385 research outputs found
The algebra of rewriting for presentations of inverse monoids
We describe a formalism, using groupoids, for the study of rewriting for
presentations of inverse monoids, that is based on the Squier complex
construction for monoid presentations. We introduce the class of pseudoregular
groupoids, an example of which now arises as the fundamental groupoid of our
version of the Squier complex. A further key ingredient is the factorisation of
the presentation map from a free inverse monoid as the composition of an
idempotent pure map and an idempotent separating map. The relation module of a
presentation is then defined as the abelianised kernel of this idempotent
separating map. We then use the properties of idempotent separating maps to
derive a free presentation of the relation module. The construction of its
kernel - the module of identities - uses further facts about pseudoregular
groupoids.Comment: 22 page
Relation modules and identities for presentations of inverse monoids
We investigate the Squier complexes of presentations of groups and inverse monoids using the theory semiregular, regular, and pseudoregular groupoids. Our main interest is the class of regular groupoids, and the new class of pseudoregular groupoids.
Our study of group presentations uses monoidal, regular groupoids. These are equivalent to crossed modules, and we recover the free crossed module usually associated to a group presentation, and a free presentation of the relation module with kernel the fundamental group of the Squier complex, the module of identities among the relations. We carry out a similar study of inverse monoid presentations using pseudoregular groupoids. The relation module is defined via an intermediate construction – the derivation module of a homomorphism, – and a key ingredient is the factorisation of the presentation map from a free inverse monoid as the composition of an idempotent pure map and an idempotent separating map. We can then use the properties of idempotent separating maps, and properties of the derivation module as a left adjoint, to derive a free presentation of the relation module. The construction of its kernel – the module of identities – uses further key facts about pseudoregular groupoids
On The Homotopy Type of Higher Orbifolds and Haefliger Classifying Spaces
We describe various equivalent ways of associating to an orbifold, or more
generally a higher \'etale differentiable stack, a weak homotopy type. Some of
these ways extend to arbitrary higher stacks on the site of smooth manifolds,
and we show that for a differentiable stack X arising from a Lie groupoid G,
the weak homotopy type of X agrees with that of BG. Using this machinery, we
are able to find new presentations for the weak homotopy type of certain
classifying spaces. In particular, we give a new presentation for the Borel
construction of an almost free action of a Lie group G on a smooth manifold M
as the classifying space of a category whose objects consists of smooth maps
R^n to M which are transverse to all the G-orbits, where n=dim M - dim G. We
also prove a generalization of Segal's theorem, which presents the weak
homotopy type of Haefliger's groupoid as the classifying space of
the monoid of self-embeddings of R^q, and our generalization gives analogous
presentations for the weak homotopy type of the Lie groupoids
and which are related to the classification of
foliations with transverse symplectic forms and transverse metrics
respectively. We also give a short and simple proof of Segal's original theorem
using our machinery.Comment: 47 page
Presentation of homotopy types under a space
We compare the structure of a mapping cone in the category Top^D of spaces
under a space D with differentials in algebraic models like crossed complexes
and quadratic complexes. Several subcategories of Top^D are identified with
algebraic categories. As an application we show that there are exactly 16
essential self--maps of S^2 x S^2 fixing the diagonal.Comment: 21 page
Parabolic subgroups of Garside groups II: ribbons
We introduce and investigate the ribbon groupoid associated with a Garside
group. Under a technical hypothesis, we prove that this category is a Garside
groupoid. We decompose this groupoid into a semi-direct product of two of its
parabolic subgroupoids and provide a groupoid presentation. In order to
established the latter result, we describe quasi-centralizers in Garside
groups. All results hold in the particular case of Artin-Tits groups of
spherical type
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
Infinite root stacks and quasi-coherent sheaves on logarithmic schemes
We define and study infinite root stacks of fine and saturated logarithmic
schemes, a limit version of the root stacks introduced by Niels Borne and the
second author. We show in particular that the infinite root stack determines
the logarithmic structure, and recovers the Kummer-flat topos of the
logarithmic scheme. We also extend the correspondence between parabolic sheaves
and quasi-coherent sheaves on root stacks to this new setting.Comment: v2: 61 pages. Final version, to appear in Proc. Lond. Math. So
Gcd-monoids arising from homotopy groupoids
The interval monoid (P) of a poset P is defined by generators [x,
y], where x y in P , and relations [x, x] = 1, [x, z] = [x, y]
[y, z] for x y z. It embeds into its universal group
(P), the interval group of P , which is also the universal group of the
homotopy groupoid of the chain complex of P. We prove the following results:
The monoid (P) has finite left and right greatest common
divisors of pairs (we say that it is a gcd-monoid) iff every principal ideal
(resp., filter) of P is a join-semilattice (resp., a meet-semilattice).
For every group G, there is a poset P of length 2 such that
(P) is a gcd-monoid and G is a free factor of (P) by
a free group. Moreover, P can be taken finite iff G is finitely presented.
For every finite poset P , the monoid (P) can be embedded
into a free monoid. Some of the results above, and many related ones,
can be extended from interval monoids to the universal monoid Umon(S) of any
category S. This enables us, in particular, to characterize the embeddability
of Umon(S) into a group, by stating that it holds at the hom-set level. We thus
obtain new easily verified sufficient conditions for embeddability of a monoid
into a group. We illustrate our results by various examples and
counterexamples.Comment: 27 pages (v4). Semigroup Forum, to appea
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