6,982 research outputs found
A General Framework for the Semantics of Type Theory
We propose an abstract notion of a type theory to unify the semantics of
various type theories including Martin-L\"{o}f type theory, two-level type
theory and cubical type theory. We establish basic results in the semantics of
type theory: every type theory has a bi-initial model; every model of a type
theory has its internal language; the category of theories over a type theory
is bi-equivalent to a full sub-2-category of the 2-category of models of the
type theory
The Price equation program: simple invariances unify population dynamics, thermodynamics, probability, information and inference
The fundamental equations of various disciplines often seem to share the same
basic structure. Natural selection increases information in the same way that
Bayesian updating increases information. Thermodynamics and the forms of common
probability distributions express maximum increase in entropy, which appears
mathematically as loss of information. Physical mechanics follows paths of
change that maximize Fisher information. The information expressions typically
have analogous interpretations as the Newtonian balance between force and
acceleration, representing a partition between direct causes of change and
opposing changes in the frame of reference. This web of vague analogies hints
at a deeper common mathematical structure. I suggest that the Price equation
expresses that underlying universal structure. The abstract Price equation
describes dynamics as the change between two sets. One component of dynamics
expresses the change in the frequency of things, holding constant the values
associated with things. The other component of dynamics expresses the change in
the values of things, holding constant the frequency of things. The separation
of frequency from value generalizes Shannon's separation of the frequency of
symbols from the meaning of symbols in information theory. The Price equation's
generalized separation of frequency and value reveals a few simple invariances
that define universal geometric aspects of change. For example, the
conservation of total frequency, although a trivial invariance by itself,
creates a powerful constraint on the geometry of change. That constraint plus a
few others seem to explain the common structural forms of the equations in
different disciplines. From that abstract perspective, interpretations such as
selection, information, entropy, force, acceleration, and physical work arise
from the same underlying geometry expressed by the Price equation.Comment: Version 3: added figure illustrating geometry; added table of symbols
and two tables summarizing mathematical relations; this version accepted for
publication in Entrop
Coalgebraic completeness-via-canonicity for distributive substructural logics
We prove strong completeness of a range of substructural logics with respect
to a natural poset-based relational semantics using a coalgebraic version of
completeness-via-canonicity. By formalizing the problem in the language of
coalgebraic logics, we develop a modular theory which covers a wide variety of
different logics under a single framework, and lends itself to further
extensions. Moreover, we believe that the coalgebraic framework provides a
systematic and principled way to study the relationship between resource models
on the semantics side, and substructural logics on the syntactic side.Comment: 36 page
Modalities, Cohesion, and Information Flow
It is informally understood that the purpose of modal type constructors in
programming calculi is to control the flow of information between types. In
order to lend rigorous support to this idea, we study the category of
classified sets, a variant of a denotational semantics for information flow
proposed by Abadi et al. We use classified sets to prove multiple
noninterference theorems for modalities of a monadic and comonadic flavour. The
common machinery behind our theorems stems from the the fact that classified
sets are a (weak) model of Lawvere's theory of axiomatic cohesion. In the
process, we show how cohesion can be used for reasoning about multi-modal
settings. This leads to the conclusion that cohesion is a particularly useful
setting for the study of both information flow, but also modalities in type
theory and programming languages at large
Polynomial functors and combinatorial Dyson-Schwinger equations
We present a general abstract framework for combinatorial Dyson-Schwinger
equations, in which combinatorial identities are lifted to explicit bijections
of sets, and more generally equivalences of groupoids. Key features of
combinatorial Dyson-Schwinger equations are revealed to follow from general
categorical constructions and universal properties. Rather than beginning with
an equation inside a given Hopf algebra and referring to given Hochschild
-cocycles, our starting point is an abstract fixpoint equation in groupoids,
shown canonically to generate all the algebraic structure. Precisely, for any
finitary polynomial endofunctor defined over groupoids, the system of
combinatorial Dyson-Schwinger equations has a universal solution,
namely the groupoid of -trees. The isoclasses of -trees generate
naturally a Connes-Kreimer-like bialgebra, in which the abstract
Dyson-Schwinger equation can be internalised in terms of canonical
-operators. The solution to this equation is a series (the Green function)
which always enjoys a Fa\`a di Bruno formula, and hence generates a
sub-bialgebra isomorphic to the Fa\`a di Bruno bialgebra. Varying yields
different bialgebras, and cartesian natural transformations between various
yield bialgebra homomorphisms and sub-bialgebras, corresponding for example to
truncation of Dyson-Schwinger equations. Finally, all constructions can be
pushed inside the classical Connes-Kreimer Hopf algebra of trees by the
operation of taking core of -trees. A byproduct of the theory is an
interpretation of combinatorial Green functions as inductive data types in the
sense of Martin-L\"of Type Theory (expounded elsewhere).Comment: v4: minor adjustments, 49pp, final version to appear in J. Math. Phy
String compactifications on Calabi-Yau stacks
In this paper we study string compactifications on Deligne-Mumford stacks.
The basic idea is that all such stacks have presentations to which one can
associate gauged sigma models, where the group gauged need be neither finite
nor effectively-acting. Such presentations are not unique, and lead to
physically distinct gauged sigma models; stacks classify universality classes
of gauged sigma models, not gauged sigma models themselves. We begin by
defining and justifying a notion of ``Calabi-Yau stack,'' recall how one
defines sigma models on (presentations of) stacks, and calculate of physical
properties of such sigma models, such as closed and open string spectra. We
describe how the boundary states in the open string B model on a Calabi-Yau
stack are counted by derived categories of coherent sheaves on the stack. Along
the way, we describe numerous tests that IR physics is
presentation-independent, justifying the claim that stacks classify
universality classes. String orbifolds are one special case of these
compactifications, a subject which has proven controversial in the past;
however we resolve the objections to this description of which we are aware. In
particular, we discuss the apparent mismatch between stack moduli and physical
moduli, and how that discrepancy is resolved.Comment: 85 pages, LaTeX; v2: typos fixe
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