Higher categories, colimits, and the blob complex

Higher categories, colimits, and the blob complex.

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)...

Homotopy colimits in model categories

Calculating Limits and Colimits in Pro-categories

We present some constructions of limits and colimits in pro-categories. These are critical tools in several applications. In particular, certain technical arguments concerning strict pro-maps are essential for a theorem about étale homotopy types. Also, we show that cofiltered limits in pro-categories commute with finite colimits.

On Colimits in Various Categories of Manifolds

Homotopy theorists have traditionally studied topological spaces and used methods which rely heavily on computation, construction, and induction. Proofs often go by constructing some horrendously complicated object (usually via a tower of increasingly complicated objects) and then proving inductively that we can understand what’s going on at each step and that in the limit these steps do what i...

The Blob Complex

Warning: This draft is draftier than you might expect. More specifically, • Some sections are missing. • Many sections are incomplete. In most cases the incompleteness is noted, but occasionally it isn’t. • Some sections were written nearly two years ago, and are now outdated. • There are not yet enough citations to similar work of other people. Despite all this, there’s probably enough deciphe...