arXiv subjects

Marco Grandis

Publications and source records attributed to Marco Grandis.

18 recordsLinked to original sources

arXiv2026

Enriched categories, real metrics and Lorentz manifolds

This expository article brings together two subjects: generalised metrics based on enriched categories, on the one hand, and Lorentz manifolds, on the other, at the price of dealing with details that are well known either in category theory or in relativity. The spacetime of relativity can be given a real valued metric $ρ(x, y)$, with values in the extended real line, or better (if equivalently) a real valued `antimetric' $γ(x, y) = - ρ(x, y)$ (satisfying a reverse triangle inequality); the latter, as a function of $y$, is positive on the timecone of $x$, annihilates on its lightcone, and is $- \infty$ on all events which cannot be influenced by $x$. All this can be given a well-established base in category theory, extending Lawvere's notion of a metric space. In fact, a space with a real valued metric can be viewed as an enriched category on the extended real line, structured as a symmetric monoidal closed category.

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arXiv2026

Weighted algebraic topology, II (Real valued metrics)

Extending the `metric spaces' of Lawvere, we study `real metrics', with values in the extended real line. Formally, this ordered set is a symmetric monoidal closed category, and our structures are enriched categories on the latter. Concretely, the present goal is measuring `profits' and `losses' of a process, in any sense - possibly related to energy, or a variable in any science. In particular, linear real metrics derive from a potential function. This article is Part II in a series devoted to `weighted algebraic topology' - an enriched version of directed algebraic topology, where paths are measured. Part III will introduce a finer framework, more adequate to `quotient spaces' (as the spheres) and better related to topology.

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arXiv2025

Multiple categories of general quintets

We construct various multiple categories, based on generalised Ehresmann quintets. The main construction is a multiple category whose objects are all the `lax' multiple categories; the transversal arrows are their strict multiple functors while the arrows in a positive direction are multiple functors of a `mixed laxity', varying from the lax ones (in direction 1) to the colax ones (in direction \infty).

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arXiv2024

The topology of critical processes, II (The fundamental category)

Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The first part of this series introduced controlled spaces, examining how they can model critical processes in various domains, from the change of state in a memory cell to the action of a thermostat or a siphon. We now construct the fundamental category of these spaces. [The previous version of this Preprint was too long and has been split in Parts II and III.]

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arXiv2024

The topology of critical processes, III (Computing homotopy)

Directed Algebraic Topology studies spaces equipped with a form of direction, to include models of non-reversible processes. In the present extension we also want to cover critical processes, indecomposable and unstoppable. The previous parts of this series introduced controlled spaces and their fundamental category. Here we study how to compute the latter. The homotopy structure of these spaces will be examined in Part IV.

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arXiv2024

A note on the smash product and regular associativity

This note is about the smash product of pointed topological spaces, without relying on some convenient subcategory. We deal with its partial associativity properties and their connection with the function spaces, introducing a property of regular associativity related to the colax monoidal structure of the product. We also study a large class of triples of pointed spaces where associativity fails. Lax and colax monoidal structures are unusual and interesting, in category theory. Some parts of this note will be obvious to a topologist and others to a categorist, in order to take into account both backgrounds.

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arXiv2023

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.

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arXiv2016

Intercategories: A framework for three-dimensional category theory

We show how the notion of intercategory encompasses a wide variety of three-dimensional structures from the literature, notably duoidal categories, monoidal double categories, cubical bicategories, double bicategories and Gray categories. Variations on the notion of span provide further examples of interest, an important one being the intercategory of sets. We consider the three kinds of morphism of intercategory as well as the cells binding them with applications to the above structures. In particular hom functors are studied.

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arXiv2015

Intercategories

We introduce a 3-dimensional categorical structure which we call intercategory. This is a kind of weak triple category with three kinds of arrows, three kinds of 2-dimensional cells and one kind of 3-dimensional cells. In one dimension, the compositions are strictly associative and unitary, whereas in the other two, these laws only hold up to coherent isomorphism. The main feature is that the interchange law between the second and third compositions does not hold, but rather there is a non invertible comparison cell which satisfies some coherence conditions. We introduce appropriate morphisms of intercategory, of which there are three types, and cells relating these. We show that these fit together to produce a strict triple category of intercategories.

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arXiv2010

Homotopy spectral sequences

In homotopy theory, exact sequences and spectral sequences consist of groups and pointed sets, linked by actions. We prove that the theory of such exact and spectral sequences can be established in a categorical setting which is based on the existence of kernels and cokernels with respect to an assigned ideal of null morphisms, a generalisation of abelian categories and Puppe-exact categories.

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arXiv2008

Cubical cospans and higher cobordisms (Cospans in algebraic topology, III)

After two papers on weak cubical categories and {\it collarable} cospans, respectively, we put things together and construct a {\it weak} cubical category of cubical {\it collared} cospans of topological spaces. We also build a second structure, called a {\it quasi} cubical category, formed of arbitrary cubical cospans concatenated by homotopy pushouts. This structure, simpler but weaker, has {\it lax} identities. It contains a similar framework for cobordisms of manifolds with corners and could therefore be the basis to extend the study of TQFT's of Part II to higher cubical degree.

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arXiv2006

The fundamental weighted category of a weighted space (From directed to weighted algebraic topology)

We want to investigate 'spaces' where paths have a 'weight', or 'cost', expressing length, duration, price, energy, etc. The weight function is not assumed to be invariant up to path-reversion. Thus, 'weighted algebraic topology' can be developed as an enriched version of directed algebraic topology, where illicit paths are penalised with an infinite cost, and the licit ones are measured. Its algebraic counterpart will be 'weighted algebraic structures', equipped with a sort of directed seminorm. In the fundamental weighted category of a generalised metric space, introduced here, each homotopy class of paths has a weight (or seminorm), which is subadditive with respect to composition. We also study a more general setting, spaces with weighted paths, which has finer quotients and strong links with noncommutative geometry. Weighted homology of weighted cubical sets has already been developed in a previous work, with similar results.

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arXiv2006

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.

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arXiv2001

Directed homotopy theory, I. The fundamental category

Directed Algebraic Topology is beginning to emerge from various applications. The basic structure we shall use for such a theory, a 'd-space', is a topological space equipped with a family of 'directed paths', closed under some operations. This allows for 'directed homotopies', generally non reversible, represented by a cylinder and cocylinder functors. The existence of 'pastings' (colimits) yields a geometric realisation of cubical sets as d-spaces, together with homotopy constructs which will be developed in a sequel. Here, the 'fundamental category' of a d-space is introduced and a 'Seifert - van Kampen' theorem proved; its homotopy invariance rests on 'directed homotopy' of categories. In the process, new shapes appear, for d-spaces but also for small categories, their elementary algebraic model. Applications of such tools are briefly considered or suggested, for objects which model a directed image, or a portion of space-time, or a concurrent process.

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arXiv2001

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'.

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arXiv2001

On the monad of proper factorisation systems in categories

It is known that factorisation systems in categories can be viewed as unitary pseudo algebras for the "squaring" monad in Cat. We show in this note that an analogous fact holds for proper (i.e., epi-mono) factorisation systems and a suitable quotient of the former monad, deriving from a construct introduced by P. Freyd for stable homotopy. Structural similarities of the previous monad with the path endofunctor of topological spaces are considered.

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arXiv2000

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.

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arXiv2000

Exactness and stability in homotopical algebra

Exact sequences are a well known notion in homological algebra. We investigate here the more vague properties of 'homotopical exactness', appearing for instance in the fibre or cofibre sequence of a map. Such notions of exactness can be given for very general 'categories with homotopies' having homotopy kernels and cokernels, but become more interesting under suitable stability hypotheses, satisfied - in particular - by chain complexes. It is then possible to measure the default of homotopical exactness of a sequence by the homotopy type of a certain object, a sort of 'homotopical homology'.

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