Suppose that G is a finite group. We look at the problem of expressing the classifying space BG, up to mod p cohomology, as a homotopy colimit of classifying spaces of smaller groups. A number of interesting tools come into play, such as simplicial sets and spaces, nerves of categories, equivariant homotopy theory, and the transfer.

We extend the theory of Quillen adjunctions by combining ideas of homotopical algebra and of enriched category theory. Our results describe how the formulas for homotopy colimits of Bousfield and Kan arise from general formulas describing the derived functor of the weighted colimit functor.

For C a G-category, we give a condition on a diagram of simplicial sets indexed on C that allows us to define a natural G-action on its homotopy colimit, and in some other simplicial sets and categories defined in terms of the diagram. Well-known theorems on homeomorphisms and homotopy equivalences are generalized to an equivariant version.

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. Mathematical Subject Classification: 18D05, 55P15, 18F25.

Building on the foundations in our previous paper, we study Segal conditions that are given by finite products, determined structures call cartesian patterns. We set up Day convolution presheaves this setting and use it to give under which there is a colimit formula for free algebras other left adjoints. This specializes simple proof of Lurie's results operadic Kan extensions symmetric $\infty$...

We consider a notion of morphism of neural networks and develop its properties. We show how, given any definite logic program P , the least fixed point of the immediate consequence operator TP can be computed by the colimit of a family of neural networks determined

In this dissertation we investigate presheaf models for concurrent computation. Our aim is to provide a systematic treatment of bisimulation for a wide range of concurrent process calculi. Bisimilarity is defined abstractly in terms of open maps as in the work of Joyal, Nielsen and Winskel. Their work inspired this thesis by suggesting that presheaf categories could provide abstract models for ...

The most powerful feature of categories of posets with directed sups is the ability to solve domain equations such as D ∼= D. A crucial ingredient of this technique is the fact that for certain kinds of diagrams (in particular sequences of “projection pairs”) the limit and colimit are isomorphic. This much is known to everyone in the subject: what appears not to be generally known is (i) that w...

An object in motivic homotopy theory is called cellular if it can be built out of motivic spheres using homotopy colimit constructions. We explore some examples and consequences of cellularity. We explain why the algebraic K-theory and algebraic cobordism spectra are both cellular, and prove some Künneth theorems for cellular objects.

We establish a general method to produce cofibrant approximations in the model category US(C,D) of S-valued C-indexed diagrams with D-weak equivalences and D-fibrations. We also present explicit examples of such approximations. Here, S is an arbitrary cofibrantly generated simplicial model category and D ⊂ C are small categories. An application to the notion of homotopy colimit is presented.