A Colimit Decomposition for Homotopy Algebras in Cat

Badzioch showed that in the category of simplicial sets each homotopy algebra of a Lawvere theory is weakly equivalent to a strict algebra. In seeking to extend this result to other contexts Rosický observed a key point to be that each homotopy colimit in SSet admits a decomposition into a homotopy sifted colimit of finite coproducts, and asked the author whether a similar decomposition holds in the 2-category of categories Cat. Our purpose in the present paper is to show that this is the case.     

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.     

Numerably Contractible Categories

We develop the notion of numerably contractible category and use it for describing conditions when a homotopy associative H-category has a homotopy inverse. We prove that complex categories are numerably contractible. The results play a role in Bak’s program for constructing delooping machines for global actions, small categories and related objects. (Received: February 2006)     

Weighted limits in an $$(\infty ,1)$$ ( ∞ , 1 ) -category

We introduce the notion of weighted limit in an arbitrary quasi-category, suitably generalizing ordinary limits in a quasi-category, and classical weighted limits in an ordinary category. This is accomplished by generalizing Joyal’s approach: we identify a meaningful construction for the quasi-category of weighted cones over a diagram in a quasi-category, whose terminal object is the weighted limit of the considered diagram. We then show that each weighted limit can be expressed as an ordinary limit. When the quasi-category arises as the homotopy coherent nerve of a category enriched over Kan complexes, we generalize an argument by Riehl-Verity to show that the weighted limit agrees with the homotopy weighted limit in the sense of enriched category theory, for which explicit constructions are available. When the quasi-category is complete, tensored and cotensored over the quasi-category of spaces, we discuss a possible comparison of our definition of weighted limit with the approach by Gepner-Haugseng-Nikolaus.

     

Adjoining adjoints

We construct the 2-category obtained from a category by freely adjoining a right adjoint for each morphism and isolate its universal property. Some others basic properties are also studied. Some examples in which the category is freely generated by a graph are discussed in detail. For these categories, the 2-cells are given a geometric interpretation and shown to be similar to certain diagrams which have appeared in the literature on C∗-algebras.     

2-Groups, trialgebras and their Hopf categories of representations

A strict 2-group is a 2-category with one object in which all morphisms and all 2-morphisms have inverses. 2-Groups have been studied in the context of homotopy theory, higher gauge theory and Topological Quantum Field Theory (TQFT). In the present article, we develop the notions of trialgebra and cotrialgebra, generalizations of Hopf algebras with two multiplications and one comultiplication or vice versa, and the notion of Hopf categories, generalizations of monoidal categories with an additional functorial comultiplication. We show that each strict 2-group has a ‘group algebra’ which is a cocommutative trialgebra, and that each strict finite 2-group has a ‘function algebra’ which is a commutative cotrialgebra. Each such commutative cotrialgebra gives rise to a symmetric Hopf category of corepresentations. In the semisimple case, this Hopf category is a 2-vector space according to Kapranov and Voevodsky. We also show that strict compact topological 2-groups are characterized by their C^*-cotrialgebras of ‘complex-valued functions’, generalizing the Gel'fand representation, and that commutative cotrialgebras are characterized by their symmetric Hopf categories of corepresentations, generalizing Tannaka-Kre?ˇn reconstruction. Technically, all these results are obtained using ideas from functorial semantics, by studying models of the essentially algebraic theory of categories in various base categories of familiar algebraic structures and the functors that describe the relationships between them.     

Biset Functors as Module Mackey Functors and its Relation to Derivators

In this article, we will show that the category of biset functors can be regarded as a reflective monoidal subcategory of the category of Mackey functors on the 2-category of finite groupoids. This reflective subcategory is equivalent to the category of modules over the Burnside functor. As a consequence of the reflectivity, we can associate a biset functor to any derivator on the 2-category of finite categories.     

The formal theory of monoidal monads

We give a 3-categorical, purely formal argument explaining why on the category of Kleisli algebras for a lax monoidal monad, and dually on the category of Eilenberg–Moore algebras for an oplax monoidal monad, we always have a natural monoidal structures. The key observation is that the 2-category of lax monoidal monads in any 2-category D with finite products is isomorphic to the 2-category of monoidal objects with oplax morphisms in the 2-category of monads with lax morphisms in D. We explain at the end of the paper that a similar phenomenon occurs in many other situations.     

Homology Theory of Graphs

Higher homotopy of graphs has been defined in several articles. In Dochterman (Hom complexes and homotopy theory in the category of graphs. arXiv math/0605275 v2,28/09/2006, 2006), the authors asked for a companion homology theory. We define such a theory for the category of unoriented reflexive graphs; it exhibits a long exact sequence for a pair of graphs (G, A), satisfies an excision property and a Hurewicz theorem. This allows us to compute the top homology of the graphical n-spheres showing that the theory is not trivial and is able to detect n-dimensional holes in a graph. The long-term objective is to compare the homotopy of the topological and graphical spheres.     

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.

     

Four-Cycled Graphs with Topological Applications

We call a simple graph G a 4-cycled graph if either it has no edges or every edge of it is contained in an induced 4-cycle of G. Our interest on 4-cycled graphs is motivated by the fact that their clique complexes play an important role in the simple-homotopy theory of simplicial complexes. We prove that the minimal simple models within the category of flag simplicial complexes are exactly the clique complexes of some 4-cycled graphs. We further provide structural properties of 4-cycled graphs and describe constructions yielding such graphs. We characterize 4-cycled cographs, and 4-cycled graphs arising from finite chessboards. We introduce a family of inductively constructed graphs, the external extensions, related to an arbitrary graph, and determine the homotopy type of the independence complexes of external extensions of some graphs.     

Strong homotopy induced by adjacency structure

This paper generalizes the concept of SA-homotopy in finite topological adjacency category, which is introduced in our previous work, to graph category and discusses its properties. We prove that every SA-strong deformation retract of a simple graph G could be obtained by removing trivial vertices one by one, which makes it possible to allow an iterative algorithm of finding all SA-strong deformation retracts of G. We also obtain that two simple graphs are SA-homotopy equivalent if and only if they have graph isomorphic cores. Compared with the graph homotopy transformation defined by S.T. Yau et al. and the s-homotopy transformation defined by R. Boulet et al., the main advantage of SA-homotopy transformation is that it could reflect correspondences between vertices, and hence it more accurately describe the transformation process than the graph homotopy transformation and s-homotopy transformation. As an application of SA-homotopy on graphs, we introduce the mapping class group of a graph, which also shows its advantage over the graph homotopy transformation and the s-homotopy transformation.     

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.     

Measurable versions of the LS category on laminations

We give two new versions of the LS category for the set-up of measurable laminations defined by Bermúdez. Both of these versions must be considered as “tangential categories”. The first one, simply called (LS) category, is the direct analogue for measurable laminations of the tangential category of (topological) laminations introduced by Colman Vale and Macías Virgós. For the measurable lamination that underlies any lamination, our measurable tangential category is a lower bound of the tangential category. The second version, called the Λ-category, depends on the choice of a transverse invariant measure Λ. We show that both of these “tangential categories” satisfy appropriate versions of some well known properties of the classical category: the homotopy invariance, existence of a dimensional upper bound, a cohomological lower bound (cup length), and an upper bound given by the critical points of a smooth function. Also, we show possible applications of these invariants to variational problems.     

Triple Brackets and Lax Morphism Categories

Various aspects of the traditional homotopy theory of topological spaces may be developed in an arbitrary 2-category C with zeros. In particular certain secondary composition operations called box brackets recently have been defined for C; these are similar to, but extend, the familiar Toda brackets in the topological case. In this paper we introduce further the notion of a suspension functor in C and explore the ramifications of relativizing the theory in terms of the associated lax morphism category of C, denoted mC. Four operations associated to a 3-box diagram are introduced and relations among them are clarified. The results and insights obtained, while by nature somewhat technical, yield effective and efficient techniques for computing many operations of Toda bracket type. We illustrate by recording some computations from the homotopy groups of spheres. Also the properties of a new operation, the 2-sided matrix Toda bracket, are explored.