nLab blob n-category

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 $n \in \mathbb{N}$ be a natural number.

Definition

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

• for every $k \in \mathbb{N}$ a functor

  $$C_k : core(Ball_k) \to Set$$

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

We think of $C(B^k)$ 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 $(k-1)$-cells for $k \geq 1$. Then composition of $k$-cells on $C$ is a choice of the following structure

• a natural transformation – boundary restriction (source/target)

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

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

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

  $$\circ : C(B_1) \times_{C(B_1 \cap B_2)} C(B_2) \to C(B)$$

  satisfying some compatibility conditions

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

  $$C(X) \to C(X \times D)$$

  satisfying some compatibility conditions.

Definition (roughly)

(extension to general shapes)

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

$$\underset{\to}{C}_{k-1}(X) := {\lim_{\to}}_{({\coprod_i U_i \to X})} \left( fiber\;product\;of\;C_{k-1}(U_i)s\;over\;joint\;boundary\;labels \right)$$

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 $\Pi_{\leq n}(X)$ is the blob $n$-category which sends a $k$-ball for $k \lt n$ 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 $n \in \mathbb{N}$ the blob $n$-category of $n$-dimensional cobordisms $Bord_n$ is the blob $n$-category that sends a $k$-ball $B$ for $k \lt n$ to the set of $k$-dimensional submanifolds $W \hookrightarrow B \times \mathbb{R}^\infty$ 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)).

Related concepts

• blob homology

• (∞,n)-category with adjoints, (∞,n)-category with duals

References

Section 6 of

• Scott Morrison, Kevin Walker, The blob complex, (arxiv/1009.5025) (website)