## Paths in double categories

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Venue: | Theory Appl. Categ |

Citations: | 7 - 2 self |

### BibTeX

@ARTICLE{Dawson_pathsin,

author = {R. J. Macg Dawson and R. Paré and D. A. Pronk},

title = {Paths in double categories},

journal = {Theory Appl. Categ},

year = {},

volume = {16},

pages = {2006}

}

### OpenURL

### Abstract

Abstract. Two constructions of paths in double categories are studied, providing algebraic versions of the homotopy groupoid of a space. Universal properties of these constructions are presented. The first is seen as the codomain of the universal oplax morphism of double categories and the second, which is a quotient of the first, gives the universal normal oplax morphism. Normality forces an equivalence relation on cells, a special case of which was seen before in the free adjoint construction. These constructions are the object part of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fc-multicategories, with representable identities in the second case.

### Citations

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(Show Context)
Citation Context ...FAmn FA00 ��··· �� FAmn. Ff01 Ffmns474 R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK 1.11. Remark. Conditions (OL2) - (OL4) are the same as the coherence conditions for oplax morphisms of bicategories =-=[Bé]-=-. In the case of bicategories, condition (OL1) is vacuous as all vertical arrows are identities and 1 =id1A . However, when specifying idA the cells ϕA for bicategories one might wonder whether furthe... |

78 |
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(Show Context)
Citation Context ...mains Ti, theycanbe vertically composed with the above one, and this composition is associative. This is the kind of structure we want. It is simply the vertical dual of Leinster’s fc-multicategories =-=[L1, L2, L3]-=-, whose theory we now review. They are the double category version of what Hermida [H1] calls ‘multicategories with several objects’ and what Hermida, Makkai, and Power [HMP] call ‘multitopic categori... |

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Citation Context ... categories, because of the applications we have in mind.)s484 R. J. MACG. DAWSON, R. PARÉ, AND D. A. PRONK 2.1. Multicategories. All of these concepts are an outgrowth of the notion of multicategory =-=[Lam]-=- which was inspired by the relationship between tensor products and multilinear maps. A (small) multicategory consists of a set of objects and set of multiarrows with domains finite strings of objects... |

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(Show Context)
Citation Context ...es to get the two dimensional structure of Doub and Path. In fact, Pathisevenbetterthana2-comonad,itisone in which the structure morphisms are left adjoint to the counits, of the sort studied by Kock =-=[K]-=- and Zöberlein [Z]. Following Kelly and Lack [KL] we call them oplax idempotent comonads. (See also [M].) Let us recall the definition. A 2-comonad G =(G, E, D) on a 2-category A is oplax idempotent i... |

19 |
private communications
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(Show Context)
Citation Context ...for every object A, there are 2-cells α: DA · GEA → 1G2A and β :1G2A → DA· EGA which together with the triangle identities 1GA = GEA · DA and EGA· DA = 1GA give an adjoint triple GEA ⊣ DA ⊣ EGA. Wood =-=[W]-=- has shown that if ε: UR → 1B and η :1A → RU are the adjunctions for an �� U adjoint pair A ��B of 2-functors, then there is a transformation ϕ: ηR · Rε → 1RUR R which together with the triangle ident... |

18 |
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(Show Context)
Citation Context ...−1)···g2)g1 �� Bn α A f That we actually get a lax double category follows from the coherence theorem for (weak) double categories which is a minor variation on the coherence theorem for bicategories =-=[DP2]-=-, and in fact is equivalent to it. This is one of Hermida’s main points in [H1]. The following theorem is implicitly in [H1] (see 9.4 and 9.5 there). 2.6. Theorem. Let A and B be double categories. A ... |

18 | Doctrines whose structure forms a fully faithful adjoint string
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(Show Context)
Citation Context ...ne in which the structure morphisms are left adjoint to the counits, of the sort studied by Kock [K] and Zöberlein [Z]. Following Kelly and Lack [KL] we call them oplax idempotent comonads. (See also =-=[M]-=-.) Let us recall the definition. A 2-comonad G =(G, E, D) on a 2-category A is oplax idempotent if for every object A, there are 2-cells α: DA · GEA → 1G2A and β :1G2A → DA· EGA which together with th... |

15 |
On weak higherdimensional categories I: Part 1
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(Show Context)
Citation Context ...fc-multicategories [L1, L2, L3], whose theory we now review. They are the double category version of what Hermida [H1] calls ‘multicategories with several objects’ and what Hermida, Makkai, and Power =-=[HMP]-=- call ‘multitopic categories’. We propose to call them lax double categories for reasons hinted at above and which should become clear shortly. (Actually, we shall study the dual oplax double categori... |

13 |
Limits in double categories, Cahiers Topologie Géom. Différentielle Catég
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(Show Context)
Citation Context ...rom the definition of θ ∗ .sPATHS IN DOUBLE CATEGORIES 471 1.9. Oplax morphisms. In order to express the universal property of Path A we need the notion of oplax morphism of double categories. Recall =-=[GP]-=- that F : A → B is an oplax morphism if it assigns to objects, vertical and horizontal arrows, and cells of A corresponding elements of B, preserving domains and codomains A1 v1 • �� A3 f1 �� A2 α f3 ... |

13 | From coherent structures to universal properties - Hermida |

8 |
Representable multicategories, Adv
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(Show Context)
Citation Context .... This is the kind of structure we want. It is simply the vertical dual of Leinster’s fc-multicategories [L1, L2, L3], whose theory we now review. They are the double category version of what Hermida =-=[H1]-=- calls ‘multicategories with several objects’ and what Hermida, Makkai, and Power [HMP] call ‘multitopic categories’. We propose to call them lax double categories for reasons hinted at above and whic... |

8 | On property-like structures
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(Show Context)
Citation Context ...and Path. In fact, Pathisevenbetterthana2-comonad,itisone in which the structure morphisms are left adjoint to the counits, of the sort studied by Kock [K] and Zöberlein [Z]. Following Kelly and Lack =-=[KL]-=- we call them oplax idempotent comonads. (See also [M].) Let us recall the definition. A 2-comonad G =(G, E, D) on a 2-category A is oplax idempotent if for every object A, there are 2-cells α: DA · G... |

7 | What is a free double category like
- Dawson, Paré
(Show Context)
Citation Context ...tanding of the Π2 construction and its relationship to spans. The main results of this section, Theorems 1.17 and 1.21 below, justify our choices. Let A be a double category (see, for example, [E] or =-=[DP1]-=-). We construct a new double category Path A with the same objects and vertical arrows as in A. Ahorizontal arrow in Path A is a path A0 f1 −→ A1 of horizontal arrows in A, withm � 0. Let B0 g1 −→ B1 ... |

6 | 2002) Categories enriched on two sides - Kelly, Labella, et al. |

5 |
Adjoining adjoints
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(Show Context)
Citation Context ...of 2-comonads which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fc-multicategories, with representable identities in the second case. Introduction In =-=[DPP1]-=- and [DPP2] we studied the 2-category Π2A obtained by freely adjoining right adjoints to each arrow of a category A considered as a locally discrete 2-category. The 2-category Π2A was conceived as a m... |

4 |
Undecidability of the free adjoint construction
- Dawson, Pronk
(Show Context)
Citation Context ...ds which are shown to be oplax idempotent. The coalgebras for these comonads turn out to be Leinster’s fc-multicategories, with representable identities in the second case. Introduction In [DPP1] and =-=[DPP2]-=- we studied the 2-category Π2A obtained by freely adjoining right adjoints to each arrow of a category A considered as a locally discrete 2-category. The 2-category Π2A was conceived as a more informa... |

4 | Universal properties of span
- Dawson, Paré, et al.
(Show Context)
Citation Context ... of spans in A. The cells of Π2A are a bit more complicated, being equivalence classes of certain diagrams, which we call fences and which formally look like directed homotopies between the paths. In =-=[DPP3]-=- we studied several universal properties of the Span construction applied to categories with pullbacks, each expressing the sense in which Span (A) istheresultof adjoining right adjoints for the arrow... |

4 | Generalized enrichment of categories - Leinster - 2002 |

4 |
Doktrinen auf 2-Kategorien
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(Show Context)
Citation Context ...dimensional structure of Doub and Path. In fact, Pathisevenbetterthana2-comonad,itisone in which the structure morphisms are left adjoint to the counits, of the sort studied by Kock [K] and Zöberlein =-=[Z]-=-. Following Kelly and Lack [KL] we call them oplax idempotent comonads. (See also [M].) Let us recall the definition. A 2-comonad G =(G, E, D) on a 2-category A is oplax idempotent if for every object... |

3 |
Coalgebras in a category of algebras
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(Show Context)
Citation Context ... get a monad ( ) ∗ on Gph whose Eilenberg-Moore algebras are small categories; i.e., U is monadic. This of course is well known. What is perhaps less well known is that, in the spirit of Barr’s paper =-=[Ba]-=-, F is comonadic. Thus, a graph may be considered as a category equipped with a costructure. It will be instructive for us to work out in detail how this works. The triple (FU,ε,FηU) is a comonad on C... |

3 | Introduction to linear bicategories. The Lambek Festschrift: mathematical structures in computer science - Cockett, Koslowski, et al. - 1997 |

3 | Morphisms and modules for polybicategories - Cockett, Koslowski, et al. - 2003 |

3 |
2-categories and directed
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(Show Context)
Citation Context ... related notions of lax bicategories which have comparison cells between certain pairs of associations of composites in a preferred direction (which is chosen to model problems in concurrency theory) =-=[G]-=-. These form a special case of Leinster’s lax bicategories where some of the comparison cells are identities. We believe that our notion of lax double category is the most basic one from which the oth... |

3 | A categorical outlook on relational modalities and simulations - Hermida - 2002 |

2 |
Lax monads. Indexed monoidal monads
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(Show Context)
Citation Context ... components are all identities (so there are only 2-cells between morphisms which agree on objects). However, they are not that special, and indeed occur throughout bicategory theory (see for example =-=[CR]-=-). Let Doub(A, B) denote the category of double functors from A to B with vertical natural transformations, and DoubOpl(A, B) the category of oplax morphisms from A to B with vertical transformations.... |

2 | A monadic approach to polycategories - Koslowski - 2005 |

1 |
T -catégories, Cahiers Top
- Burroni
(Show Context)
Citation Context ...ich T preserves pullbacks and the naturality squares f A �� B ηB ηA ��TA Tf and T 2 f T 2 A µA �� �� �� TB 2 T B µB �� TA Tf �� �� TB are all pullbacks. This gives Burroni’s notion of T-multicategory =-=[Bu]-=-. In keeping with the theme of spans, let us introduce the double category T-Span(A) for any cartesian monad T =(T,η,µ) on a category A with pullbacks. The objects of T-Span(A) are those of A. A horiz... |

1 |
A convenient 2-category of bicategories, Presentation at CT2006, www.mathstat.dal.ca/∼selinger/ct2006/slides/CT06-Lack.pdf
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(Show Context)
Citation Context ...illuminating. Indeed, it is well-known that we cannot form a 2-category consisting of bicategories, oplax morphisms and any of the obvious choices of 2-cells (lax, oplax, or pseudo). (See for example =-=[Lac]-=-.) Indeed, although oplax morphisms do compose in a strictly associative way (so we would obtain a 2-category rather than a bicategory if anything) whiskering does not work for any of these choices. S... |

1 |
fc-multicategories, E-print math.CT/9903004
- Leinster
- 1999
(Show Context)
Citation Context ...mains Ti, theycanbe vertically composed with the above one, and this composition is associative. This is the kind of structure we want. It is simply the vertical dual of Leinster’s fc-multicategories =-=[L1, L2, L3]-=-, whose theory we now review. They are the double category version of what Hermida [H1] calls ‘multicategories with several objects’ and what Hermida, Makkai, and Power [HMP] call ‘multitopic categori... |