789,956 research outputs found
Deformation theory of representations of prop(erad)s
We study the deformation theory of morphisms of properads and props thereby
extending to a non-linear framework Quillen's deformation theory for
commutative rings. The associated chain complex is endowed with a Lie algebra
up to homotopy structure. Its Maurer-Cartan elements correspond to deformed
structures, which allows us to give a geometric interpretation of these
results.
To do so, we endow the category of prop(erad)s with a model category
structure. We provide a complete study of models for prop(erad)s. A new
effective method to make minimal models explicit, that extends Koszul duality
theory, is introduced and the associated notion is called homotopy Koszul.
As a corollary, we obtain the (co)homology theories of (al)gebras over a
prop(erad) and of homotopy (al)gebras as well. Their underlying chain complex
is endowed with a canonical Lie algebra up to homotopy structure in general and
a Lie algebra structure only in the Koszul case. In particular, we explicit the
deformation complex of morphisms from the properad of associative bialgebras.
For any minimal model of this properad, the boundary map of this chain complex
is shown to be the one defined by Gerstenhaber and Schack. As a corollary, this
paper provides a complete proof of the existence of a Lie algebra up to
homotopy structure on the Gerstenhaber-Schack bicomplex associated to the
deformations of associative bialgebras.Comment: Version 4 : Statement about the properad of (non-commutative)
Frobenius bialgebras fixed in Section 4. [82 pages
A general framework for homotopic descent and codescent
In this paper we elaborate a general homotopy-theoretic framework in which to
study problems of descent and completion and of their duals, codescent and
cocompletion. Our approach to homotopic (co)descent and to derived
(co)completion can be viewed as -category-theoretic, as our framework
is constructed in the universe of simplicially enriched categories, which are a
model for -categories.
We provide general criteria, reminiscent of Mandell's theorem on
-algebra models of -complete spaces, under which homotopic
(co)descent is satisfied. Furthermore, we construct general descent and
codescent spectral sequences, which we interpret in terms of derived
(co)completion and homotopic (co)descent.
We show that a number of very well-known spectral sequences, such as the
unstable and stable Adams spectral sequences, the Adams-Novikov spectral
sequence and the descent spectral sequence of a map, are examples of general
(co)descent spectral sequences. There is also a close relationship between the
Lichtenbaum-Quillen conjecture and homotopic descent along the
Dwyer-Friedlander map from algebraic K-theory to \'etale K-theory. Moreover,
there are intriguing analogies between derived cocompletion (respectively,
completion) and homotopy left (respectively, right) Kan extensions and their
associated assembly (respectively, coassembly) maps.Comment: Discussion of completeness has been refined; statement of the theorem
on assembly has been corrected; numerous small additions and minor
correction
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Homotopy Theory of Monoids and Group Completion
This thesis presents several complete and partial models for the homotopy theory of monoids and the derived functor of group completion. We show that there is a simplicial model structure on the category of reduced simplicial sets that is Quillen equivalent to the Quillen model structure of simplicial monoids. Using this Quillen equivalence we recover the fact that the derived functor of group completion is isomorphic to the homotopy type of loops on the classifying space of a monoid. We use the Street nerve to show that the derived functor of group completion of monoids in the category of ω-groupoids for the Gray tensor product is isomorphic to group completion for simplicial monoids in low degrees. Finally we exploit the connection of ω-groupoids with the theory of rewriting for presentations of monoids to calculate the second homotopy group of the classifying space BM of a monoid M in terms of a chosen presentation by generators and relations.Department of Pure Mathematics and Mathematical Statistics
Cambridge Commonwealth Trust
Natural Sciences and Engineering Research Council of Canad
Structure and semantics
Algebraic theories describe mathematical structures that are defined in terms of operations
and equations, and are extremely important throughout mathematics. Many generalisations
of the classical notion of an algebraic theory have sprung up for use in different mathematical
contexts; some examples include Lawvere theories, monads, PROPs and operads. The first
central notion of this thesis is a common generalisation of these, which we call a proto-theory.
The purpose of an algebraic theory is to describe its models, which are structures in which
each of the abstract operations of the theory is given a concrete interpretation such that the
equations of the theory hold. The process of going from a theory to its models is called
semantics, and is encapsulated in a semantics functor. In order to define a model of a theory in
a given category, it is necessary to have some structure that relates the arities of the operations in
the theory with the objects of the category. This leads to the second central notion of this thesis,
that of an interpretation of arities, or aritation for short. We show that any aritation gives rise
to a semantics functor from the appropriate category of proto-theories, and that this functor
has a left adjoint called the structure functor, giving rise to a structure{semantics adjunction.
Furthermore, we show that the usual semantics for many existing notions of algebraic theory
arises in this way by choosing an appropriate aritation.
Another aim of this thesis is to find a convenient category of monads in the following sense.
Every right adjoint into a category gives rise to a monad on that category, and in fact some
functors that are not right adjoints do too, namely their codensity monads. This is the structure
part of the structure{semantics adjunction for monads. However, the fact that not every functor
has a codensity monad means that the structure functor is not defined on the category of all
functors into the base category, but only on a full subcategory of it.
This deficiency is solved when passing to general proto-theories with a canonical choice of
aritation whose structure{semantics adjunction restricts to the usual one for monads. However,
this comes at a cost: the semantics functor for general proto-theories is not full and faithful,
unlike the one for monads. The condition that a semantics functor be full and faithful can be
thought of as a kind of completeness theorem | it says that no information is lost when passing
from a theory to its models. It is therefore desirable to retain this property of the semantics of
monads if possible.
The goal then, is to find a notion of algebraic theory that generalises monads for which
the semantics functor is full and faithful with a left adjoint; equivalently the semantics functor
should exhibit the category of theories as a re
ective subcategory of the category of all functors
into the base category. We achieve this (for well-behaved base categories) with a special kind of
proto-theory enriched in topological spaces, which we call a complete topological proto-theory.
We also pursue an analogy between the theory of proto-theories and that of groups. Under
this analogy, monads correspond to finite groups, and complete topological proto-theories
correspond to profinite groups. We give several characterisations of complete topological proto-theories
in terms of monads, mirroring characterisations of profinite groups in terms of finite
groups
Deterministic Pomsets
This paper is about partially ordered multisets (pomsets for short). We investigate a particular class of pomsets that we call deterministic, properly including all partially ordered sets, which satisfies a number of interesting properties: among other things, it forms a distributive lattice under pomset prefix (hence prefix closed sets of deterministic pomsets are prime algebraic), and it constitutes a reflective subcategory of the category of all pomsets. For the deterministic pomsets we develop an algebra with a sound and (ω-)complete equational theory. The operators in the algebra are concatenation and join, the latter being a variation on the more usual disjoint union of pomsets with the special property that it yields the least upper bound with respect to pomset prefix.\ud
\ud
This theory is then extended in several ways. We capture refinement of pomsets by incorporating homomorphisms between models as objects in the algebra and homomorphism application as a new operator. This in turn allows to formulate distributed termination and sequential composition of pomsets, where the latter is different from concatenation in that it is right-distributive over union. To contrast this we also formulate a notion of global termination. Each variation is captured equationally by a sound and ω-complete theory.\u
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