792 research outputs found
Eilenberg Theorems for Free
Eilenberg-type correspondences, relating varieties of languages (e.g. of
finite words, infinite words, or trees) to pseudovarieties of finite algebras,
form the backbone of algebraic language theory. Numerous such correspondences
are known in the literature. We demonstrate that they all arise from the same
recipe: one models languages and the algebras recognizing them by monads on an
algebraic category, and applies a Stone-type duality. Our main contribution is
a variety theorem that covers e.g. Wilke's and Pin's work on
-languages, the variety theorem for cost functions of Daviaud,
Kuperberg, and Pin, and unifies the two previous categorical approaches of
Boja\'nczyk and of Ad\'amek et al. In addition we derive a number of new
results, including an extension of the local variety theorem of Gehrke,
Grigorieff, and Pin from finite to infinite words
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
Behavior of Quillen (co)homology with respect to adjunctions
This paper aims to answer the following question: Given an adjunction between
two categories, how is Quillen (co)homology in one category related to that in
the other? We identify the induced comparison diagram, giving necessary and
sufficient conditions for it to arise, and describe the various comparison
maps. Examples are given. Along the way, we clarify some categorical
assumptions underlying Quillen (co)homology: cocomplete categories with a set
of small projective generators provide a convenient setup.Comment: Minor corrections. To appear in Homology, Homotopy and Application
Confinement studies in lattice QCD
We describe the current search for confinement mechanisms in lattice QCD. We
report on a recent derivation of a lattice Ehrenfest-Maxwell relation for the
Abelian projection of SU(2) lattice gauge theory. This gives a precise lattice
definition of field strength and electric current due to static sources,
charged dynamical fields, gauge fixing and ghosts. In the maximal Abelian gauge
the electric charge is anti-screened analogously to the non-Abelian charge.Comment: Talk given at the Symposium in Honor of Dick Slansky, Los Alamos, May
20-21 1998, to be published in Phys. Rep., Latex 2e uses class file
elsart.cl
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