## The shape of a category up to directed homotopy (2004)

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Venue: | Theory Appl. Categ |

Citations: | 12 - 4 self |

### BibTeX

@ARTICLE{Grandis04theshape,

author = {Marco Grandis},

title = {The shape of a category up to directed homotopy},

journal = {Theory Appl. Categ},

year = {2004},

volume = {15},

pages = {95--146}

}

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### Abstract

This work is a contribution to a recent field, Directed Algebraic Topology. Categories which appear as fundamental categories of ‘directed structures’, e.g. ordered topological spaces, have to be studied up to appropriate notions of directed homotopy equivalence, which are more general than ordinary equivalence of categories. Here we introduce past and future equivalences of categories—sort of symmetric versions of an adjunction—and use them and their combinations to get ‘directed models ’ of a category; in the simplest case, these are the join of the least full reflective and the least full coreflective subcategory.

### Citations

968 |
Categories for the Working Mathematician
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- 1971
(Show Context)
Citation Context ...pological space ↑R (the euclidean line with the natural order), whose fundamental category is r. The classical properties of adjunctions and equivalences of categories are used without reference (see =-=[18]-=-). Cat denotes the category of small categories. Acknowledgements. The author is grateful to the referee for many helpful suggestions, meant to make the exposition clearer. 1. Directed homotopies and ... |

42 | Algebraic topology and concurrency
- Fajstrup, Goubault, et al.
(Show Context)
Citation Context ...tive model of M, its full subcategory on the objects α, β. Directed homotopies have been studied in various structures: differential graded algebras [8], ordered or locally ordered topological spaces =-=[4, 6, 7]-=-, simplicial, precubical and cubical sets [4, 6, 9, 11], directed simplicial complexes [9], directed topological spaces [10], inequilogical spaces [12], small categories [10], etc. Their present appli... |

42 | Topological deformation of higher dimensional automata
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- 2003
(Show Context)
Citation Context ...tive model of M, its full subcategory on the objects α, β. Directed homotopies have been studied in various structures: differential graded algebras [8], ordered or locally ordered topological spaces =-=[4, 6, 7]-=-, simplicial, precubical and cubical sets [4, 6, 9, 11], directed simplicial complexes [9], directed topological spaces [10], inequilogical spaces [12], small categories [10], etc. Their present appli... |

23 |
A New Category? Domains, Spaces and Equivalence Relations. Unpublished manuscript
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- 1996
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Citation Context ...urrency [4, 6, 7], ‘d-spaces’ [10] or—perhaps more simply—‘inequilogical spaces’. The category pEql of inequilogical spaces, introduced in [12], is a directed version of D. Scott’s equilogical spaces =-=[20, 19, 1]-=-; an object of this category is a preordered topological space equipped with an equivalence relation, while a morphism is an equivalence class of preorder-preserving continuous mappings which respect ... |

21 | Directed homotopy theory, i. the fundamental category, Cahiers Top
- Grandis
- 2001
(Show Context)
Citation Context ...ial graded algebras [8], ordered or locally ordered topological spaces [4, 6, 7], simplicial, precubical and cubical sets [4, 6, 9, 11], directed simplicial complexes [9], directed topological spaces =-=[10]-=-, inequilogical spaces [12], small categories [10], etc. Their present applications deal mostly with concurrency (see [4, 6, 7] and references there). The present study has similarities with a recent ... |

17 |
Adjoint for double categories
- Grandis
(Show Context)
Citation Context ... ′ y ′ )=(h ′ x ′ ,k ′ y ′ ; h ′ u ′ : h ′ x ′ → h ′ g ′ y ′ = gk ′ y ′ ). and constructs an adjunction r ⊣ r ′ which gives commutative squares in (67). A similar factorisation has been introduced in =-=[13]-=-, for a colax-lax adjunction between double categories; the present result is likely known, but we have not been able to find a reference. 4.5. Faithful adjunctions. We shall say that the adjunction f... |

13 | Directed combinatorial homology and noncommutative - Grandis - 2004 |

13 | Functorial factorization, well-pointedness and separability
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- 1999
(Show Context)
Citation Context ...ding of a full coreflective subcategory) followed by a future retraction (the reflection onto a full reflective subcategory). Within the category of adjunctions, this factorisation is functorial (cf. =-=[14]-=-) and mono-epi, but we shall not need these facts. Let f: X ⇄ Y :g be an adjunction, with η:1 → gf and ε: fg → 1. We shall factor it through the following comma category, the graph of the adjunction W... |

12 | Higher fundamental functors for simplicial sets, Cahiers Topologie Geom. Differentielle Categ
- Grandis
(Show Context)
Citation Context ...s α, β. Directed homotopies have been studied in various structures: differential graded algebras [8], ordered or locally ordered topological spaces [4, 6, 7], simplicial, precubical and cubical sets =-=[4, 6, 9, 11]-=-, directed simplicial complexes [9], directed topological spaces [10], inequilogical spaces [12], small categories [10], etc. Their present applications deal mostly with concurrency (see [4, 6, 7] and... |

11 |
Geometry and concurrency: a user's guide, in: Geometry and concurrency
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(Show Context)
Citation Context ...tive model of M, its full subcategory on the objects α, β. Directed homotopies have been studied in various structures: differential graded algebras [8], ordered or locally ordered topological spaces =-=[4, 6, 7]-=-, simplicial, precubical and cubical sets [4, 6, 9, 11], directed simplicial complexes [9], directed topological spaces [10], inequilogical spaces [12], small categories [10], etc. Their present appli... |

10 |
Cubical homotopical algebra and cochain
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- 1996
(Show Context)
Citation Context ...l: iterating the procedure, we get a smaller projective model of M, its full subcategory on the objects α, β. Directed homotopies have been studied in various structures: differential graded algebras =-=[8]-=-, ordered or locally ordered topological spaces [4, 6, 7], simplicial, precubical and cubical sets [4, 6, 9, 11], directed simplicial complexes [9], directed topological spaces [10], inequilogical spa... |

7 | Inequilogical spaces, directed homology and noncommutative geometry
- Grandis
(Show Context)
Citation Context ...dered or locally ordered topological spaces [4, 6, 7], simplicial, precubical and cubical sets [4, 6, 9, 11], directed simplicial complexes [9], directed topological spaces [10], inequilogical spaces =-=[12]-=-, small categories [10], etc. Their present applications deal mostly with concurrency (see [4, 6, 7] and references there). The present study has similarities with a recent one [5], using categories o... |

6 |
Equilogical spaces and filter spaces, Categorical studies in Italy (Perugia
- Rosolini
- 1997
(Show Context)
Citation Context ...urrency [4, 6, 7], ‘d-spaces’ [10] or—perhaps more simply—‘inequilogical spaces’. The category pEql of inequilogical spaces, introduced in [12], is a directed version of D. Scott’s equilogical spaces =-=[20, 19, 1]-=-; an object of this category is a preordered topological space equipped with an equivalence relation, while a morphism is an equivalence class of preorder-preserving continuous mappings which respect ... |

5 | On locales of localizations
- Borceux, Kelly
- 1987
(Show Context)
Citation Context ...ote the symmetry of this presentation. 3.7. Split pf-projections. The dual notion of split pf-projection is well-known in category theory: it has been studied under the name of essential localisation =-=[16, 2]-=-, or ‘unity and identity of adjoint opposites’ [17]; presently, the term ‘adjoint reflexive graph’ is also used by F.W. Lawvere. It can be presented as a pf-equivalence p: X ← → ←M :i α where the natu... |

4 |
Localization of universal problems
- Ehresmann
(Show Context)
Citation Context ...he same time (as in 9.2). Finally, the present analysis of a category through minimal past and future models has unexpectedly appeared to be related with notions recently introduced by A.C. Ehresmann =-=[3]-=-, for investigating biosystems, neural systems, etc.—the ’root’ and ’coroot’ of a category. Such relations, having likely at their basis a common purpose of studying non-reversible phenomena, will be ... |

4 |
On the ordered set of reflective subcategories
- Kelly
- 1987
(Show Context)
Citation Context ...ples of Section 9. The categories r and c have minimal future retracts, but do not have a least one (5.5, 5.6). The ordered set of (replete) reflective subcategories of a category was investigated in =-=[15]-=- (see also its references). Such results are generally based on the existence of limits in the original category and cannot be applied here. 7.2. Spectra. Recall that we have defined, in the set of ob... |

4 |
Unity and identity of opposites in calculus and physics. Applied categorical structures
- Lawvere
- 1996
(Show Context)
Citation Context ...rojections. The dual notion of split pf-projection is well-known in category theory: it has been studied under the name of essential localisation [16, 2], or ‘unity and identity of adjoint opposites’ =-=[17]-=-; presently, the term ‘adjoint reflexive graph’ is also used by F.W. Lawvere. It can be presented as a pf-equivalence p: X ← → ←M :i α where the natural transformations pi − → 1M and 1M → pi + are ide... |

2 |
On the complete lattice of essential localizations, Actes du Colloque en l’Honneur du Soixantième Anniversaire de R. Lavendhomme (Louvain–la-Neuve
- Kelly, Lawvere
- 1989
(Show Context)
Citation Context ...ote the symmetry of this presentation. 3.7. Split pf-projections. The dual notion of split pf-projection is well-known in category theory: it has been studied under the name of essential localisation =-=[16, 2]-=-, or ‘unity and identity of adjoint opposites’ [17]; presently, the term ‘adjoint reflexive graph’ is also used by F.W. Lawvere. It can be presented as a pf-equivalence p: X ← → ←M :i α where the natu... |