CATEGORICAL HOMOTOPY

We present a homotopy theory of small categories. In a work of this nature there is a need to give a theory which is clear and which shows the methods of work in this field. It is also necessary to prove theorems which place the theory within the general framework of homotopy, i.e. particularly to liaise with the homotopy of topological spaces and with abstract homotopy theories. Firstly we define the important notion of finite functor on which the theory is based. Next we introduce a type of fibred category fitting to the work on homotopy. After having studied the paths and loops of a category, we consider homotopy between functors. Finally, we demonstrate the possibility of obtaining homotopy groups before taking into consideration the relations between categorical and topological homotopy.     

Three models for the homotopy theory of homotopy theories

Given any model category, or more generally any category with weak equivalences, its simplicial localization is a simplicial category which can rightfully be called the “homotopy theory” of the model category. There is a model category structure on the category of simplicial categories, so taking its simplicial localization yields a “homotopy theory of homotopy theories”. In this paper we show that there are two different categories of diagrams of simplicial sets, each equipped with an appropriate definition of weak equivalence, such that the resulting homotopy theories are each equivalent to the homotopy theory arising from the model category structure on simplicial categories. Thus, any of these three categories with the respective weak equivalences could be considered a model for the homotopy theory of homotopy theories. One of them in particular, Rezk’s complete Segal space model category structure on the category of simplicial spaces, is much more convenient from the perspective of making calculations and therefore obtaining information about a given homotopy theory.     

Rational homotopy theory for non-simply connected spaces

We construct an algebraic rational homotopy theory for all connected CW spaces (with arbitrary fundamental group) whose universal cover is rationally of finite type. This construction extends the classical theory in the simply connected case and has two basic properties: (1) it induces a natural equivalence of the corresponding homotopy category to the homotopy category of spaces whose universal cover is rational and of finite type and (2) in the algebraic category, homotopy equivalences are isomorphisms. This algebraisation introduces a new homotopy invariant: a rational vector bundle with a distinguished class of linear connections.

     

Model structures on exact categories

We define model structures on exact categories, which we call exact model structures. We look at the relationship between these model structures and cotorsion pairs on the exact category. In particular, when the underlying category is weakly idempotent complete, we get Hovey’s one-to-one correspondence between model structures and complete cotorsion pairs. We classify the right and the left homotopy relation in terms of the cotorsion pairs and look at examples of exact model structures. In particular, we see that given any hereditary abelian model category, the full subcategories of cofibrant, fibrant and cofibrant-fibrant subobjects each has natural exact model structures equivalent to the original model structure. These model structures each has interesting characteristics. For example, the cofibrant-fibrant subobjects form a Frobenius category, whose stable category is the same as the homotopy category of its model structure.     

Strong cofibrations and fibrations in enriched categories

We define strong cofibrations and fibrations in suitably enriched categories using the relative homotopy extension resp. lifting property. We prove a general pairing result, which for topological spaces specializes to the well-known pushout-product theorem for cofibrations. Strong cofibrations and fibrations give rise to cofibration and fibration categories in the sense of homotopical algebra. We discuss various examples; in particular, we deduce that the category of chain complexes with chain equivalences and the category of categories with equivalences are symmetric monoidal proper closed model categories. Eine überarbeitete Fassung ging am 5. 12. 2001 ein     

Classifying spaces for braided monoidal categories and lax diagrams of bicategories

This work contributes to clarifying several relationships between certain higher categorical structures and the homotopy type of their classifying spaces. Bicategories (in particular monoidal categories) have well-understood simple geometric realizations, and we here deal with homotopy types represented by lax diagrams of bicategories, that is, lax functors to the tricategory of bicategories. In this paper, it is proven that, when a certain bicategorical Grothendieck construction is performed on a lax diagram of bicategories, then the classifying space of the resulting bicategory can be thought of as the homotopy colimit of the classifying spaces of the bicategories that arise from the initial input data given by the lax diagram. This result is applied to produce bicategories whose classifying space has a double loop space with the same homotopy type, up to group completion, as the underlying category of any given (non-necessarily strict) braided monoidal category. Specifically, it is proven that these double delooping spaces, for categories enriched with a braided monoidal structure, can be explicitly realized by means of certain genuine simplicial sets characteristically associated to any braided monoidal categories, which we refer to as their (Street's) geometric nerves.     

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 Eckmann-Hilton argument and higher operads

The classical Eckmann-Hilton argument shows that two monoid structures on a set, such that one is a homomorphism for the other, coincide and, moreover, the resulting monoid is commutative. This argument immediately gives a proof of the commutativity of the higher homotopy groups. A reformulation of this argument in the language of higher categories is: suppose we have a one object, one arrow 2-category, then its Hom-set is a commutative monoid. A similar argument due to A. Joyal and R. Street shows that a one object, one arrow tricategory is ‘the same’ as a braided monoidal category.In this paper we begin to investigate how one can extend this argument to arbitrary dimension. We provide a simple categorical scheme which allows us to formalise the Eckmann-Hilton type argument in terms of the calculation of left Kan extensions in an appropriate 2-category. Then we apply this scheme to the case of n-operads in the author's sense and classical symmetric operads. We demonstrate that there exists a functor of symmetrisation Symn from a certain subcategory of n-operads to the category of symmetric operads such that the category of one object, one arrow, … , one (n−1)-arrow algebras of A is isomorphic to the category of algebras of Symn(A). Under some mild conditions, we present an explicit formula for Symn(A) which involves taking the colimit over a remarkable categorical symmetric operad.We will consider some applications of the methods developed to the theory of n-fold loop spaces in the second paper of this series.     

Homotopy fiber sequences induced by 2-functors

This paper contains some contributions to the study of the relationship between 2-categories and the homotopy types of their classifying spaces. Mainly, generalizations are given of both Quillen’s Theorem B and Thomason’s Homotopy Colimit Theorem to 2-functors.     

An operadic model for a mapping space and its associated spectral sequence

Let X and Y be simplicial sets and K a field. In [B. Fresse, Derived division functors and mapping spaces, 2002, Preprint arXiv:math.At/0208091], Fresse has constructed an algebra model over an E_∞K-operad E for the mapping space F(X,Y), whose source X is finite, provided the homotopy groups of the target Y are finite. In this paper, we show that if the underlying field K is the closure of the finite field F_p and the given mapping space is connected, then the finiteness assumption of the homotopy group of Y can be dropped in constructing the E-algebra model. Moreover, we give a spectral sequence converging to the cohomology of F(X,Y) with coefficients in , whose E₂-term is expressed via Lannes’ division functor in the category of unstable -algebra over the Steenrod algebra.     

A counter-example to a conjecture of Félix

If X is a simply connected space of finite type, then the rational homotopy groups of the based loop space of X possess the structure of a graded Lie algebra, denoted L_X. The radical of L_X, which is an important rational homotopy invariant of X, is of finite total dimension if the Lusternik-Schnirelmann category of X is finite.Let X be a simply connected space with finite Lusternik-Schnirelmann category. If dimL_X<∞, i.e., if X is elliptic, then L_X is its own radical, and therefore the total dimension of the radical of L_X in odd degrees is less than or equal to its total dimension in even degrees (Friedlander and Halperin (1979) [8]). Félix conjectured that this inequality should hold for all simply connected spaces with finite Lusternik-Schnirelmann category.We prove Félix’s conjecture in some interesting special cases, then provide a counter-example to the general case.     

The homotopy categorical crossed module of a CW-complex

Crossed modules have longstanding uses in homotopy theory and the cohomology of groups. The corresponding notion in the setting of categorical groups, that is, categorical crosses modules, allowed the development of a low-dimensional categorical group cohomology. Now, its relevance is also shown here to homotopy types by associating, to any pointed CW-complex (X,∗), a categorical crossed module that algebraically represents the homotopy 3-type of X.     

A folk model structure on omega-cat

The primary aim of this work is an intrinsic homotopy theory of strict ω-categories. We establish a model structure on ωCat, the category of strict ω-categories. The constructions leading to the model structure in question are expressed entirely within the scope of ωCat, building on a set of generating cofibrations and a class of weak equivalences as basic items. All objects are fibrant while free objects are cofibrant. We further exhibit model structures of this type on n-categories for arbitrary n∈N, as specializations of the ω-categorical one along right adjoints. In particular, known cases for n=1 and n=2 nicely fit into the scheme.     

A projective homotopy theory of crossedG-modules

In this article we extend Hilton's projective homotopy theory of modules (Hilton, 1967) to a homotopy theory of crossed modules, and then reduce some resulting homotopy classification problems to problems in group homology. We also observe that our homotopy theory satisfies the axioms of a Baues fibration category (Baues, 1989).This author would like to thank the University of Cape Town for its hospitality.     

Weighted limits in simplicial homotopy theory

By combining ideas of homotopical algebra and of enriched category theory, we explain how two classical formulas for homotopy colimits, one arising from the work of Quillen and one arising from the work of Bousfield and Kan, are instances of general formulas for the derived functor of the weighted colimit functor.     

On the subdivision of small categories

We present an intrinsic and concrete development of the subdivision of small categories, give some simple examples and derive its fundamental properties. As an application, we deduce an alternative way to compare the homotopy categories of spaces and small categories, by using partially ordered sets. This yields a new conceptual proof to the well-known fact that these two homotopy categories are equivalent.