nLab vine

Idea

Lavers (1997) introduced the monoid $V_n$ of $n$-vines, whose elements are thought of as paths between two sets of $n$ distinct points in $\mathbb{R}^3$ which are allowed to merge into a single path, but not separate again. Some examples, from Lavers’ original paper, are depicted below.

Notice that $V_n$ is not a group because a general vine cannot be untangled to give the trivial vine which is $n$ paths straight down. However, via pre- and post-composition, the vine monoid is acted on by the braid group on $n$ strands. When two $n$-vines are composed any resulting strands which are only attached to one endpoint get retracted down to that endpoint, as in the following image (again from Lavers 97):

More generally, one can consider vines from $n$ points to $m$ points. As a result vines can be assembled into a category, as described below.

Definition

What is called the category of vines is the free braided strict monoidal category containing a braided monoid (i.e. an $E_2$-algebra). It is also the PROB for braided monoids.

Concretely:

Definition

Let

• $P_I \coloneqq \big\{ (i,0,1) \,\big\vert\, i=1, 2, \ldots, m \big\}$

• $P_T \coloneqq \big\{ (i,0,0) \,\big\vert\, i=1, 2, \ldots,n \big\}$

be linearly ordered collections of points in $\mathbb{R}^3$.

Then an $(m,n)$-vine is a set of arcs $\{v_1,\ldots,v_m\}$ (i.e. piecewise linear maps from $[0,1]$ to $\mathbb{R}^3$) with the following properties:

1. $v_i(0)=(i,0,1)$,

2. $v_i(1)\in P_T$,

3. for all $h\in [0,1]$, $v_i(h)$ has $z$-coordinate $1-h$,

4. if $v_i(t)=v_j(t)$ for any $t\in (0,1]$ then $v_i(s)=v_j(s)$ for all $t\leq s\leq 1$.

Definition

Let $\mathbb{V}$ be the category with set of objects the natural numbers with set of morphisms $\mathbb{V}(m,n)$ being $(m,n)$-vines modulo a suitable notion of ambient isotopy that allows arcs to be deformed up to homotopy but not pass through one another.

Properties

The monoidal structure of $\mathbb{V}$ is given by addition of natural numbers and juxtaposition of vines. Because $\mathbb{V}$ contains the braid category as a subcategory, it cannot be symmetric monoidal, but it is braided monoidal. The braiding is the same as that of the braid category, the “$m$-over-$n$” braid depicted below.

Exercise

Show that the above is a braided monoidal structure on $\mathbb{V}$.

References

• T. G. Lavers, The theory of vines, Comm. Algebra 25 4 (1997) 1257–1284 [doi:10.1080/00927879708825919]

• Mark Weber, §6.3 of: Internal algebra classifiers as codescent objects of crossed internal categories. Theory and Applications of Categories, 30 50 (2015) 1713–1792 [tac:30/50, pdf]