, as automata and objects for object
oriented programming languages in computer science, etc. Hence, one would like a
uni(cid:12)ed theory of coalgebras to play a role analogous to that of the theory of algebras.
This goal is aided by the duality between algebras and coalgebras. Statements about
categories of algebras yield dual statements about categories of coalgebras. One can
then investigate whether there are reasonable assumptions about the categories of
coalgebras that yield the dual theorems.
Algebras, intheircommonest form, canbeunderstoodasasettogetherwithsome
operations on the set. In other words, algebras are structures for a signature. The
termalgebrasareexamplesoffreealgebras, wherefreeness iseasilyexpressed interms
of adjoint functors. Such free algebras (which are initial objects in a related category
of algebras) come with the proof principle of induction, which can be understood in
terms of minimality. That is, the principle of induction is equivalent to the property
thatanalgebrahasnonon-trivialsubalgebras. Thepropertyofde(cid:12)nitionbyrecursion
is exactly the property that an algebra is an initial object. Thus, these familiar topics
of universal algebra are well-suited for a categorical setting. We can use the tools
of category theory to investigate freeness, induction and recursion as special cases
of adjointness, minimality and initiality, respectively. In particular, these algebraic
properties can be represented as standard categorical properties applied to categories
of algebras (in which the structure of the category leads to the well-known algebraic
properties).
Coalgebras can also be regarded as a set together with certain operations on it,
but with a key di(cid:11)erence. Where an algebra is intended to model combinatorial op-
erations, a coalgebra models a set with various unary operations whose codomain is a
(typically) more complex structure. These operations can be viewed as \destructors"
which take an element of the coalgebra to its constituent parts. Compare this view
1
2 INTRODUCTION
with the notion that an algebras operations give a means (not necessarily unique) of
\constructing" an element out of a tuple.
Consider, for instance, a set S of A-labeled binary trees1 which is closed under
the \childOf" relation. That is, if x 2 S, then both the left and right subtrees of x
(if they exist) are also in S. Then S has a natural coalgebraic structure consisting
of three destructor functions. Given any x 2 S, we may ask for the label of x. We
may also ask for the left child or right child of x, assuming that there is an \error
state" which can be returned if x has no such child . These three structure maps
de(cid:12)ne a signature (cid:6) for a category of coalgebras in the same way that a set with
some combinatorial operations de(cid:12)ne a signature for a category of algebras (i.e., a
similarity type). Any set X, together with three operations,
a:X //A;
l:X //X +1;
r:X //X +1;
is a (cid:6)-coalgebra. Equivalently, any set X with a single map
ha; l; ri:X //A(cid:2)(X +1)(cid:2)(X +1)
is a coalgebra of the same type as our set S of binary trees. Indeed, any such
structured set can be regarded as a set of trees itself.
We can use the theory of algebras in order to develop the theory of coalgebras.
The duality is apparent in the distinguished initial algebra/(cid:12)nal coalgebra. The
initial algebra is the initial (i.e., \least") (cid:12)xed point of the associated functor, while
the (cid:12)nal coalgebra is the (cid:12)nal (i.e., \greatest") (cid:12)xed point. The initial algebra comes
equipped with principles of recursion and induction, while the (cid:12)nal coalgebra satis(cid:12)es
theprinciples ofcorecursionandcoinduction, thatis, principles whichareappropriate
to collections of non-well-founded structures. Intuitively, the elements of the initial
algebra are those which can be constructed from some set of basic elements in a (cid:12)nite
number ofsteps, while the elements of the (cid:12)nal coalgebra are allof those structures of
theappropriate signature, including those forwhich no(cid:12)nite construction is apparent
(think of the distinction between well-founded binary trees and non-well-founded
binary trees). Of course, the extent to which this intuition is appropriate depends
on the functor (i.e., signature) at hand. But the point of this comparison remains:
To construct a theory of coalgebras, one may take the theory of algebras and dualize
the central theorems. One then interprets the result in order to make sense of it {
the traditional statement of the principle of coinduction, for instance, does not make
apparent its duality with induction. Similarly, the description of a cofree coalgebra
1In this example, we do not require that a tree have both a left and a right child if it has any
children.
