nLab blob n-category

Context

Higher category theory

higher category theory

Contents

Idea

The notion of blob $n$-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 $n\in ℕ$ be a natural number.

Definition

A blob $n$-graph $C$ is given by

• for every $k\in ℕ$ a functor

${C}_{k}:\mathrm{core}\left({\mathrm{Ball}}_{k}\right)\to \mathrm{Set}$C_k : core(Ball_k) \to Set

from the groupoid of topological spaces homeomorphic to a $k$-ball and homeomorphisms between them to Set.

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

Definition (roughly)

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

Assume we have a blob $n$-graph $C$ with composition for $\left(k-1\right)$-cells for $k\ge 1$. Then composition of $k$-cells on $C$ is a choice of the following structure

• a natural transformationboundary restriction (source/target)

$\partial :{C}_{k}\left(X\right)\to {\underset{\to }{C}}_{k-1}\left(\partial X\right)\phantom{\rule{thinmathspace}{0ex}},$\partial : C_k(X) \to \underset{\to}{C}_{k-1}(\partial X) \,,

where on the right we have the extension to $\left(k-1\right)$ spheres of ${C}_{k-1}$ described below;

• for all balls $B={B}_{1}{\cup }_{{B}_{1}\cap {B}_{2}}{B}_{2}$ and $E:=\partial \left({B}_{1}\cap {B}_{2}\right)$ a natural transformation – composition

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

satisfying some compatibility conditions

• for all balls $X$, $D$ a natural map – identity

$C\left(X\right)\to C\left(X×D\right)$C(X) \to C(X \times D)

satisfying some compatibility conditions.

Definition (roughly)

(extension to general shapes)

For $C$ a blob $n$-graph with composition for $\left(k-1\right)$-cells and $X$ any $\left(k-1\right)$-dimensional manifold with $k, define ${\underset{\to }{C}}_{k-1}\left(X\right)$ to be the colimit

${\underset{\to }{C}}_{k-1}\left(X\right):={\underset{\to }{\mathrm{lim}}}_{\left(\coprod _{i}{U}_{i}\to X\right)}\left(\mathrm{fiber}\phantom{\rule{thickmathspace}{0ex}}\mathrm{product}\phantom{\rule{thickmathspace}{0ex}}\mathrm{of}\phantom{\rule{thickmathspace}{0ex}}{C}_{k-1}\left({U}_{i}\right)s\phantom{\rule{thickmathspace}{0ex}}\mathrm{over}\phantom{\rule{thickmathspace}{0ex}}\mathrm{joint}\phantom{\rule{thickmathspace}{0ex}}\mathrm{boundary}\phantom{\rule{thickmathspace}{0ex}}\mathrm{labels}\right)$\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 $X$, where the composition operation in $C$ is used to label refinements of permissible decompositions.

This is (MorrisonWalker, def. 6.3.2).

Examples

Definition

For $X$ a topological space, its fundamental blob $n$-category ${\Pi }_{\le n}\left(X\right)$ is the blob $n$-category which sends a $k$-ball for $k to the set of continuous maps of the ball into $X$, and an $n$-ball to the set of homotopy-classes of such maps, relative boundary.

This is (MorrisonWalker, example 6.2.1)).

Definition

For $n\in ℕ$ the blob $n$-category of $n$-dimensional cobordisms ${\mathrm{Bord}}_{n}$ is the blob $n$-category that sends a $k$-ball $B$ for $k to the set of $k$-dimensional submanifolds $W↪B×{ℝ}^{\infty }$ such that the projection $W\to B$ is transverse to $\partial B$. An $n$-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)