30,438 research outputs found
Fiat categorification of the symmetric inverse semigroup IS_n and the semigroup F^*_n
Starting from the symmetric group , we construct two fiat
-categories. One of them can be viewed as the fiat "extension" of the
natural -category associated with the symmetric inverse semigroup
(considered as an ordered semigroup with respect to the natural order). This
-category provides a fiat categorification for the integral semigroup
algebra of the symmetric inverse semigroup. The other -category can be
viewed as the fiat "extension" of the -category associated with the maximal
factorizable subsemigroup of the dual symmetric inverse semigroup (again,
considered as an ordered semigroup with respect to the natural order). This
-category provides a fiat categorification for the integral semigroup
algebra of the maximal factorizable subsemigroup of the dual symmetric inverse
semigroup.Comment: v2: minor revisio
Stacky Lie groups
Presentations of smooth symmetry groups of differentiable stacks are studied
within the framework of the weak 2-category of Lie groupoids, smooth principal
bibundles, and smooth biequivariant maps. It is shown that principality of
bibundles is a categorical property which is sufficient and necessary for the
existence of products. Stacky Lie groups are defined as group objects in this
weak 2-category. Introducing a graphic notation, it is shown that for every
stacky Lie monoid there is a natural morphism, called the preinverse, which is
a Morita equivalence if and only if the monoid is a stacky Lie group. As
example we describe explicitly the stacky Lie group structure of the irrational
Kronecker foliation of the torus.Comment: 40 pages; definition of group objects in higher categories added;
coherence relations for groups in 2-categories given (section 4
A new model for pro-categories
In this paper we present a new way to construct the pro-category of a
category. This new model is very convenient to work with in certain situations.
We present a few applications of this new model, the most important of which
solves an open problem of Isaksen [Isa] concerning the existence of functorial
factorizations in what is known as the strict model structure on a
pro-category. Additionally we explain and correct an error in one of the
standard references on pro-categories.Comment: Substantial overlap with arXiv:1305.4607. Accepted for publication in
the Journal of Pure and Applied Algebra, reference: JPAA-504
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