Drawing wiring diagrams in TikZ

Catlab can draw morphism expressions as TikZ pictures. To use this feature, LaTeX must be installed and the Julia package TikzPictures.jl must be loaded.

For best results, it is recommended to load the packages Convex.j and SCS.jl. When available they are used to optimize the layout of the outer ports.

In [1]:
using Catlab.WiringDiagrams, Catlab.Graphics

import Convex, SCS
import TikzPictures

Examples

Symmetric monoidal category

In [2]:
using Catlab.Theories

A, B, C, D = Ob(FreeSymmetricMonoidalCategory, :A, :B, :C, :D)
f, g = Hom(:f, A, B), Hom(:g, B, A);

To start, here are a few very simple examples.

In [3]:
to_tikz(f, labels=true)
Out[3]:
In [4]:
to_tikz(f⋅g, labels=true)
Out[4]:
In [5]:
to_tikz(f⊗g, labels=true, orientation=TopToBottom)
Out[5]:

Here is a more complex example, involving generators with compound domains and codomains.

In [6]:
h, k = Hom(:h, C, D),  Hom(:k, D, C)
m, n = Hom(:m, B⊗A, A⊗B), Hom(:n, D⊗C, C⊗D)
q = Hom(:l, A⊗B⊗C⊗D, D⊗C⊗B⊗A)

to_tikz((f⊗g⊗h⊗k)(m⊗n)q⋅(n⊗m)(h⊗k⊗f⊗g))
Out[6]:

Identities and braidings appear as wires.

In [7]:
to_tikz(id(A), labels=true)
Out[7]:
In [8]:
to_tikz(braid(A,B), labels=true, labels_pos=0.25)
Out[8]:
In [9]:
to_tikz(braid(A,B)  (g⊗f)  braid(A,B))
Out[9]:

The isomorphism $A \otimes B \otimes C \to C \otimes B \otimes A$ induced by the permutation $(3\ 2\ 1)$ is a composite of braidings and identities.

In [10]:
σ = (braid(A,B)  id(C))  (id(B)  braid(A,C)  (braid(B,C)  id(A)))

to_tikz(σ, arrowtip="Stealth", arrowtip_pos="-0.1pt",
        labels=true, labels_pos="0.1pt")
Out[10]:

By default, anchor points are added along identity and braiding wires to reproduce the expression structure in the layout. The anchors can be disabled to get a more "unbiased" layout.

In [11]:
to_tikz(σ, anchor_wires=false, arrowtip="Stealth", arrowtip_pos="-0.1pt",
        labels=true, labels_pos="0.1pt")
Out[11]:

Biproduct category

In [12]:
A, B, C = Ob(FreeBiproductCategory, :A, :B, :C)
f = Hom(:f, A, B)

to_tikz(mcopy(A), labels=true)
Out[12]:
In [13]:
to_tikz(delete(A), labels=true)
Out[13]:
In [14]:
to_tikz(mcopy(A)(f⊗f)mmerge(B), labels=true)
Out[14]:
In [15]:
to_tikz(mcopy(A⊗B), orientation=TopToBottom, labels=true)
Out[15]:
In [16]:
to_tikz(mcopy(A⊗B⊗C), orientation=TopToBottom, labels=true)
Out[16]:

Compact closed category

The unit and co-unit of a compact closed category appear as caps and cups.

In [17]:
A, B = Ob(FreeCompactClosedCategory, :A, :B)

to_tikz(dunit(A), arrowtip="Stealth", labels=true)
Out[17]:
In [18]:
to_tikz(dcounit(A), arrowtip="Stealth", labels=true)
Out[18]:

In a self-dual compact closed category, such as a bicategory of relations, every morphism $f: A \to B$ has a transpose $f^\dagger: B \to A$ given by bending wires:

In [19]:
A, B = Ob(FreeBicategoryRelations, :A, :B)
f = Hom(:f, A, B)

to_tikz((dunit(A)  id(B))  (id(A)  f  id(B))  (id(A)  dcounit(B)))
Out[19]:

Abelian bicategory of relations

In an abelian bicategory of relations, such as the category of linear relations, the duplication morphisms $\Delta_X: X \to X \oplus X$ and addition morphisms $\blacktriangledown_X: X \oplus X \to X$ belong to a bimonoid. Among other things, this means that the following two morphisms are equal.

In [20]:
X = Ob(FreeAbelianBicategoryRelations, :X)

to_tikz(plus(X)  mcopy(X))
Out[20]:
In [21]:
to_tikz((mcopy(X)⊕mcopy(X))  (id(X)⊕swap(X,X)⊕id(X))  (plus(X)⊕plus(X)))
Out[21]:

Custom styles

The visual appearance of wiring diagrams can be customized using the builtin options or by redefining the TikZ styles for the boxes or wires.

In [22]:
A, B, = Ob(FreeSymmetricMonoidalCategory, :A, :B)
f, g = Hom(:f, A, B), Hom(:g, B, A)

pic = to_tikz(f⋅g, styles=Dict(
  "box" => ["draw", "fill"=>"{rgb,255: red,230; green,230; blue,250}"],
))
Out[22]:
In [23]:
X = Ob(FreeAbelianBicategoryRelations, :X)

to_tikz(plus(X)  mcopy(X), styles=Dict(
  "junction" => ["circle", "draw", "fill"=>"red", "inner sep"=>"0"],
  "variant junction" => ["circle", "draw", "fill"=>"blue", "inner sep"=>"0"],
))
Out[23]:

By default, the boxes are rectangular (:rectangle). Other available shapes include circles (:circle), ellipses (:ellipse), triangles (:triangle, :invtriangle), and trapezoids (:trapezium, :invtrapezium).

In [24]:
to_tikz(f⋅g, default_box_shape=:circle)
Out[24]:
In [25]:
to_tikz(f⋅g, rounded_boxes=false, box_shapes=Dict(
  f => :triangle, g => :invtriangle,
))
Out[25]:
In [26]:
to_tikz(f⋅g, orientation=TopToBottom, rounded_boxes=false, box_shapes=Dict(
  f => :triangle, g => :invtriangle,
))
Out[26]:
In [27]:
to_tikz(f⋅g, box_shapes=Dict(
  f => :invtrapezium, g => :trapezium,
))
Out[27]:

Output formats

The function to_tikz returns an object of type TikZ.Document, representing a TikZ picture and its TikZ library dependencies as an abstract syntax tree. When displayed interactively, this object is compiled by LaTeX to PDF and then converted to SVG.

To generate the LaTeX source code, use the builtin pretty-printer. This feature does not require LaTeX or TikzPictures.jl to be installed.

In [28]:
import Catlab.Graphics: TikZ

doc = to_tikz(f⋅g)
TikZ.pprint(doc)
\usepackage{amssymb}
\usetikzlibrary{calc}
\usetikzlibrary{shapes.geometric}
\begin{tikzpicture}[unit length/.code={{\newdimen\tikzunit}\setlength{\tikzunit}{#1}},unit length=4mm,x=\tikzunit,y=\tikzunit,semithick,box/.style={rectangle,draw,solid,rounded corners},outer box/.style={draw=none},wire/.style={draw}]
  \node[outer box,minimum width=10\tikzunit,minimum height=4\tikzunit] (root) at (0,0) {};
  \node[box,minimum size=2\tikzunit] (n1) at (-2,0) {$f$};
  \node[box,minimum size=2\tikzunit] (n2) at (2,0) {$g$};
  \path[wire] (root.west) to[out=0,in=-180] (n1.west);
  \path[wire] (n1.east) to[out=0,in=-180] (n2.west);
  \path[wire] (n2.east) to[out=0,in=180] (root.east);
\end{tikzpicture}