nLab vine



Monoidal categories

monoidal categories

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Internal monoids



In higher category theory



Lavers (1997) introduced the monoid V nV_n of nn-vines, whose elements are thought of as paths between two sets of nn distinct points in 3\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 nV_n is not a group because a general vine cannot be untangled to give the trivial vine which is nn paths straight down. However, via pre- and post-composition, the vine monoid is acted on by the braid group on n n strands. When two nn-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 nn points to mm points. As a result vines can be assembled into a category, as described below.


What is called the category of vines is the free braided strict monoidal category containing a braided monoid (i.e. an E 2 E_2 -algebra). It is also the PROB for braided monoids.




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

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

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

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

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

  2. v i(1)P Tv_i(1)\in P_T,

  3. for all h[0,1]h\in [0,1], v i(h)v_i(h) has zz-coordinate 1h1-h,

  4. if v i(t)=v j(t)v_i(t)=v_j(t) for any t(0,1]t\in (0,1] then v i(s)=v j(s)v_i(s)=v_j(s) for all ts1t\leq s\leq 1.


Let 𝕍\mathbb{V} be the category with set of objects the natural numbers with set of morphisms 𝕍(m,n)\mathbb{V}(m,n) being (m,n)(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.


The monoidal structure of 𝕍\mathbb{V} is given by addition of natural numbers and juxtaposition of vines. Because 𝕍\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 “mm-over-nn” braid depicted below.


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


  • 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]

Last revised on July 5, 2023 at 06:43:22. See the history of this page for a list of all contributions to it.