Begin with a small category C. The goal of this short note is to point out that there is such a thing as a ‘universal model category built from C’. We describe applications of this to the study of homotopy colimits, the Dwyer-Kan theory of framings, to sheaf theory, and to the homotopy theory of schemes.

Let G be a compact, simply-connected, simple Lie group and T ⊂ G a maximal torus. The purpose of this paper is to study the connection between various fibrations over BG (where G is a compact, simply-connected, simple Lie group) associated to the adjoint representation and homotopy colimits over poset categories C, hocolimCBGI where GI are certain connected maximal rank subgroups of G.

1 Monoidal ∞-Categories 4 1.1 Monoidal Structures and Algebra Objects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Cartesian Monoidal Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Subcategories of Monoidal ∞-Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 1.4 Free Algebras . . . . . . . . . . . . . . . . . . ....

1. Categories and functors 2. Standard (boring) examples 3. Initial and final objects 4. Categories of diagrams: products and coproducts 5. Example: sets 6. Example: topological spaces 7. Example: products of groups 8. Example: coproducts of abelian groups 9. Example: vectorspaces and duality 10. Limits 11. Colimits 12. Example: nested intersections of sets 13. Example: ascending unions of sets...

Self-* is widely considered as a foundation for autonomic computing. The notion of autonomic systems (ASs) and self-* serves as a basis on which to build our intuition about category of ASs in general. In this paper we will specify ASs and self-* and then move on to consider finite limits and colimits in ASs. All of this material is taken as an investigation of our category, the category of ASs...

in addition to exploring constructions and properties of limits and colimits in categories of topologicalalgebras, we study special subcategories of topological algebras and their properties. in particular, undercertain conditions, reflective subcategories when paired with topological structures give rise to reflectivesubcategories and epireflective subcategories give rise to epireflective subc...

elementary classes (Shelah 1987) form a special case of the former (Lieberman 2011, Beke and JR 2012) and generalize Lκω-elementary classes. Surprisingly, I do not know any abstract elementary class which does not have an Lκω-axiomatization (as an abstract category). With Makkai, we failed to prove that uncountable sets with injective mappings form such an example. It seemed a long time that th...

Abstract Given a group G and -category $${\textbf{C}}$$ C , we give condition on diagram of simplicial sets indexed by that allows us to define natural action its homotopy colimit, some other defined in terms the diagram. Well-known theorems homeomorphisms equivalences are generalized equivariant versions.