47 research outputs found
A Quillen model structure for Gray-categories
A Quillen model structure on the category Gray-Cat of Gray-categories is
described, for which the weak equivalences are the triequivalences. It is shown
to restrict to the full subcategory Gray-Gpd of Gray-groupoids. This is used to
provide a functorial and model-theoretic proof of the unpublished theorem of
Joyal and Tierney that Gray-groupoids model homotopy 3-types. The model
structure on Gray-Cat is conjectured to be Quillen equivalent to a model
structure on the category Tricat of tricategories and strict homomorphisms of
tricategories.Comment: v2: fuller discussion of relationship with work of Berger;
localizations are done directly with simplicial set
Higher quasi-categories vs higher Rezk spaces
We introduce a notion of n-quasi-categories as fibrant objects of a model
category structure on presheaves on Joyal's n-cell category \Theta_n. Our
definition comes from an idea of Cisinski and Joyal. However, we show that this
idea has to be slightly modified to get a reasonable notion. We construct two
Quillen equivalences between the model category of n-quasi-categories and the
model category of Rezk \Theta_n-spaces showing that n-quasi-categories are a
model for (\infty, n)-categories. For n = 1, we recover the two Quillen
equivalences defined by Joyal and Tierney between quasi-categories and complete
Segal spaces.Comment: 44 pages, v2: terminology changed (see Remark 5.27), Corollary 7.5
added, appendix A added, references added, v3: reorganization of Sections 5
and 6, more informal comments, new section characterizing strict n-categories
whose nerve is an n-quasi-category, numbering has change
Segal-type algebraic models of n-types
For each n\geq 1 we introduce two new Segal-type models of n-types of
topological spaces: weakly globular n-fold groupoids, and a lax version of
these. We show that any n-type can be represented up to homotopy by such models
via an explicit algebraic fundamental n-fold groupoid functor. We compare these
models to Tamsamani's weak n-groupoids, and extract from them a model for
(k-1)connected n-typesComment: Added index of terminology and notation. Minor amendments and added
details is some definitions and proofs. Some typos correcte
Surveying drifting icebergs and ice islands: Deterioration detection and mass estimation with aerial photogrammetry and laser scanning
Icebergs and ice islands (large, tabular icebergs) are challenging targets to survey due to their size, mobility, remote locations, and potentially difficult environmental conditions. Here, we assess the precision and utility of aerial photography surveying with structure-from-motion multi-view stereo photogrammetry processing (SfM) and vessel-based terrestrial laser scanning (TLS) for iceberg deterioration detection and mass estimation. For both techniques, we determine the minimum amount of change required to reliably resolve iceberg deterioration, the deterioration detection threshold (DDT), using triplicate surveys of two iceberg survey targets. We also calculate their relative uncertainties for iceberg mass estimation. The quality of deployed Global Positioning System (GPS) units that were used for drift correction and scale assignment was a major determinant of point cloud precision. When dual-frequency GPS receivers were deployed, DDT values of 2.5 and 0.40 m were calculated for the TLS and SfM point clouds, respectively. In contrast, values of 6.6 and 3.4 m were calculated when tracking beacons with lower-quality GPS were used. The SfM dataset was also more precise when used for iceberg mass estimation, and we recommend further development of this technique for iceberg-related end-uses
Monads with arities and their associated theories
After a review of the concept of "monad with arities" we show that the
category of algebras for such a monad has a canonical dense generator. This is
used to extend the correspondence between finitary monads on sets and Lawvere's
algebraic theories to a general correspondence between monads and theories for
a given category with arities. As application we determine arities for the free
groupoid monad on involutive graphs and recover the symmetric simplicial nerve
characterisation of groupoids.Comment: New introduction; Section 1 shortened and redispatched with Section
2; Subsections on symmetric operads (3.14) and symmetric simplicial sets
(4.17) added; Bibliography complete
Model Structures on the Category of Small Double Categories
In this paper we obtain several model structures on {\bf DblCat}, the
category of small double categories. Our model structures have three sources.
We first transfer across a categorification-nerve adjunction. Secondly, we view
double categories as internal categories in {\bf Cat} and take as our weak
equivalences various internal equivalences defined via Grothendieck topologies.
Thirdly, {\bf DblCat} inherits a model structure as a category of algebras over
a 2-monad. Some of these model structures coincide and the different points of
view give us further results about cofibrant replacements and cofibrant
objects. As part of this program we give explicit descriptions and discuss
properties of free double categories, quotient double categories, colimits of
double categories, several nerves, and horizontal categorification.Comment: 103 pages. Included Quillen adjunctions with Cat, improved
characterization of flexible double categories, proved 2-cocompleteness of
DblCat_v, proved horizontal nerve is 2-coskeletal, cut double
categorification for a future article, removed identity squares from double
derivation schemes, improved counterexample to Reedy transfer
General Reversibility
The first and the second author introduced reversible ccs (rccs) in order to model concurrent computations where certain actions are allowed to be reversed. Here we show that the core of the construction can be analysed at an abstract level, yielding a theorem of pure category theory which underlies the previous results. This opens the way to several new examples; in particular we demonstrate an application to Petri nets.
The fundamental pro-groupoid of an affine 2-scheme
A natural question in the theory of Tannakian categories is: What if you
don't remember \Forget? Working over an arbitrary commutative ring , we
prove that an answer to this question is given by the functor represented by
the \'etale fundamental groupoid \pi_1(\spec(R)), i.e.\ the separable
absolute Galois group of when it is a field. This gives a new definition
for \'etale \pi_1(\spec(R)) in terms of the category of -modules rather
than the category of \'etale covers. More generally, we introduce a new notion
of "commutative 2-ring" that includes both Grothendieck topoi and symmetric
monoidal categories of modules, and define a notion of for the
corresponding "affine 2-schemes." These results help to simplify and clarify
some of the peculiarities of the \'etale fundamental group. For example,
\'etale fundamental groups are not "true" groups but only profinite groups, and
one cannot hope to recover more: the "Tannakian" functor represented by the
\'etale fundamental group of a scheme preserves finite products but not all
products.Comment: 46 pages + bibliography. Diagrams drawn in Tik
A Thomason Model Structure on the Category of Small n-fold Categories
We construct a cofibrantly generated Quillen model structure on the category
of small n-fold categories and prove that it is Quillen equivalent to the
standard model structure on the category of simplicial sets. An n-fold functor
is a weak equivalence if and only if the diagonal of its n-fold nerve is a weak
equivalence of simplicial sets. This is an n-fold analogue to Thomason's
Quillen model structure on Cat. We introduce an n-fold Grothendieck
construction for multisimplicial sets, and prove that it is a homotopy inverse
to the n-fold nerve. As a consequence, we completely prove that the unit and
counit of the adjunction between simplicial sets and n-fold categories are
natural weak equivalences.Comment: More details added. 23 new pages for a total of 77 pages
Understanding the small object argument
The small object argument is a transfinite construction which, starting from
a set of maps in a category, generates a weak factorisation system on that
category. As useful as it is, the small object argument has some problematic
aspects: it possesses no universal property; it does not converge; and it does
not seem to be related to other transfinite constructions occurring in
categorical algebra. In this paper, we give an "algebraic" refinement of the
small object argument, cast in terms of Grandis and Tholen's natural weak
factorisation systems, which rectifies each of these three deficiencies.Comment: 42 pages; supersedes the earlier arXiv preprint math/0702290; v2:
final journal version, minor corrections onl