1,438 research outputs found
Reflections on hermeneutics and translocality
"This paper reflects on the issues that were brought
to the Roundtable on Hermeneutics and Translocality
held at the ZMO in 2006. I review the successive
ways in which I have drawn on the hermeneutic
philosophical tradition as an anthropologist,
emphasing the ethical dimension. Translocality
heightens the hermeneutic problem but does not
radically change it; it may entail recognizing that
everything is always already pretranslated. In reflecting
on the task and means of anthropology, I
briefly juxtapose Gadamer’s admirable deference
or modesty to Ricoeur’s dialectic of appropriation
and distanciation and to what Cavell calls the arrogation
of voice." [author´s abstract
Tensor envelopes of regular categories
We extend the calculus of relations to embed a regular category A into a
family of pseudo-abelian tensor categories T(A,d) depending on a degree
function d. Under the condition that all objects of A have only finitely many
subobjects, our main results are as follows:
1. Let N be the maximal proper tensor ideal of T(A,d). We show that T(A,d)/N
is semisimple provided that A is exact and Mal'cev. Thereby, we produce many
new semisimple, hence abelian, tensor categories.
2. Using lattice theory, we give a simple numerical criterion for the
vanishing of N.
3. We determine all degree functions for which T(A,d) is Tannakian. As a
result, we are able to interpolate the representation categories of many series
of profinite groups such as the symmetric groups S_n, the hyperoctahedral
groups S_n\semidir Z_2^n, or the general linear groups GL(n,F_q) over a fixed
finite field.
This paper generalizes work of Deligne, who first constructed the
interpolating category for the symmetric groups S_n. It also extends (and
provides proofs for) a previous paper math.CT/0605126 on the special case of
abelian categories.Comment: v1: 52 pages; v2: 52 pages, proof of Lemma 7.2 fixed, otherwise minor
change
Distributional Sentence Entailment Using Density Matrices
Categorical compositional distributional model of Coecke et al. (2010)
suggests a way to combine grammatical composition of the formal, type logical
models with the corpus based, empirical word representations of distributional
semantics. This paper contributes to the project by expanding the model to also
capture entailment relations. This is achieved by extending the representations
of words from points in meaning space to density operators, which are
probability distributions on the subspaces of the space. A symmetric measure of
similarity and an asymmetric measure of entailment is defined, where lexical
entailment is measured using von Neumann entropy, the quantum variant of
Kullback-Leibler divergence. Lexical entailment, combined with the composition
map on word representations, provides a method to obtain entailment relations
on the level of sentences. Truth theoretic and corpus-based examples are
provided.Comment: 11 page
Classifying Serre subcategories via atom spectrum
In this paper, we introduce the atom spectrum of an abelian category as a
topological space consisting of all the equivalence classes of monoform
objects. In terms of the atom spectrum, we give a classification of Serre
subcategories of an arbitrary noetherian abelian category. Moreover we show
that the atom spectrum of a locally noetherian Grothendieck category is
homeomorphic to its Ziegler spectrum.Comment: 15 page
Permutative categories, multicategories, and algebraic K-theory
We show that the -theory construction of arXiv:math/0403403, which
preserves multiplicative structure, extends to a symmetric monoidal closed
bicomplete source category, with the multiplicative structure still preserved.
The source category of arXiv:math/0403403, whose objects are permutative
categories, maps fully and faithfully to the new source category, whose objects
are (based) multicategories
Abstract Tensor Systems as Monoidal Categories
The primary contribution of this paper is to give a formal, categorical
treatment to Penrose's abstract tensor notation, in the context of traced
symmetric monoidal categories. To do so, we introduce a typed, sum-free version
of an abstract tensor system and demonstrate the construction of its associated
category. We then show that the associated category of the free abstract tensor
system is in fact the free traced symmetric monoidal category on a monoidal
signature. A notable consequence of this result is a simple proof for the
soundness and completeness of the diagrammatic language for traced symmetric
monoidal categories.Comment: Dedicated to Joachim Lambek on the occasion of his 90th birthda
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