@Article{kavanagh_2020:_param_fixed_point,
author = {Kavanagh, Ryan},
title = {Parametrized Fixed Points and Their Applications To
Session Types},
volume = 352,
pages = {149-172},
doi = {10.1016/j.entcs.2020.09.008},
abstract = {Parametrized fixed points are of particular interest
to denotational semantics and are often given by
``dagger
operations''~\cite{bloom_esik_1993:_iterat_theor,bloom_esik_1996:_fixed_point_operat,bloom_esik_1995:_some_equat_laws_initial}.
Dagger operations that satisfy the Conway
identities~\cite{bloom_esik_1996:_fixed_point_operat}
are particularly useful, because these identities
imply a large class of identities used in semantic
reasoning. We generalize existing techniques to
define dagger operations on $\omega$-categories and
on $\textbf{O}$-categories. These operations enjoy
a 2-categorical structure that implies the Conway
identities. We illustrate these operators by
considering applications to the semantics of
session-typed languages.},
date = {2020-10},
issn = {1571-0661},
issuetitle = {The 36th Mathematical Foundations of Programming
Semantics Conference, 2020},
journaltitle = {Electronic Notes in Theoretical Computer Science},
keywords = {O-categories, $\omega$-categories, fixed points,
dagger operations, Conway identities,
rak-conference},
}
@Book{bloom_esik_1993:_iterat_theor,
author = {Bloom, Stephen L. and {\'E}sik, Zolt{\'a}n},
title = {Iteration Theories},
publisher = {Springer-Verlag Berlin Heidelberg},
abstract = {This monograph contains the results of our joint
research over the last ten years on the logic of the
fixed point operation. The intended audience
consists of graduate students and research
scientists interested in mathematical treatments of
semantics. We assume the reader has a good
mathematical background, although we provide some
preliminary facts in Chapter 1. Written both for
graduate students and research scientists in
theoretical computer science and mathematics, the
book provides a detailed investigation of the
properties of the fixed point or iteration
operation. Iteration plays a fundamental role in the
theory of computation: for example, in the theory of
automata, in formal language theory, in the study of
formal power series, in the semantics of flowchart
algorithms and programming languages, and in
circular data type definitions. It is shown that in
all structures that have been used as semantical
models, the equational properties of the fixed point
operation are cap-tured by the axioms describing
iteration theories. These structures include ordered
algebras, partial functions, relations, finitary and
infinitary regular languages, trees, synchronization
trees, 2-categories, and others.},
date = 1993,
doi = {10.1007/978-3-642-78034-9},
isbn = {978-3-642-78034-9},
pagetotal = {xv+630~\ppno},
series = {EATCS Monographs on Theoretical Computer Science},
subtitle = {The Equational Logic of Iterative Processes},
}
@Article{bloom_esik_1995:_some_equat_laws_initial,
author = {Bloom, Stephen L. and {\'E}sik, Zolt{\'a}n},
title = {Some Equational Laws of Initiality in {2CCC}'s},
volume = 6,
number = 2,
pages = {95-118},
doi = {10.1142/S0129054195000081},
abstract = {A result obtained in Ref. 2 for least prefixed
points in order enriched cartesian closed categories
is generalized to initiality in cartesian closed
2-categories. In brief, the result is that if a
fixed point operation on a cartesian closed
2-category is defined by initiality, then under a
mild condition, the operation satisfies the
cartesian Conway identities and the abstraction
identity. In addition, we show that the operation
satisfies the power identities, and hence, except
for the law $a^{**}=a^*$, the analogues of Conway's
classical identities for the regular sets.},
date = 1995,
journaltitle = {International Journal of Foundations of Computer
Science},
}
@Article{bloom_esik_1996:_fixed_point_operat,
author = {Bloom, Stephen L. and {\'E}sik, Zolt{\'a}n},
title = {Fixed-Point Operations on {ccc's}. Part I},
volume = 155,
number = 1,
pages = {1-38},
doi = {10.1016/0304-3975(95)00010-0},
date = {1996-02-25},
issn = {0304-3975},
journaltitle = {Theoretical Computer Science},
}