7,896 research outputs found
What is the Jacobian of a Riemann surface with boundary?
We define the Jacobian of a Riemann surface with analytically parametrized
boundary components. These Jacobians belong to a moduli space of ``open abelian
varieties'' which satisfies gluing axioms similar to those of Riemann surfaces,
and therefore allows a notion of ``conformal field theory'' to be defined on
this space. We further prove that chiral conformal field theories corresponding
to even lattices factor through this moduli space of open abelian varieties.Comment: 27 pages. Minor explanation and motivation added
Monads in Double Categories
We extend the basic concepts of Street's formal theory of monads from the
setting of 2-categories to that of double categories. In particular, we
introduce the double category Mnd(C) of monads in a double category C and
define what it means for a double category to admit the construction of free
monads. Our main theorem shows that, under some mild conditions, a double
category that is a framed bicategory admits the construction of free monads if
its horizontal 2-category does. We apply this result to obtain double
adjunctions which extend the adjunction between graphs and categories and the
adjunction between polynomial endofunctors and polynomial monads.Comment: 30 pages; v2: accepted for publication in the Journal of Pure and
Applied Algebra; added hypothesis in Theorem 3.7 that source and target
functors preserve equalizers; on page 18, bottom, in the statement concerning
the existence of a left adjoint, "if and only if" was replaced by "a
sufficient condition"; acknowledgements expande
Euler Characteristics of Categories and Homotopy Colimits
In a previous article, we introduced notions of finiteness obstruction, Euler
characteristic, and L^2-Euler characteristic for wide classes of categories. In
this sequel, we prove the compatibility of those notions with homotopy colimits
of I-indexed categories where I is any small category admitting a finite
I-CW-model for its I-classifying space. Special cases of our Homotopy Colimit
Formula include formulas for products, homotopy pushouts, homotopy orbits, and
transport groupoids. We also apply our formulas to Haefliger complexes of
groups, which extend Bass--Serre graphs of groups to higher dimensions. In
particular, we obtain necessary conditions for developability of a finite
complex of groups from an action of a finite group on a finite category without
loops.Comment: 44 pages. This final version will appear in Documenta Mathematica.
Remark 8.23 has been improved, discussion of Grothendieck construction has
been slightly expanded at the beginning of Section 3, and a few other minor
improvements have been incoporate
Finiteness obstructions and Euler characteristics of categories
We introduce notions of finiteness obstruction, Euler characteristic,
L^2-Euler characteristic, and M\"obius inversion for wide classes of
categories. The finiteness obstruction of a category Gamma of type (FP) is a
class in the projective class group K_0(RGamma); the functorial Euler
characteristic and functorial L^2-Euler characteristic are respectively its
RGamma-rank and L^2-rank. We also extend the second author's K-theoretic
M\"obius inversion from finite categories to quasi-finite categories. Our main
example is the proper orbit category, for which these invariants are
established notions in the geometry and topology of classifying spaces for
proper group actions. Baez-Dolan's groupoid cardinality and Leinster's Euler
characteristic are special cases of the L^2-Euler characteristic. Some of
Leinster's results on M\"obius-Rota inversion are special cases of the
K-theoretic M\"obius inversion.Comment: Final version, accepted for publication in the Advances in
Mathematics. Notational change: what was called chi(Gamma) in version 1 is
now called chi(BGamma), and chi(Gamma) now signifies the sum of the
components of the functorial Euler characteristic chi_f(Gamma). Theorem 5.25
summarizes when all Euler characteristics are equal. Minor typos have been
corrected. 88 page
Laplaza Sets, or How to Select Coherence Diagrams for Pseudo Algebras
We define a general concept of pseudo algebras over theories and 2-theories.
A more restrictive such notion was introduced by Hu and Kriz, but as noticed by
M. Gould, did not capture the desired examples. The approach taken in this
paper corrects the mistake by introducing a more general concept, allowing more
flexibility in selecting coherence diagrams for pseudo algebras.Comment: 21 pages. To appear in the Advances of Mathematics. Exposition
improved, notion of operad with degeneracies removed as a simplification,
definition of pseudo algebra improved
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