Hom complexes and homotopy in the category of graphs

In this extended abstract we develop a notion of ×-homotopy of graph maps that is based on the internal hom associated to the categorical product. We show that graph ×-homotopy is characterized by the topological properties of the so-called Hom complex, a functorial way to assign a poset to a pair of graphs. Along the way we establish some structural properties of Hom complexes involving products and exponentials of graphs, as well as a symmetry result which can be used to reprove a theorem of Kozlov involving foldings of graphs. We end with a discussion of graph homotopies arising from other internal homs, including the construction of ‘A-theory’ associated to the cartesian product in the category of reflexive graphs. For proofs and further discussions we refer the reader to the full paper [Anton Dochtermann. Hom complexes and homotopy theory in the category of graphs. arXiv:math.CO/0605275].     

A normalization theorem for regular homotopy of finite directed graphs

Dati uno spazio topologico normaleS ed un grafo finito ed orientatoG, si dimostra che ogni funzione regolare diS inG è omotopa ad una funzione fortemente regolare.     

The homotopy category of parametrized spectra

A homotopy category of spectra parametrized over some space B is constructed, which has useful properties for applications. It is a symmetric monoidal category with multiplication given by the smash product. In the original construction the objects are coordinate free spectra indexed by inner product spaces. A translation of the results to a category whose objects are the more familiar spectra indexed by natural numbers is also given.     

The homotopy type of the cobordism category

The embedded cobordism category under study in this paper generalizes the category of conformal surfaces, introduced by G. Segal in [S2] in order to formalize the concept of field theories. Our main result identifies the homotopy type of the classifying space of the embedded d-dimensional cobordism category for all d. For d = 2, our results lead to a new proof of the generalized Mumford conjecture, somewhat different in spirit from the original one, presented in [MW].     

On the homotopy category of AC-injective complexes

Let R be any ring. We motivate the study of a class of chain complexes of injective R-modules that we call AC-injective complexes, showing that K(AC-Inj), the chain homotopy category of all AC-injective complexes, is always a compactly generated triangulated category. In general, all DGinjective complexes are AC-injective and in fact there is a recollement linking K(AC-Inj) to the usual derived category D(R). This is based on the author’s recent work inspired by work of Krause and Stovicek. Our focus here is on giving straightforward proofs that our categories are compactly generated.     

Rational homotopy theory and differential graded category

We propose a generalization of Sullivan’s de Rham homotopy theory to non-simply connected spaces. The formulation is such that the real homotopy type of a manifold should be the closed tensor dg-category of flat bundles on it much the same as the real homotopy type of a simply connected manifold is the de Rham algebra in original Sullivan’s theory. We prove the existence of a model category structure on the category of small closed tensor dg-categories and as a most simple case, confirm an equivalence between the homotopy category of spaces whose fundamental groups are finite and whose higher homotopy groups are finite dimensional rational vector spaces and the homotopy category of small closed tensor dg-categories satisfying certain conditions.     

Geometrical symmetric powers in the motivic homotopy category

Symmetric powers of quasi-projective schemes can be extended via Kan extensions to geometrical symmetric powers of motivic spaces. In this work, we study geometrical symmetric powers and compare them with various symmetric powers in the unstable and stable ${\mathbb{A}}^1$-homotopy category of schemes over a field.     

Enriched simplicial presheaves and the motivic homotopy category

We construct models for the motivic homotopy category based on simplicial functors from smooth schemes over a field to simplicial sets. These spaces are homotopy invariant and therefore one does not have to invert the affine line in order to get a model for the motivic homotopy category.     

The homotopy category of complexes of projective modules

The homotopy category of complexes of projective left-modules over any reasonably nice ring is proved to be a compactly generated triangulated category, and a duality is given between its subcategory of compact objects and the finite derived category of right-modules.     

Duality theorems for regular homotopy of finite directed graphs

Dati uno spazio topologico normale e numerabilmente paracompattoS ed un grafo finito ed orientatoG si prova che tra gli insiemiQ(S, G) eQ ^*(S, G) delle classi dio-omotopia e dio ^*-omotopia esiste una biiezione naturale. Nelle stesse condizioni, seS′ è un sottospazio chiuso diS eG′ un sottografo diG, esiste ancora una biiezione naturale tra gli insiemiQ (S, S′; G, G′) eQ ^* (S, S′; G, G′) delle classi di omotopia. Si mostra infine che in condizioni meno restrittive per lo spazioS le precedenti biiezioni possono non sussistere.     

The homotopy category of pseudofunctors and translation cohomology

We develop the obstruction theory of the 2-category of abelian track categories, pseudofunctors and pseudonatural transformations by using the cohomology of categories. The obstructions are defined in Baues-Wirsching cohomology groups. We introduce translation cohomology to classify endomorphisms in the 2-category of abelian track categories. In a sequel to this paper we will show, under certain conditions which are satisfied by all homotopy categories, that a translation cohomology class determines the exact triangles of a triangulated category.     

The coherent homotopy category over a fixed space is a category of fractions

In this note we prove that the coherent homotopy category over a fixed space B with morphisms represented by certain homotopy commutative squares (see [8]) is isomorphic to the category obtained by formally inverting those maps in the category Top_B of topological spaces over B which are ordinary homotopy equivalences.