## Dynamical systems and diagrams (2010)

### BibTeX

@MISC{Jardine10dynamicalsystems,

author = {J. F. Jardine},

title = {Dynamical systems and diagrams},

year = {2010}

}

### OpenURL

### Abstract

In general, a dynamical system consists of an action

### Citations

121 |
Simplicial presheaves
- Jardine
(Show Context)
Citation Context ... m+n −→ sd (sd X) = sd X (2) which is natural in simplicial sets X, by adjointness. Suppose that C is a small Grothendieck site, and let s Pre(C) be the category of simplicial presheaves on C. Recall =-=[2]-=-, [3] that the category s Pre((C)) has a proper closed simplicial model structure, for which the cofibrations are the monomorphisms, the weak equivalences are the local weak equivalences, and the fibr... |

94 |
Simplicial homotopy theory, volume 174
- Goerss, Jardine
- 1999
(Show Context)
Citation Context ...all categories 17 4 Presheaves of simplicial categories 23 5 Diagrams for simplicial categories 27 1 Subdivision model structures We begin by recalling some of the basic features of subdivisions [4], =-=[1]-=-. Every simplicial set X has a poset NX of non-degerate simplices, ordered by the face relationship. The assignment X ↦→ NX is functorial in X: if f : X → Y is a simplicial set map and σ is a non-dege... |

15 |
Boolean localization in practice
- Jardine
- 1996
(Show Context)
Citation Context ...−→ sd (sd X) = sd X (2) which is natural in simplicial sets X, by adjointness. Suppose that C is a small Grothendieck site, and let s Pre(C) be the category of simplicial presheaves on C. Recall [2], =-=[3]-=- that the category s Pre((C)) has a proper closed simplicial model structure, for which the cofibrations are the monomorphisms, the weak equivalences are the local weak equivalences, and the fibration... |

12 |
Cat as a closed model category, Cahiers Topologie Géom. Différentielle XXI
- Thomason
- 1980
(Show Context)
Citation Context ...m as a bisimplicial set map X → BE. The homotopy theory for presheaves of simplicial categories which appears here in Theorem 13 is a generalization of Thomason’s model structure for small categories =-=[10]-=-. It is Quillen equivalent to an “sd 2,0 -model structure” on the category of bisimplicial presheaves, in which the weak equivalences are diagonal local weak equivalences, and the cofibrations are gen... |

6 | Simplicial approximation
- Jardine
(Show Context)
Citation Context ...of small categories 17 4 Presheaves of simplicial categories 23 5 Diagrams for simplicial categories 27 1 Subdivision model structures We begin by recalling some of the basic features of subdivisions =-=[4]-=-, [1]. Every simplicial set X has a poset NX of non-degerate simplices, ordered by the face relationship. The assignment X ↦→ NX is functorial in X: if f : X → Y is a simplicial set map and σ is a non... |

6 |
Higher principal bundles
- Jardine, Luo
(Show Context)
Citation Context ...tors Lh : Ho(Set A ) ⇆ Ho(s 2 Set/BA) : d · pb . It also follows that there is a natural weak equivalence d(γφ∗) d(γ) −−→ d(φ) The maps α, γ : holim −−−→ A B(A/?) → BA are homotopic (see Remark 16 of =-=[7]-=-), and so the functors Ho(sSet/d(holim −−−→ AB(A/?))) → Ho(sSet/d(BA)) which are defined by composition with the maps d(α) and d(γ) coincide. follows that there is a natural isomorphism It d(Lhd(pb(φ)... |

2 | Diagonal model structures
- Jardine
- 2010
(Show Context)
Citation Context ... taking double subdivisions of ordinary cofibrations in all vertical degrees. This sd 2,0 -model structure for bisimplicial presheaves is Quillen equivalent to the “ordinary” diagonal model structure =-=[8]-=-, but also gives a setting for the model structure for presheaves of simplical categories: a morphism f : C → D of such objects is a weak equivalence if the map BC → BD of bisimplicial nerves is a dia... |

2 | Path categories and resolutions
- Jardine
(Show Context)
Citation Context ...he induced functor i∗ : P (A) → P (X) of path categories is a full imbedding. Here, the path category functor P : sSet → Cat is defined to be the left adjoint of the nerve functor B : Cat → sSet. See =-=[9]-=-. Proof. Condition 1) implies that all induced functions i∗ : P (A)(a, b) → P (X)(a, b) are surjective, so that the functor i∗ is full. Suppose that a ∈ A0. Then the 1-simplex a s0a −−→ a of X is in A... |