## Homotopy equivalence of isospectral graphs. (906)

### BibTeX

@MISC{Bisson906homotopyequivalence,

author = {Terrence Bisson and Aristide Tsemo},

title = {Homotopy equivalence of isospectral graphs.},

year = {906}

}

### OpenURL

### Abstract

to this model structure. We endow the categories of N-sets and Z-sets with related model structures, and show that their homotopy categories are Quillen equivalent to the homotopy category Ho(Gph). This enables us to show that Ho(Gph) is equivalent to the category cZSet of periodic Z-sets, and to show that two finite directed graphs are almost-isospectral if and only if they are homotopy-equivalent in our sense. §0. Introduction. Mathematicians often study complicated categories by means of invariants (which are equal for isomorphic objects in the category). Sometimes a complicated category can be replaced by a (perhaps simpler) homotopy category which is better related to the various invariants used to study it. In topology, this was first achieved by declaring two continuous functions to be equivalent when one could be deformed into the other. But it

### Citations

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An Introduction to Symbolic Dynamics and Coding
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(Show Context)
Citation Context ...t functor of an adjunction. The first general statement of such a transfer in the literature is due to Crans.” Here is their informal statement of this “transfer principle”. The reference is to Crans =-=[1995]-=-. Transport Theorem: Let E be a model category which is cofibrantly generated, with cofibrations generated by I and acyclic cofibrations generated by J. Let E ′ be a category with all limits and colim... |

265 |
Categories for the working mathematician, Graduate Texts
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(Show Context)
Citation Context ... that TGph is a “reflective and coreflective subcategory” of Gph. These are best 3described in the language of adjoint functors. Let us give a quick review of some standard definitions (see Mac Lane =-=[1971]-=-, for instance). An adjunction between categories X and Y is a pair (L, R) of functors L : X → Y and R : Y → X together with a natural bijection of morphism sets Y(L(X), Y ) → X(X, R(Y )). In this cas... |

200 |
Sheaves in geometry and logic. A first introduction to topos theory. Corrected reprint of the 1992 edition. Universitext
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(Show Context)
Citation Context ...r Lawvere and Schanuel [1997] for fascinating discussions. It follows that Gph is a topos, and thus a category with many nice geometric and algebraic and logical properties; see Mac Lane and Moerdijk =-=[1994]-=-, for instance. In this paper we want to consider some very special kinds of graphs, as follows. Definition: A graph X is 1): an N-graph when each node of X has exactly one arc entering. 2): a Z-graph... |

135 |
Homotopy theories and model categories, in Handbook of Algebraic Topology
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(Show Context)
Citation Context ...t functor of an adjunction. The first general statement of such a transfer in the literature is due to Crans.” Here is their informal statement of this “transfer principle”. The reference is to Crans =-=[1995]-=-. Transport Theorem: Let E be a model category which is cofibrantly generated, with cofibrations generated by I and acyclic cofibrations generated by J. Let E ′ be a category with all limits and colim... |

91 |
Conceptual mathematics: a first introduction to categories
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(Show Context)
Citation Context ...f1 = f0 ◦ s and t ◦ f1 = f0 ◦ t. This defines the particular category Gph that we study here. In fact, Gph is the category of presheafs on a small category; see Lawvere [1989] or Lawvere and Schanuel =-=[1997]-=- for fascinating discussions. It follows that Gph is a topos, and thus a category with many nice geometric and algebraic and logical properties; see Mac Lane and Moerdijk [1994], for instance. In this... |

54 | Axiomatic homotopy theory for operads
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(Show Context)
Citation Context ...s “creating model structures along a right adjoint” (by Hirschhorn, Hopkins, Beke, etc), or as “transferring model structures along adjoint functors” (by Crans, etc). According to Berger and Moerdijk =-=[2003]-=- : “Cofibrantly generated model structures may be transferred along the left adjoint functor of an adjunction. The first general statement of such a transfer in the literature is due to Crans.” Here i... |

30 |
et al.: Théorie des topos et cohomologie étale. Tome 3
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Citation Context ...re are adjoint in the sense that A → φ ∗ (X) in C Sets corresponds to φ!(A) → X in D Sets, and φ ∗ (X) → A in C Sets corresponds to X → φ∗(A) in D Sets. See the analysis in Expose I.5 of Grothendieck =-=[1972]-=- This concept is related to that of “essential geometric morphism” φ : C Set ⇒ D Set in topos theory; see Mac Lane and Moerdijk [1994], for instance. Almost all the adjunctions used in this paper come... |

22 |
The Burnside ring of profinite groups and the Witt vector construction
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(Show Context)
Citation Context ...ent to the category cZSet of periodic Z-sets. In section 5 we use the functor H to associate a zeta series ZX(u) to each almost-finite graph X. This fits very well with work of Dress and Siebeneicher =-=[1988]-=- on the Burnside ring of the category of almost-finite Z-sets. As a consequence of our calculation of Ho(Gph), we show that finite graphs are almost-isospectral if and only if they are homotopy equiva... |

22 |
Qualitative Distinctions between some Toposes of Generalized Graphs
- Lawvere
- 1989
(Show Context)
Citation Context ...and f0 : X0 → Y0 such that s ◦ f1 = f0 ◦ s and t ◦ f1 = f0 ◦ t. This defines the particular category Gph that we study here. In fact, Gph is the category of presheafs on a small category; see Lawvere =-=[1989]-=- or Lawvere and Schanuel [1997] for fascinating discussions. It follows that Gph is a topos, and thus a category with many nice geometric and algebraic and logical properties; see Mac Lane and Moerdij... |

21 |
The Burnside Ring of the Infinite Cyclic Group and Its Relations to the Necklace Algebra, λ-Rings, and the Universal Ring of Witt Vectors
- Dress, Siebeneicher
- 1989
(Show Context)
Citation Context ...and f0 : X0 → Y0 such that s ◦ f1 = f0 ◦ s and t ◦ f1 = f0 ◦ t. This defines the particular category Gph that we study here. In fact, Gph is the category of presheafs on a small category; see Lawvere =-=[1989]-=- or Lawvere and Schanuel [1997] for fascinating discussions. It follows that Gph is a topos, and thus a category with many nice geometric and algebraic and logical properties; see Mac Lane and Moerdij... |

15 | Quillen closed model structures for sheaves
- Crans
- 1995
(Show Context)
Citation Context ...t functor of an adjunction. The first general statement of such a transfer in the literature is due to Crans.” Here is their informal statement of this “transfer principle”. The reference is to Crans =-=[1995]-=-. Transport Theorem: Let E be a model category which is cofibrantly generated, with cofibrations generated by I and acyclic cofibrations generated by J. Let E ′ be a category with all limits and colim... |

12 | Quasi-categories vs Segal spaces - Joyal, Tierney |

4 |
On the Galois theory of Grothendieck
- Dubuc, Vega
- 1998
(Show Context)
Citation Context ...), and shows that it gives an equivalences of categories. The proof can also be understood as an example of (the representable case of) Grothendieck’s Galois theory. The paper by Dubuc and de la Vega =-=[2000]-=- gives a self-contained exposition which seems relevant to our examples here. It is interesting to note that the calculation of products is the same at each level of the inclusions TGph ⊂ ZGph ⊂ NGph ... |

3 | A homotopical algebras of graphs related to zeta series
- Bisson, Tsemo
(Show Context)
Citation Context ...l structure; he defined the morphism sets for this homotopy category by using the classes of fibrations and cofibrations for the model structure. We review this in section 2 here. In Bisson and Tsemo =-=[2008]-=- we gave a model structure for a particular category Gph of directed and possibly infinite graphs, with loops and multiple arcs allowed (we give a precise definition of Gph in section 1 here). We focu... |

1 |
A convenient category for directed homotopy,Theory and
- Fajstrup, Rosick´y
(Show Context)
Citation Context ...ed as a graph with one node and no arcs). Here we will interpret these morphism classes, and describe our model structure for Gph, in terms of the following standard notions (see section 2.1 in Hovey =-=[1999]-=-, for instance). A model structure (C, W, F) is cofibrantly generated if there are sets I and J of morphisms such that J † = F and I † = F, so that C = † (I † ) and C = † (J † ). In short, a cofibrant... |