Related papers: Indeterminacies and models of homotopy limits
We explain how the notion of homotopy colimits gives rise to that of mapping spaces, even in categories which are not simplicial. We apply the technique of model approximations and use elementary properties of the category of spaces to be…
We study the homotopy theory of a certain type of diagram categories whose vertices are in variable categories with a functorial path, leading to a good calculation of the homotopy category in terms of cofibrant objects. The theory is…
In this work, we introduce {\em topological representations of a quiver} as a system consisting of topological spaces and its relationships determined by the quiver. Such a setting gives a natural connection between topological…
Hidden variable graphical models can sometimes imply constraints on the observable distribution that are more complex than simple conditional independence relations. These observable constraints can falsify assumptions of the model that…
We gather conditions on a class H of continuous maps of topological spaces that allow a reasonable theory of fibrations up to an equivalence (a map from this class) which we call H-fibrations. The weak homotopy equivalences recover…
Consider a push-out diagram of spaces C <-- A --> B, construct the homotopy push-out, and then the homotopy pull-back of the diagram one gets by forgetting the initial object A. We compare the difference between A and this homotopy…
Many important theorems in differential topology relate properties of manifolds to properties of their underlying homotopy types -- defined e.g. using the total singular complex or the \v{C}ech nerve of a good open cover. Upon embedding the…
We describe a category, the objects of which may be viewed as models for homotopy theories. We show that for such models, ``functors between two homotopy theories form a homotopy theory'', or more precisely that the category of such models…
For a Hopf DG-algebra corresponding to a derived algebraic group, we compute the homotopy limit of the associated cosimplicial system of DG-algebras given by the classifying space construction. The homotopy limit is taken in the model…
We introduce the notion of a logical model category which is a Quillen model category satisfying some additional conditions. Those conditions provide enough expressive power that one can soundly interpret dependent products and sums in it.…
If a Quillen model category can be specified using a certain logical syntax (intuitively, ``is algebraic/combinatorial enough''), so that it can be defined in any category of sheaves, then the satisfaction of Quillen's axioms over any site…
We prove an extension of the Quillen Theorem Bn for homotopy fibres to a similar result for homotopy pullbacks and use this to obtain sufficient conditions on a pullback diagram of categories to guarantee that it be a homotopy pullback.
Given a diagram of rings, one may consider the category of modules over them. We are interested in the homotopy theory of categories of this type: given a suitable diagram of model categories M(s) (as s runs through the diagram), we…
The theory of parity quasi-complexes (PQC) is developed, preparing a set up for defining derived functors using resolutions in the nonabelian case. A homotopy structure on the category of PQC is defined, yielding a 2-category structure. The…
Centers of categories capture the natural operations on their objects. Homotopy coherent centers are introduced here as an extension of this notion to categories with an associated homotopy theory. These centers can also be interpreted as…
We define two model structures on the category of bicomplexes concentrated in the right half plane. The first model structure has weak equivalences detected by the totalisation functor. The second model structure's weak equivalences are…
We give the definitions of model bicategory and $q$-homotopy, which are natural generalizations of the notions of model category and homotopy to the context of bicategories. For any model bicategory $\mathcal{C}$, denote by…
We define the notion of {\em classifying space} of a topological stack and show that every topological stack \X has a classifying space X which is a topological space well-defined up to weak homotopy equivalence. Under a certain…
Homotopic morphisms of $\mathbb E$-triangles in extriangulated categories are introduced. Any morphism of $\mathbb E$-triangles is a composition of homotopic morphisms. Any morphism $(\alpha_1, \alpha_2, \alpha_3)$ of $\mathbb E$-triangles…
Without the axiom of choice, the free exact completion of the category of sets (i.e. the category of setoids) may not be complete or cocomplete. We will show that nevertheless, it can be enhanced to a derivator: the formal structure of…