## A homotopical algebra of graphs related to zeta series

Venue: | Homology, Homotopy and its Applications 10 (2008), 1–13. MR2506131 (2010f:18010). [C95] Crans, Sjoerd E. Quillen |

Citations: | 3 - 3 self |

### BibTeX

@INPROCEEDINGS{Bisson_ahomotopical,

author = {Terrence Bisson and Aristide Tsemo},

title = {A homotopical algebra of graphs related to zeta series},

booktitle = {Homology, Homotopy and its Applications 10 (2008), 1–13. MR2506131 (2010f:18010). [C95] Crans, Sjoerd E. Quillen},

year = {}

}

### OpenURL

### Abstract

Abstract: The purpose of this paper is to develop a homotopical algebra for graphs, relevant to zeta series and spectra of finite graphs. More precisely, we define a Quillen model structure in a category of graphs (directed and possibly infinite, with loops and multiple arcs allowed). The weak equivalences for this model structure are the Acyclics (graph morphisms which preserve cycles). The cofibrations and fibrations for the model are determined from the class of Whiskerings (graph morphisms produced by grafting trees). Our model structure seems to fit well with the importance of acyclic directed graphs in many applications. In addition to the weak factorization systems which form this model structure, we also describe two Freyd-Kelly factorization systems based on Folding, Injecting, and Covering graph morphisms. 0. Introduction. In this paper we develop a notion of homotopy within graphs, and demonstrate its relevance to the study of zeta series and spectrum of a finite graph. We will work throughout with a particular category of graphs, described in Section 1 below. Our graphs will be directed and possibly infinite, with loops and multiple arcs allowed.

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Citation Context ...th t(a) = y. Stallings [1983] called this type of graph morphism a “folding”. Note that the word “folding” is used for other types of morphisms in literature on graphs; see the book Hell and Nesetril =-=[2004]-=-, for instance. We express the Stallings notion of folding as follows. Definition: Let V be the graph with three nodes, 0 and 1 ′ and 1 ′′ , and two arcs, a ′ and a ′′ , with a ′ from 0 to 1 ′ and a ′... |

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Citation Context ...bjects 0 and 1 and two non-identity morphisms from 0 to 1. 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. We just call attention to a few aspects here. The category Gph has all products, and all coproducts (sums); it also has pull-backs (fiber products) and pushouts. Also, products distrib... |

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Citation Context ...dimension of A/I as a finite dimensional vector space over Fq, so that ζ(s) = Z(u)| u=q −s P p for Z(u) = ∏ (1 − u ν(P) ) −1 . The zeta function for a projective variety over a finite field from Weil =-=[1949]-=- is a completed version of this. The form of zeta series that we use in section 5 seems ultimately based on the following observation. For a square matrix of a fixed size, the knowledge of the trace o... |

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Citation Context ...classes L and R with L ⊥ = R and L = ⊥ R, such that for any morphism c in S, there exist ℓ ∈ L and r ∈ R with c = r ◦ ℓ. 5This notion of factorization system is given in section 2 of Freyd and Kelly =-=[1972]-=-. A Freyd-Kelly factorization system is often just called a factorization system. We may use notation ((L , R)) to indicate a Freyd-Kelly factorization system. A basic example is the epimorphic, monom... |

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Citation Context ...h X can be identified with the set of graph morphisms from N to X, and the set of arcs of X can be identified with the set of graph morphisms from A to X. There is a stimulating discussion in Lawvere =-=[1989]-=- of Gph as the category of presheafs on the small category with objects 0 and 1 and two non-identity morphisms from 0 to 1. It follows that Gph is a topos, and thus a category with many nice geometric... |

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Citation Context ...h X can be identified with the set of graph morphisms from N to X, and the set of arcs of X can be identified with the set of graph morphisms from A to X. There is a stimulating discussion in Lawvere =-=[1989]-=- of Gph as the category of presheafs on the small category with objects 0 and 1 and two non-identity morphisms from 0 to 1. It follows that Gph is a topos, and thus a category with many nice geometric... |

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Citation Context ...tors as m ◦ e with m ∈ M and e ∈ E. We will describe two interesting Freyd-Kelly factorization systems in Gph (although we will not need them in this paper). Our first system is inspired by Stallings =-=[1983]-=-. We will sketch a complete proof, since it involves a nice use of a “small object argument” (for this idea, see Section 2.1 in Hovey [1999], for instance). Any pair {a ′ , a ′′ } of distinct arcs wit... |

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Citation Context ...plicial sets, topos theory, and small categories (including monoids, groups, groupoids, and posets). References include Thomason [1980], Joyal and Tierney [1991], Dwyer and Spalinski [1995], Cisinski =-=[2002]-=-, and many others. Also, recent proofs of the Bloch-Kato and Milnor conjectures are based upon development of a homotopical algebra for schemes; see Voevodsky and Morel [1999]. A central part of givin... |

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Citation Context ...pments in settings such as chain complexes and homological algebra, simplicial sets, topos theory, and small categories (including monoids, groups, groupoids, and posets). References include Thomason =-=[1980]-=-, Joyal and Tierney [1991], Dwyer and Spalinski [1995], Cisinski [2002], and many others. Also, recent proofs of the Bloch-Kato and Milnor conjectures are based upon development of a homotopical algeb... |

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Citation Context ...plicial sets, topos theory, and small categories (including monoids, groups, groupoids, and posets). References include Thomason [1980], Joyal and Tierney [1991], Dwyer and Spalinski [1995], Cisinski =-=[2002]-=-, and many others. Also, recent proofs of the Bloch-Kato and Milnor conjectures are based upon development of a homotopical algebra for schemes; see Voevodsky and Morel [1999]. A central part of givin... |

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Citation Context ...lmost isospectral. Proof: This follows immediately from the preceding two propositions. QED An appendix on the zeta series and its Euler product expansion. Here is a little history, taken from Thomas =-=[1977]-=-, of how our zeta series for finite directed graphs relates to the famous zeta functions from number theory. The zeta function of Euler and Riemann. Let p range over the prime numbers. Then ζ(s) = ∑ n... |

3 |
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Citation Context ...alinski [1995], Cisinski [2002], and many others. Also, recent proofs of the Bloch-Kato and Milnor conjectures are based upon development of a homotopical algebra for schemes; see Voevodsky and Morel =-=[1999]-=-. A central part of giving a model structure in a category is the specification of which morphisms in the category are to be called “weak equivalences”. In most applications, the weak equivalences are... |

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1 |
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Citation Context ...alinski [1995], Cisinski [2002], and many others. Also, recent proofs of the Bloch-Kato and Milnor conjectures are based upon development of a homotopical algebra for schemes; see Voevodsky and Morel =-=[1999]-=-. A central part of giving a model structure in a category is the specification of which morphisms in the category are to be called “weak equivalences”. In most applications, the weak equivalences are... |