127 research outputs found
Categories, norms and weights
The well-known Lawvere category R of extended real positive numbers comes
with a monoidal closed structure where the tensor product is the sum. But R has
another such structure, given by multiplication, which is *-autonomous.
Normed sets, with a norm in R, inherit thus two symmetric monoidal closed
structures, and categories enriched on one of them have a 'subadditive' or
'submultiplicative' norm, respectively. Typically, the first case occurs when
the norm expresses a cost, the second with Lipschitz norms.
This paper is a preparation for a sequel, devoted to 'weighted algebraic
topology', an enrichment of directed algebraic topology. The structure of R,
and its extension to the complex projective line, might be a first step in
abstracting a notion of algebra of weights, linked with physical measures.Comment: Revised version, 16 pages. Some minor correction
Finite Sets And Symmetric Simplicial Sets
The category of finite cardinals (or, equivalently, of finite sets) is the symmetric analogue of the category of finite ordinals, and the ground category of a relevant category of presheaves, the augmented symmetric simplicial sets. We prove here that this ground category has characterisations similar to the classical ones for the category of finite ordinals, by the existence of a universal symmetric monoid, or by generators and relations. The latter provides a definition of symmetric simplicial sets by faces, degeneracies and transpositions, under suitable relations
Higher fundamental functors for simplicial sets
This is a sequel to a previous paper, developing an intrinsic, combinatorial
homotopy theory for simplicial complexes; the latter form the cartesian closed
subcategory of 'simple presheaves' in !Smp, the topos of symmetric simplicial
sets, or presheaves on the category of finite, positive cardinals.
We show here how this homotopy theory can be extended to the topos itself,
!Smp. As a crucial advantage, the fundamental groupoid functor !Smp --> Gpd is
left adjoint to a natural functor Gpd --> !Smp, the symmetric nerve of a
groupoid, and preserves all colimits - a strong van Kampen property. Similar
results hold in all higher dimensions. Analogously, a notion of
(non-reversible) directed homotopy can be developed in the ordinary simplicial
topos Smp, with applications to image analysis as in the previous paper. We
have now a 'homotopy n-category functor' Smp --> n-Cat, left adjoint to a nerve
functor.
This construction can be applied to various presheaf categories; the basic
requirements seem to be: finite products of representables are finitely
presentable and there is a representable 'standard interval'.Comment: Revised version with minor changes 36 pages, 428
An intrinsic homotopy theory for simplicial complexes, with applications to image analysis
A simplicial complex is a set equipped with a down-closed family of
distinguished finite subsets. This structure, usually viewed as codifying a
triangulated space, is used here directly, to describe "spaces" whose geometric
realisation can be misleading. An intrinsic homotopy theory, not based on such
realisation but agreeing with it, is introduced.
The applications developed here are aimed at image analysis in metric spaces
and have connections with digital topology and mathematical morphology. A
metric space X has a structure of simplicial complex at each (positive)
resolution e; the resulting n-homotopy group detects those singularities which
can be captured by an n-dimensional grid, with edges bound by e; this works
equally well for continuous or discrete regions of euclidean spaces. Its
computation is based on direct, intrinsic methods.Comment: 46 page
The topology of critical processes, I (Processes and Models)
This article belongs to a subject, Directed Algebraic Topology, whose general
aim is including non-reversible processes in the range of topology and
algebraic topology. Here, as a further step, we also want to cover "critical
processes", indivisible and unstoppable. This introductory article is devoted
to fixing the new framework and representing processes of diverse domains, with
minimal mathematical prerequisites. The fundamental category and singular
homology in the present setting will be dealt with in a sequel.Comment: 23 page
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