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## ‣ Generalized Frobenius Algebras and the Theory of Hopf Algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Quantum Algebra#Mathematics - Category Theory#Mathematics - K-Theory and Homology#Mathematics - Rings and Algebras#Mathematics - Representation Theory#16T15, 18G35, 16T05, 20N99, 18D10, 05E10

"Co-Frobenius" coalgebras were introduced as dualizations of Frobenius
algebras. Recently, it was shown in \cite{I} that they admit left-right
symmetric characterizations analogue to those of Frobenius algebras: a
coalgebra $C$ is co-Frobenius if and only if it is isomorphic to its rational
dual. We consider the more general quasi-co-Frobenius (QcF) coalgebras; in the
first main result we show that these also admit symmetric characterizations: a
coalgebra is QcF if it is weakly isomorphic to its (left, or equivalently
right) rational dual $Rat(C^*)$, in the sense that certain coproduct or product
powers of these objects are isomorphic. These show that QcF coalgebras can be
viewed as generalizations of both co-Frobenius coalgebras and Frobenius
algebras. Surprisingly, these turn out to have many applications to fundamental
results of Hopf algebras. The equivalent characterizations of Hopf algebras
with left (or right) nonzero integrals as left (or right) co-Frobenius, or QcF,
or semiperfect or with nonzero rational dual all follow immediately from these
results. Also, the celebrated uniqueness of integrals follows at the same time
as just another equivalent statement. Moreover, as a by-product of our methods,
we observe a short proof for the bijectivity of the antipode of a Hopf algebra
with nonzero integral. This gives a purely representation theoretic approach to
many of the basic fundamental results in the theory of Hopf algebras.; Comment: New Version...

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## ‣ Symmetric homotopy theory for operads

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 09/03/2015
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#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - K-Theory and Homology#Mathematics - Quantum Algebra#Mathematics - Representation Theory#18D50, 18G55

The purpose of this foundational paper is to introduce various notions and
constructions in order to develop the homotopy theory for differential graded
operads over any ring. The main new idea is to consider the action of the
symmetric groups as part of the defining structure of an operad and not as the
underlying category. We introduce a new dual category of higher cooperads, a
new higher bar-cobar adjunction with the category of operads, and a new higher
notion of homotopy operads, for which we establish the relevant homotopy
properties. For instance, the higher bar-cobar construction provides us with a
cofibrant replacement functor for operads over any ring. All these
constructions are produced conceptually by applying the curved Koszul duality
for colored operads. This paper is a first step toward a new Koszul duality
theory for operads, where the action of the symmetric groups is properly taken
into account.; Comment: 40 pages. Comments are welcome

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## ‣ Idempotent (Asymptotic) Mathematics and the Representation Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 04/06/2002
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A brief survey of some basic ideas of the so-called Idempotent Mathematics is
presented; an "idempotent" version of the representation theory is discussed.
The Idempotent Mathematics can be treated as a result of a dequantization of
the traditional mathematics over numerical fields in the limit of the vanishing
"imaginary Planck constant"; there is a correspondence, in the spirit of N.
Bohr's correspondence principle, between constructions and results in
traditional mathematics over the fields of real and complex numbers and similar
constructions and results over idempotent semirings. In particular, there is an
"idempotent" version of the theory of linear representations of groups. Some
basic concepts and results of the "idempotent" representation theory are
presented. In the framework of this theory the well-known Legendre transform
can be treated as an idempotent version of the traditional Fourier transform.
Some unexpected versions of the Engel theorem are given.; Comment: 10 pages

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## ‣ Parametrized K-Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 01/04/2013
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#Mathematics - K-Theory and Homology#Mathematics - Commutative Algebra#Mathematics - Algebraic Geometry#Mathematics - Category Theory#18F25, 19D99 (Primary) 13D15, 14F05, 14F20, 18D10, 18D30, 18D99,
18E10, 18F10, 19E08 (Secondary)

In nature, one observes that a K-theory of an object is defined in two steps.
First a "structured" category is associated to the object. Second, a K-theory
machine is applied to the latter category to produce an infinite loop space. We
develop a general framework that deals with the first step of this process. The
K-theory of an object is defined via a category of "locally trivial" objects
with respect to a pretopology. We study conditions ensuring an exact structure
on such categories. We also consider morphisms in K-theory that such contexts
naturally provide. We end by defining various K-theories of schemes and
morphisms between them.; Comment: 31 pages

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## ‣ Knot Homology from Refined Chern-Simons Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#High Energy Physics - Theory#Mathematics - Algebraic Geometry#Mathematics - Geometric Topology#Mathematics - Representation Theory

We formulate a refinement of SU(N) Chern-Simons theory on a three-manifold
via the refined topological string and the (2,0) theory on N M5 branes. The
refined Chern-Simons theory is defined on any three-manifold with a semi-free
circle action. We give an explicit solution of the theory, in terms of a
one-parameter refinement of the S and T matrices of Chern-Simons theory,
related to the theory of Macdonald polynomials. The ordinary and refined
Chern-Simons theory are similar in many ways; for example, the Verlinde formula
holds in both. We obtain new topological invariants of Seifert three-manifolds
and torus knots inside them. We conjecture that the knot invariants we compute
are the Poincare polynomials of the sl(n) knot homology theory. The latter
includes the Khovanov-Rozansky knot homology, as a special case. The conjecture
passes a number of nontrivial checks. We show that, for a large number of torus
knots colored with the fundamental representation of SU(N), our knot invariants
agree with the Poincare polynomials of Khovanov-Rozansky homology. As a
byproduct, we show that our theory on S^3 has a large-N dual which is the
refined topological string on X=O(-1)+O(-1)->P^1; this supports the conjecture
by Gukov, Schwarz and Vafa relating the spectrum of BPS states on X to sl(n)
knot homology. We also provide a matrix model description of some amplitudes of
the refined Chern-Simons theory on S^3.; Comment: 73 pages...

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## ‣ Algebraic K-theory is stable and admits a multiplicative structure for module objects

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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After recognizing higher homotopy coherences, algebraic K-theory can be
regarded as a functor from stable $\infty$-categories to $\infty$-categories.
We establish the stability theoremm which states that the algebraic K-theory of
a stable $\infty$-category is actually a stable $\infty$-category itself. This
is a generalization of the statement that algebraic K-theory is a functor from
spectra to spectra. We then prove a result which provides a simpler
interpretation of the algebraic K-theory of ring spectra. In order to do this,
we compute the algebraic K-theory of an $\infty$-category of modules, and
establish that it is an $\infty$-category of modules itself. This result, known
as the multiplicativity theorem, vastly generalizes results obtained by
Elmendorf and Mandell. Since the algebraic K-theory of a ring spectrum $R$ is
the algebraic K-theory of the $\infty$-category of perfect modules over $R$,
this provides a simpler interpretation of the algebraic K-theory of ring
spectra. Using this result, we prove an $\infty$-categorical counterpart of the
derived Morita context for flat rings, which shows that algebraic K-theory is a
homotopy coherent version of Morita theory.; Comment: 12 pages. Any updates to this version of the paper will be available
from the author's webpage. Comments are welcome!

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## ‣ The Eta-invariant and Pontryagin duality in K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Analysis of PDEs#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - Operator Algebras#Mathematics - Spectral Theory#58J28, 19L64 (Primary)#58J22, 19K56, 58J40 (Secondary)

The topological significance of the spectral Atiyah-Patodi-Singer
eta-invariant is investigated under the parity conditions of P. Gilkey. We show
that twice the fractional part of the invariant is computed by the linking
pairing in K-theory with the orientation bundle of the manifold. The Pontrjagin
duality implies the nondegeneracy of the linking form. An example of a
nontrivial fractional part for an even-order operator is presented. This result
answers the question of P. Gilkey (1989) concerning the existence of even-order
operators on odd-dimensional manifolds with nontrivial fractional part of
eta-invariant.; Comment: 24 pages, 1 figure; final version; see
http://www.kluweronline.com/issn/0001-4346/contents

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## ‣ A1-homotopy invariance of algebraic K-theory with coefficients and Kleinian singularities

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Rings and Algebras#Mathematics - Representation Theory#13F35, 14A22, 14H20, 19D25, 19D35, 19E08, 30F50

C. Weibel and Thomason-Trobaugh proved (under some assumptions) that
algebraic K-theory with coefficients is A1-homotopy invariant. In this article
we generalize this result from schemes to the broad setting of dg categories.
Along the way, we extend Bass-Quillen's fundamental theorem as well as
Stienstra's foundational work on module structures over the big Witt ring to
the setting of dg categories. Among other cases, the above A1-homotopy
invariance result can now be applied to sheaves of (not necessarily
commutative) dg algebras over stacks. As an application, we compute the
algebraic K-theory with coefficients of dg cluster categories using solely the
kernel and cokernel of the Coxeter matrix. This leads to a complete computation
of the algebraic K-theory with coefficients of the Kleinian singularities
parametrized by the simply laced Dynkin diagrams. As a byproduct, we obtain
some vanishing and divisibility properties of algebraic K-theory (without
coefficients).; Comment: 20 pages. Revised version

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## ‣ Algebraic K-theory of strict ring spectra

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 24/03/2014
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#Mathematics - Algebraic Topology#Mathematics - Geometric Topology#Mathematics - K-Theory and Homology#Mathematics - Number Theory#19D10, 55P43, 19F27, 57R50

We view strict ring spectra as generalized rings. The study of their
algebraic K-theory is motivated by its applications to the automorphism groups
of compact manifolds. Partial calculations of algebraic K-theory for the sphere
spectrum are available at regular primes, but we seek more conceptual answers
in terms of localization and descent properties. Calculations for ring spectra
related to topological K-theory suggest the existence of a motivic cohomology
theory for strictly commutative ring spectra, and we present evidence for
arithmetic duality in this theory. To tie motivic cohomology to Galois
cohomology we wish to spectrally realize ramified extensions, which is only
possible after mild forms of localization. One such mild localization is
provided by the theory of logarithmic ring spectra, and we outline recent
developments in this area.; Comment: Contribution to the proceedings of the ICM 2014 in Seoul

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## ‣ A Theory of Adjoint Functors--with some Thoughts about their Philosophical Significance

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/11/2005
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The question "What is category theory" is approached by focusing on universal
mapping properties and adjoint functors. Category theory organizes mathematics
using morphisms that transmit structure and determination. Structures of
mathematical interest are usually characterized by some universal mapping
property so the general thesis is that category theory is about determination
through universals. In recent decades, the notion of adjoint functors has moved
to center-stage as category theory's primary tool to characterize what is
important and universal in mathematics. Hence our focus here is to present a
theory of adjoint functors, a theory which shows that all adjunctions arise
from the birepresentations of "chimeras" or "heteromorphisms" between the
objects of different categories. Since representations provide universal
mapping properties, this theory places adjoints within the framework of
determination through universals. The conclusion considers some unreasonably
effective analogies between these mathematical concepts and some central
philosophical themes.; Comment: 58 pages. Forthcoming in: What is Category Theory? Giandomenico Sica
ed., Milan: Polimetrica

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## ‣ On determinant functors and $K$-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - Number Theory#19A99, 19B99, 18F25, 18G50, 18G55, 18E10, 18E30

In this paper we introduce a new approach to determinant functors which
allows us to extend Deligne's determinant functors for exact categories to
Waldhausen categories, (strongly) triangulated categories, and derivators. We
construct universal determinant functors in all cases by original methods which
are interesting even for the known cases. Moreover, we show that the target of
each universal determinant functor computes the corresponding $K$-theory in
dimensions 0 and 1. As applications, we answer open questions by Maltsiniotis
and Neeman on the $K$-theory of (strongly) triangulated categories and a
question of Grothendieck to Knudsen on determinant functors. We also prove
additivity theorems for low-dimensional $K$-theory and obtain generators and
(some) relations for various $K_{1}$-groups.; Comment: 73 pages. We have deeply revised the paper to make it more
accessible, it contains now explicit examples and computations. We have
removed the part on localization, it was correct but we didn't want to make
the paper longer and we thought this part was the less interesting one.
Nevertheless it will remain here in the arXiv, in version 1. If you need it
in your research, please let us know

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## ‣ The homotopy fixed point theorem and the Quillen-Lichtenbaum conjecture in hermitian K-theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Number Theory

Let X be a noetherian scheme of finite Krull dimension, having 2 invertible
in its ring of regular functions, an ample family of line bundles, and a global
bound on the virtual mod-2 cohomological dimensions of its residue fields. We
prove that the comparison map from the hermitian K-theory of X to the homotopy
fixed points of K-theory under the natural Z/2-action is a 2-adic equivalence
in general, and an integral equivalence when X has no formally real residue
field. We also show that the comparison map between the higher
Grothendieck-Witt (hermitian K-) theory of X and its \'etale version is an
isomorphism on homotopy groups in the same range as for the Quillen-Lichtenbaum
conjecture in K-theory. Applications compute higher Grothendieck-Witt groups of
complex algebraic varieties and rings of 2-integers in number fields, and hence
values of Dedekind zeta-functions.; Comment: 17 pages, to appear in Adv. Math

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## ‣ K-Theory in Quantum Field Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/06/2002
Português

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#Mathematical Physics#High Energy Physics - Theory#Mathematics - Algebraic Topology#Mathematics - Differential Geometry#Mathematics - K-Theory and Homology#81T30, 81T45, 81T50, 19L99

We survey three different ways in which K-theory in all its forms enters
quantum field theory. In Part 1 we give a general argument which relates
topological field theory in codimension two with twisted K-theory, and we
illustrate with some finite models. Part 2 is a review of pfaffians of Dirac
operators, anomalies, and the relationship to differential K-theory. Part 3 is
a geometric exposition of Dirac charge quantization, which in superstring
theories also involves differential K-theory. Parts 2 and 3 are related by the
Green-Schwarz anomaly cancellation mechanism. An appendix, joint with Jerry
Jenquin, treats the partition function of Rarita-Schwinger fields.; Comment: 56 pages, expanded version of lectures at "Current Developments in
Mathematics"

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## ‣ Comparison between algebraic and topological K-theory of locally convex algebras

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - K-Theory and Homology#Mathematics - Rings and Algebras#18G, 19K, 46H, 46L80, 46M, 58B34

This paper is concerned with the algebraic K-theory of locally convex
algebras stabilized by operator ideals, and its comparison with topological
K-theory. We show that the obstruction for the comparison map between algebraic
and topological K-theory to be an isomorphism is (absolute) algebraic cyclic
homology and prove the existence of an 6-term exact sequence.
We show that cyclic homology vanishes in the case when J is the ideal of
compact operators and L is a Frechet algebra with bounded app. unit. This
proves the generalized version of Karoubi's conjecture due to Mariusz Wodzicki
and announced in his paper "Algebraic K-theory and functional analysis", First
European Congress of Mathematics, Vol. II (Paris, 1992), 485--496, Progr.
Math., 120, Birkh\"auser, Basel, 1994.
We also consider stabilization with respect to a wider class of operator
ideals, called sub-harmonic. We study the algebraic K-theory of the tensor
product of a sub-harmonic ideal with an arbitrary complex algebra and prove
that the obstruction for the periodicity of algebraic K-theory is again cyclic
homology.
The main technical tools we use are the diffeotopy invariance theorem of
Cuntz and the second author (which we generalize), and the excision theorem for
infinitesimal K-theory...

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## ‣ Homotopy invariance of higher K-theory for abelian categories

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#Mathematics - Algebraic Topology#Mathematics - Category Theory#Mathematics - K-Theory and Homology

The main theorem in this paper is that the base change functor from a
noetherian abelian category to its noetherian polynomial category induces an
isomorphism on K-theory. The main theorem implies the well-known fact that
A^1-homotopy invariance of K'-theory for noetherian schemes.; Comment: arXiv admin note: substantial text overlap with arXiv:1104.4240

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## ‣ Unified Foundations for Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/03/2004
Português

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There are different meanings of foundation of mathematics: philosophical,
logical, and mathematical. Here foundations are considered as a theory that
provides means (concepts, structures, methods etc.) for the development of
whole mathematics. Set theory has been for a long time the most popular
foundation. However, it was not been able to win completely over its rivals:
logic, the theory of algorithms, and theory of categories. Moreover, practical
applications of mathematics and its inner problems caused creation of different
generalization of sets: multisets, fuzzy sets, rough sets etc. Thus, we
encounter a problem: Is it possible to find the most fundamental structure in
mathematics? The situation is similar to the quest of physics for the most
fundamental "brick" of nature and for a grand unified theory of nature. It is
demonstrated that in contrast to physics, which is still in search for a
unified theory, in mathematics such a theory exists. It is the theory of named
sets.

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## ‣ Tilting theory for trees via stable homotopy theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Topology#Mathematics - Algebraic Geometry#Mathematics - Category Theory#Mathematics - Representation Theory#55U35 (Primary) 16E35, 18E30, 55U40 (Secondary)

We show that variants of the classical reflection functors from quiver
representation theory exist in any abstract stable homotopy theory, making them
available for example over arbitrary ground rings, for quasi-coherent modules
on schemes, in the differential-graded context, in stable homotopy theory as
well as in the equivariant, motivic, and parametrized variant thereof. As an
application of these equivalences we obtain abstract tilting results for trees
valid in all these situations, hence generalizing a result of Happel.
The main tools introduced for the construction of these reflection functors
are homotopical epimorphisms of small categories and one-point extensions of
small categories, both of which are inspired by similar concepts in homological
algebra.; Comment: To appear in J. Pure Appl. Algebra, it is a sequel to arXiv:1401.6451
and continues the development of abstract tilting theory. Version 2: various
improvements in the presentation. Version 3: a detailed explanation added (in
Construction 9.13 and Lemma 9.15) for the key fact that both the branches of
Figure 2 lead to the same category

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## ‣ Almost ring theory - sixth release

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Português

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#Mathematics - Algebraic Geometry#Mathematics - Commutative Algebra#Mathematics - Number Theory#Mathematics - Rings and Algebras#13D03, 12J20, 14A99, 14F99, 18D10, 14G22

We develop almost ring theory, which is a domain of mathematics somewhere
halfway between ring theory and category theory (whence the difficulty of
finding appropriate MSC-class numbers). We apply this theory to valuation
theory and to p-adic analytic geometry. You should really have a look at the
introductions (each chapter has one).; Comment: This is the sixth - and assuredly final - release of "Almost ring
theory". It is about 230 page long; it is written in AMSLaTeX and uses XYPic
and a few not so standard fonts. Any future corrections (mainly typos, I
expect) will be found on my personal web page:
http://www.math.u-bordeaux.fr/~ramero/

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## ‣ Homotopy Type Theory: Univalent Foundations of Mathematics

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 03/08/2013
Português

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#Mathematics - Logic#Computer Science - Programming Languages#Mathematics - Algebraic Topology#Mathematics - Category Theory

Homotopy type theory is a new branch of mathematics, based on a recently
discovered connection between homotopy theory and type theory, which brings new
ideas into the very foundation of mathematics. On the one hand, Voevodsky's
subtle and beautiful "univalence axiom" implies that isomorphic structures can
be identified. On the other hand, "higher inductive types" provide direct,
logical descriptions of some of the basic spaces and constructions of homotopy
theory. Both are impossible to capture directly in classical set-theoretic
foundations, but when combined in homotopy type theory, they permit an entirely
new kind of "logic of homotopy types". This suggests a new conception of
foundations of mathematics, with intrinsic homotopical content, an "invariant"
conception of the objects of mathematics -- and convenient machine
implementations, which can serve as a practical aid to the working
mathematician. This book is intended as a first systematic exposition of the
basics of the resulting "Univalent Foundations" program, and a collection of
examples of this new style of reasoning -- but without requiring the reader to
know or learn any formal logic, or to use any computer proof assistant.; Comment: 465 pages. arXiv v1: first-edition-257-g5561b73...

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## ‣ Categorical Foundations for K-Theory

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 14/11/2011
Português

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#Mathematics - K-Theory and Homology#Mathematics - Algebraic Geometry#Mathematics - Algebraic Topology#Mathematics - Category Theory#19-02, 18F25 (Primary) 18D05, 18D10, 18D30, 18D35, 18F10, 19E08,
19L99, 55N15 (Secondary)

Recall that the definition of the $K$-theory of an object C (e.g., a ring or
a space) has the following pattern. One first associates to the object C a
category A_C that has a suitable structure (exact, Waldhausen, symmetric
monoidal, ...). One then applies to the category A_C a "$K$-theory machine",
which provides an infinite loop space that is the $K$-theory K(C) of the object
C.
We study the first step of this process. What are the kinds of objects to be
studied via $K$-theory? Given these types of objects, what structured
categories should one associate to an object to obtain $K$-theoretic
information about it? And how should the morphisms of these objects interact
with this correspondence?
We propose a unified, conceptual framework for a number of important examples
of objects studied in $K$-theory. The structured categories associated to an
object C are typically categories of modules in a monoidal (op-)fibred
category. The modules considered are "locally trivial" with respect to a given
class of trivial modules and a given Grothendieck topology on the object C's
category.; Comment: 176 + xi pages. This monograph is a revised and augmented version of
my PhD thesis. The official thesis is available at
http://library.epfl.ch/en/theses/?nr=4861

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