papers

papers

[John Baez]

Dear Categorists -

Here are some new papers on category theory:


Higher-Dimensional Algebra VII: Groupoidification
John C. Baez, Alexander E. Hoffnung and Christopher D. Walker
http://arxiv.org/abs/0908.4305

Groupoidification is a form of categorification in which vector spaces are
replaced by groupoids, and linear operators are replaced by spans of
groupoids.
We introduce this idea with a detailed exposition of "degroupoidification":
a
systematic process that turns groupoids and spans into vector spaces and
linear
operators. Then we present three applications of groupoidification. The
first
is to Feynman diagrams. The Hilbert space for the quantum harmonic
oscillator
arises naturally from degroupoidifying the groupoid of finite sets and
bijections. This allows for a purely combinatorial interpretation of
creation
and annihilation operators, their commutation relations, field operators,
their
normal-ordered powers, and finally Feynman diagrams. The second application
is
to Hecke algebras. We explain how to groupoidify the Hecke algebra
associated
to a Dynkin diagram whenever the deformation parameter q is a prime power.
We
illustrate this with the simplest nontrivial example, coming from the A2
Dynkin
diagram. In this example we show that the solution of the Yang-Baxter
equation
built into the A2 Hecke algebra arises naturally from the axioms of
projective
geometry applied to the projective plane over the finite field with q
elements.
The third application is to Hall algebras. We explain how the standard
construction of the Hall algebra from the category of representations of a
simply-laced quiver can be seen as an example of degroupoidification. This
in
turn provides a new way to categorify - or more precisely, groupoidify - the
positive part of the quantum group associated to the quiver.


A Prehistory of n-Categorical Physics
John Baez and Aaron Lauda
http://arxiv.org/abs/0908.2469

This paper traces the growing role of categories and n-categories in
physics, starting with groups and their role in relativity, and leading up
to more sophisticated concepts which manifest themselves in Feynman
diagrams, spin networks, string theory, loop quantum gravity, and
topological quantum field theory. Our chronology ends around 2000, with just
a taste of later developments such as open-closed topological string theory,
the categorification of quantum groups, Khovanov homology, and Lurie's work
on the classification of topological quantum field theories.


Physics, Topology, Logic and Computation: a Rosetta Stone
John Baez and Mike Stay
http://arxiv.org/abs/0903.0340

In physics, Feynman diagrams are used to reason about quantum processes. In
the 1980s, it became clear that underlying these diagrams is a powerful
analogy between quantum physics and topology: namely, a linear operator
behaves very much like a "cobordism". Similar diagrams can be used to reason
about logic, where they represent proofs, and computation, where they
represent programs. With the rise of interest in quantum cryptography and
quantum computation, it became clear that there is extensive network of
analogies between physics, topology, logic and computation. In this
expository paper, we make some of these analogies precise using the concept
of "closed symmetric monoidal category". We assume no prior knowledge of
category theory, proof theory or computer science.


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papers

[Eduardo Dubuc]

I call to your atention the following yet unpublished papers:

1. arXiv:0706.1771 [ps, pdf, other]
    Title: The fundamental progroupoid of a general topos
    Authors: Eduardo J. Dubuc
    Comments: 19 pages
    Subjects: Category Theory (math.CT); Algebraic Topology (math.AT)

Here we introduce a new notion of covering projections in a topos E. They
are a particular kind of locally constant objects. When the topos is
locally connected, all locally constant objects are covering projections.
Given a fix cover U = {U_i}, we show how to construct the set of connected
components in the topos E_U of covering projections trivialized by U, even
if the topos E is not locally connected. Our theory generalize to an
arbritrary topos the whole theory of the fundamental progroupoid known for
locally connected topos.

The following two papers are in the area of categorical topology, or
topological categories.

arXiv:math/0612727 [ps, pdf, other]
    Title: Quasitopoi over a base category
    Authors: Eduardo J. Dubuc, Luis Español
    Comments: 23 pages
    Subjects: Category Theory (math.CT)
3. arXiv:math/0611701 [ps, pdf, other]
    Title: Topological functors as familiarly-fibrations
    Authors: Eduardo J. Dubuc, Luis Español
    Comments: 16 pages
    Subjects: Category Theory (math.CT)