nLab
blob n-category

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Idea

The notion of blob nn-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.

Definition

Let nn \in \mathbb{N} be a natural number.

Definition

A blob nn-graph CC is given by

We think of C(B k)C(B^k) as the set of k-morphisms in the nn-graph CC. 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 kk-cells of a blob nn-graph by induction over kk. Given a blob nn-graph with composition for kk-cells, it can be extended from balls to arbitrary manifolds by the definition extension to general shapes below.

Definition (roughly)

Say that a blob nn-graph is a blob nn-graph with composition for 0-cells.

Assume we have a blob nn-graph CC with composition for (k1)(k-1)-cells for k1k \geq 1. Then composition of kk-cells on CC is a choice of the following structure

  • a natural transformationboundary restriction (source/target)

    :C k(X)C k1(X), \partial : C_k(X) \to \underset{\to}{C}_{k-1}(\partial X) \,,

    where on the right we have the extension to (k1)(k-1) spheres of C k1C_{k-1} described below;

  • for all balls B=B 1 B 1B 2B 2B = B_1 \cup_{B_1 \cap B_2} B_2 and E:=(B 1B 2)E := \partial (B_1 \cap B_2) a natural transformation – composition

    :C(B 1)× C(B 1B 2)C(B 2)C(B) \circ : C(B_1) \times_{C(B_1 \cap B_2)} C(B_2) \to C(B)

    satisfying some compatibility conditions

  • for all balls XX, DD a natural map – identity

    C(X)C(X×D) C(X) \to C(X \times D)

    satisfying some compatibility conditions.

Definition (roughly)

(extension to general shapes)

For CC a blob nn-graph with composition for (k1)(k-1)-cells and XX any (k1)(k-1)-dimensional manifold with k<nk \lt n, define C k1(X)\underset{\to}{C}_{k-1}(X) to be the colimit

C k1(X):=lim ( iU iX)(fiberproductofC k1(U i)soverjointboundarylabels) \underset{\to}{C}_{k-1}(X) := {\lim_{\to}}_{({\coprod_i U_i \to X})} \left( fiber\;product\;of\;C_{k-1}(U_i)s\;over\;joint\;boundary\;labels \right)

over the category of permissible decompositions (…) of XX, where the composition operation in CC is used to label refinements of permissible decompositions.

This is (MorrisonWalker, def. 6.3.2).

Examples

Definition

For XX a topological space, its fundamental blob nn-category Π n(X)\Pi_{\leq n}(X) is the blob nn-category which sends a kk-ball for k<nk \lt n to the set of continuous maps of the ball into XX, and an nn-ball to the set of homotopy-classes of such maps, relative boundary.

This is (MorrisonWalker, example 6.2.1)).

Definition

For nn \in \mathbb{N} the blob nn-category of nn-dimensional cobordisms Bord nBord_n is the blob nn-category that sends a kk-ball BB for k<nk \lt n to the set of kk-dimensional submanifolds WB× W \hookrightarrow B \times \mathbb{R}^\infty such that the projection WBW \to B is transverse to B\partial B. An nn-ball is sent to homeomorphism classes rel boundary of such submanifolds.

This is (MorrisonWalker, example 6.2.6)).

References

Section 6 of

Revised on October 31, 2012 22:49:07 by Urs Schreiber (82.169.65.155)