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Second-Order Algebraic Theories
Second-order universal algebra and second-order equational logic respectively provide a model theory
and a formal deductive system for languages with variable binding and parameterised metavariables.
This dissertation completes the algebraic foundations of second-order languages from the viewpoint of
categorical algebra.
In particular, the dissertation introduces the notion of second-order algebraic theory. A main role in
the definition is played by the second-order theory of equality M, representing the most elementary
operators and equations present in every second-order language. We show that M can be described
abstractly via the universal property of being the free cartesian category on an exponentiable object.
Thereby, in the tradition of categorical algebra, a second-order algebraic theory consists of a cartesian
category M and a strict cartesian identity-on-objects functor M: M →M that preserves the universal
exponentiable object of M.
At the syntactic level, we establish the correctness of our definition by showing a categorical equivalence
between second-order equational presentations and second-order algebraic theories. This equivalence,
referred to as the Second-Order Syntactic Categorical Type Theory Correspondence, involves distilling
a notion of syntactic translation between second-order equational presentations that corresponds to the canonical notion of morphism between second-order algebraic theories. Syntactic translations provide a mathematical formalisation of notions such as encodings and transforms for second-order languages.
On top of the aforementioned syntactic correspondence, we furthermore establish the Second-Order
Semantic Categorical Type Theory Correspondence. This involves generalising Lawvere’s notion of
functorial model of algebraic theories to the second-order setting. By this semantic correspondence,
second-order functorial semantics is shown to correspond to the model theory of second-order universal algebra.
We finally show that the core of the theory surrounding Lawvere theories generalises to the second order as well. Instances of this development are the existence of algebraic functors and monad morphisms in the second-order universe. Moreover, we define a notion of translation homomorphism that allows us to establish a 2-categorical type theory correspondence
Second-Order Algebraic Theories
Fiore and Hur recently introduced a conservative extension of universal
algebra and equational logic from first to second order. Second-order universal
algebra and second-order equational logic respectively provide a model theory
and a formal deductive system for languages with variable binding and
parameterised metavariables. This work completes the foundations of the subject
from the viewpoint of categorical algebra. Specifically, the paper introduces
the notion of second-order algebraic theory and develops its basic theory. Two
categorical equivalences are established: at the syntactic level, that of
second-order equational presentations and second-order algebraic theories; at
the semantic level, that of second-order algebras and second-order functorial
models. Our development includes a mathematical definition of syntactic
translation between second-order equational presentations. This gives the first
formalisation of notions such as encodings and transforms in the context of
languages with variable binding
Functorial Semantics for Petri Nets under the Individual Token Philosophy
Although the algebraic semantics of place/transition Petri nets under the collective token philosophy has been fully explained in terms of (strictly) symmetric (strict) monoidal categories, the analogous construction under the individual token philosophy is not completely satisfactory because it lacks universality and also functoriality. We introduce the notion of pre-net to recover these aspects, obtaining a fully satisfactory categorical treatment centered on the notion of adjunction. This allows us to present a purely logical description of net behaviours under the individual token philosophy in terms of theories and theory morphisms in partial membership equational logic, yielding a complete match with the theory developed by the authors for the collective token view of net
Strongly Complete Logics for Coalgebras
Coalgebras for a functor model different types of transition systems in a
uniform way. This paper focuses on a uniform account of finitary logics for
set-based coalgebras. In particular, a general construction of a logic from an
arbitrary set-functor is given and proven to be strongly complete under
additional assumptions. We proceed in three parts. Part I argues that sifted
colimit preserving functors are those functors that preserve universal
algebraic structure. Our main theorem here states that a functor preserves
sifted colimits if and only if it has a finitary presentation by operations and
equations. Moreover, the presentation of the category of algebras for the
functor is obtained compositionally from the presentations of the underlying
category and of the functor. Part II investigates algebras for a functor over
ind-completions and extends the theorem of J{\'o}nsson and Tarski on canonical
extensions of Boolean algebras with operators to this setting. Part III shows,
based on Part I, how to associate a finitary logic to any finite-sets
preserving functor T. Based on Part II we prove the logic to be strongly
complete under a reasonable condition on T
`Third' Quantization of Vacuum Einstein Gravity and Free Yang-Mills Theories
Based on the algebraico-categorical (:sheaf-theoretic and sheaf
cohomological) conceptual and technical machinery of Abstract Differential
Geometry, a new, genuinely background spacetime manifold independent, field
quantization scenario for vacuum Einstein gravity and free Yang-Mills theories
is introduced. The scheme is coined `third quantization' and, although it
formally appears to follow a canonical route, it is fully covariant, because it
is an expressly functorial `procedure'. Various current and future Quantum
Gravity research issues are discussed under the light of 3rd-quantization. A
postscript gives a brief account of this author's personal encounters with
Rafael Sorkin and his work.Comment: 43 pages; latest version contributed to a fest-volume celebrating
Rafael Sorkin's 60th birthday (Erratum: in earlier versions I had wrongly
written that the Editor for this volume is Daniele Oriti, with CUP as
publisher. I apologize for the mistake.
Towards a homotopy theory of process algebra
This paper proves that labelled flows are expressive enough to contain all
process algebras which are a standard model for concurrency. More precisely, we
construct the space of execution paths and of higher dimensional homotopies
between them for every process name of every process algebra with any
synchronization algebra using a notion of labelled flow. This interpretation of
process algebra satisfies the paradigm of higher dimensional automata (HDA):
one non-degenerate full -dimensional cube (no more no less) in the
underlying space of the time flow corresponding to the concurrent execution of
actions. This result will enable us in future papers to develop a
homotopical approach of process algebras. Indeed, several homological
constructions related to the causal structure of time flow are possible only in
the framework of flows.Comment: 33 pages ; LaTeX2e ; 1 eps figure ; package semantics included ; v2
HDA paradigm clearly stated and simplification in a homotopical argument ; v3
"bug" fixed in notion of non-twisted shell + several redactional improvements
; v4 minor correction : the set of labels must not be ordered ; published at
http://intlpress.com/HHA/v10/n1/a16
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