Higher category theory
higher category theory
Extra properties and structure
The notion of blob -category captures the notion of an n-category with all duals. It is formulated in the style of hyperstructure: without any distinction between source and targets.
The definition is well-adapted to describing the (∞,n)-category of cobordisms in the spirit of blob homology.
Let be a natural number.
A blob -graph is given by
We think of as the set of k-morphisms in the -graph . This means that the geometric shape for higher structures used here is the globe. Therefore the term blob .
We define now a notion of composition on -cells of a blob -graph by induction over . Given a blob -graph with composition for -cells, it can be extended from balls to arbitrary manifolds by the definition extension to general shapes below.
Say that a blob -graph is a blob -graph with composition for 0-cells.
Assume we have a blob -graph with composition for -cells for . Then composition of -cells on is a choice of the following structure
a natural transformation – boundary restriction (source/target)
where on the right we have the extension to spheres of described below;
for all balls and a natural transformation – composition
satisfying some compatibility conditions
for all balls , a natural map – identity –
satisfying some compatibility conditions.
(extension to general shapes)
For a blob -graph with composition for -cells and any -dimensional manifold with , define to be the colimit
over the category of permissible decompositions (…) of , where the composition operation in is used to label refinements of permissible decompositions.
This is (MorrisonWalker, def. 6.3.2).
For a topological space, its fundamental blob -category is the blob -category which sends a -ball for to the set of continuous maps of the ball into , and an -ball to the set of homotopy-classes of such maps, relative boundary.
This is (MorrisonWalker, example 6.2.1)).
For the blob -category of -dimensional cobordisms is the blob -category that sends a -ball for to the set of -dimensional submanifolds such that the projection is transverse to . An -ball is sent to homeomorphism classes rel boundary of such submanifolds.
This is (MorrisonWalker, example 6.2.6)).
Section 6 of