We introduce a general construction on 2-monads. develop background maps of 2-monads, their left semi-algebras, and colimits in 2-category. Then, we the colimit induced by map show that obtain structure 2-monad give characterisation its algebras. Finally, apply to 2-monads between free symmetric monoidal cartesian combine them into linear-non-linear 2-monad.

We study the homotopy theory of locally ordered spaces, that is manifolds with boundary whose charts are partially ordered in a compatible way. Their category is not particularly well-behaved with respect to colimits. However, this category turns out to be a certain full subcategory of a topos of sheaves over a simpler site. The ambient topos makes available some general homotopical machinery.

We develop a homotopy theory of categories enriched in a monoidal model category V. In particular, we deal with homotopy weighted limits and colimits, and homotopy local presentability. The main result, which was known for simplicially-enriched categories, links homotopy locally presentable V-categories with combinatorial model V-categories, in the case where all objects of V are cofibrant.

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.

An introduction to the notions of homotopy limit and colimit is given. It is explained how they can be used to neatly describe the “old” distinguished triangles and shift functors of derived categories resp. cofiber and fiber sequences in algebraic topology. One of the goals is to motivate the language of derivators from the perspective of classical homological algebra. Another one is to give e...

We show that homotopy pullbacks of sheaves of simplicial sets over a Grothendieck topology distribute over homotopy colimits; this generalizes a result of Puppe about topological spaces. In addition, we show that inverse image functors between categories of simplicial sheaves preserve homotopy pullback squares. The method we use introduces the notion of a sharp map, which is analogous to the no...

We summarize our axioms for higher categories, and describe the "blob complex." Fixing an n-category , the blob complex associates a chain complex B(*)(W;C) to any n-manifold W. The zeroth homology of this chain complex recovers the usual topological quantum field theory invariants of W. The higher homology groups should be viewed as generalizations of Hochschild homology (indeed, when W = S(1)...

Let κQnt be the category of of κ-quantales, quantales closed under κ-joins in which the monoid identity is the largest element. (κ is an infinite regular cardinal.) Although the lack of lattice completeness in this setting would seem to mitigate against the techniques which lend themselves so readily to the calculation of frame quotients, we show how to easily compute κQnt quotients by applying...

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 . . . . . . . . . . . . . . . . . . ....

We propose to use Grothendieck institutions based on 2categorical diagrams as a basis for heterogeneous specification. We prove a number of results about colimits and (some weak variants of) exactness. This framework can also be used for obtaining proof systems for heterogeneous theories involving institution semi-morphisms.