README.md

ARCHIVED

We started again in Rust: https://github.com/homotopy-io/homotopy-rs.

Build instructions

1. install purescript compiler and spago build tool:
   • nix-shell can do this for you with the provided shell.nix
   • alternatively, both of these dependencies can be installed with npm install -g purescript spago
2. run spago build in the core and webclient directories

Tooling

purty is used to format the code in this repository, and this is incorporated as a git pre-commit hook (please install with git config core.hooksPath .githooks).

purescript-language-server provides IDE-like features via the Language Server Protocol. Among other features, this provides facilities to:

• trigger builds (e.g.\ on save) --- this is vastly preferred to running spago build;
  • diagnostics information (warnings, errors) resulting from the build are passed back to the editor
• apply compiler suggestions to fix warnings;
• provide autocompletion of symbols from installed packages;
  • automatically adding the corresponding import line afterwards
• provide information about a symbol on hover (type, documentation);
• provide refactoring tools;
• auto-insert case splits.

Data structure overview

In this library, we encode zigzags in the Diagram type:

data Diagram
  = Diagram0 Generator
  | DiagramN DiagramN

newtype DiagramN
  = InternalDiagram { source :: Diagram, cospans :: List Cospan }

Diagram is a sum type which distinguishes between $0$-dimensional diagrams and $>0$-dimensional diagrams. A $0$-dimensional diagram is merely a Generator (an element of the signature), whereas for $n >0$, a $n$-dimensional diagram is given by its source boundary (A $n-1$-dimensional diagram), and a list of Cospans:

type Cospan
  = { forward :: Rewrite, backward :: Rewrite }

data Rewrite
  = Rewrite0 { source :: Generator, target :: Generator }
  | RewriteI
  | RewriteN { dimension :: Int, cones :: List Cone }

type Cone
  = { index :: Int
    , source :: List Cospan
    , target :: Cospan
    , slices :: List Rewrite
    }

A Cospan encodes a cospan comprising of:

• a singular level $sᵢ$ --- the tip of the cospan;
• zigzag maps $rᵢ \xrightarrow{\texttt{forward}} sᵢ \xleftarrow{\texttt{backward}} r_{i+1}$.

Rewrite similarly encodes maps between diagrams (zigzag maps), with identity rewrites being represented by RewriteI. A rewrite between $0$-dimensional diagrams simply maps Generators to Generators (Rewrite0). A $>0$-dimensional rewrite (RewriteN) is a non-trivial zigzag map, which is sparsely encoded as a list of Cones.

Recall that in general, a zigzag map $f\colon Z → Z′$ is given by:

• a monotone function from the singular levels of $Z$ to the singular levels of $Z′$;
• built from morphisms of $𝒞$ (if we are working in the category $Z_𝒞$ --- zigzags of $𝒞$);
• identities between regular levels;
• such that all squares commute.

A Cone is a sparse encoding of the monotone functions between singular levels. Suppose that we have a monotone function from a linearly ordered set with $m$ elements (denoted $[m]$) to one with $n$ elements: $[m] \xrightarrow{f} [n]$; this can be decomposed into a sequence of Cones as follows:

An example

Let us see how we can encode the following string diagram:

When an $n$-dimensional diagram is projected onto 2D, the $n$-cells are represented as vertices, the $n-1$-cells as edges, and the $n-2$-cells as faces. Supposing that this string diagram is 2D, this means that we have might have a signature comprising of the following:

• $0$-cells: a single $0$-cell representing the regions, which we will name “Space”;
• $1$-cells: a single $1$-cell representing the wires, $\text{Space} \xrightarrow{\text{Wire}} \text{Space}$;
• $2$-cells:
  • for the monoid, a $2$-cell $\text{Wire} ⊗ \text{Wire} → \text{Wire}$;
  • for the comonoid, a $2$-cell $\text{Wire} → \text{Wire} ⊗ \text{Wire}$;
  • for the scalar, a $2$-cell $\text{Identity}(\text{Space}) → \text{Identity}(\text{Space})$.

In the data structure, this is represented as:

-- first, we fix generators for each cell
spaceG        = Generator { id: 0, dimension: 0 }
wireG         = Generator { id: 1, dimension: 1 }
monoidG       = Generator { id: 2, dimension: 2 }
comonoidG     = Generator { id: 3, dimension: 2 }
scalarG       = Generator { id: 4, dimension: 2 }
endomorphismG = Generator { id: 5, dimension: 2 }

-- 0-diagrams
space = Diagram0 spaceG :: Diagram

-- 1-diagrams
wire = fromGenerator space space wireG :: DiagramN
      -- InternalDiagram
      --   { source: space
      --   , cospans:
      --       { forward: Rewrite0 { source: spaceG, target: wireG }
      --       , backward: Rewrite0 { source: spaceG, target: wireG }
      --       } : Nil
      --   }
wire⊗wire = attach Target Nil wire wire :: Maybe DiagramN
      -- Just $ InternalDiagram
      --   { source: space
      --   , cospans:
      --       { forward: Rewrite0 { source: spaceG, target: wireG }
      --       , backward: Rewrite0 { source: spaceG, target: wireG }
      --       } :
      --       { forward: Rewrite0 { source: spaceG, target: wireG }
      --       , backward: Rewrite0 { source: spaceG, target: wireG }
      --       } : Nil
      --   }

-- 2-diagrams
monoid = fromGenerator (DiagramN $ fromJust $ wire⊗wire) (DiagramN wire) monoidG :: DiagramN
      -- InternalDiagram
      --   { source: (DiagramN $ fromJust $ wire⊗wire)
      --   , cospans: { forward: RewriteN { dimension: 1
      --                                  , cones: { index: 0
      --                                           , source: { forward: Rewrite0 { source: spaceG
      --                                                                         , target: wireG
      --                                                                         }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: wireG
      --                                                                          }
      --                                                     } : { forward: Rewrite0 { source: spaceG
      --                                                                             , target: wireG
      --                                                                             }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: wireG
      --                                                                          }
      --                                                     } : Nil
      --                                           , target: { forward: Rewrite0 { source: spaceG
      --                                                                         , target: monoidG
      --                                                                         }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: monoidG
      --                                                                          }
      --                                                     }
      --                                           , slices: (Rewrite0 { source: wireG
      --                                                               , target: monoidG
      --                                                               })
      --                                                     : (Rewrite0 { source: wireG
      --                                                                 , target: monoidG
      --                                                                 })
      --                                                     : Nil
      --                                           } : Nil
      --                                  }
      --              , backward: RewriteN { dimension: 1
      --                                   , cones: { index: 0
      --                                            , source: { forward: Rewrite0 { source: spaceG
      --                                                                          , target: wireG
      --                                                                          }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: wireG
      --                                                                           }
      --                                                      } : Nil
      --                                            , target: { forward: Rewrite0 { source: spaceG
      --                                                                          , target: monoidG
      --                                                                          }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: monoidG
      --                                                                           }
      --                                                      }
      --                                            , slices: (Rewrite0 { source: wireG
      --                                                                , target: monoidG
      --                                                                }) : Nil
      --                                            } : Nil
      --                                   }
      --              } : Nil
      --   }
comonoid = fromGenerator (DiagramN wire) (DiagramN $ fromJust $ wire⊗wire) comonoidG :: DiagramN
      -- InternalDiagram
      --   { source: (DiagramN $ wire)
      --   , cospans: { forward: RewriteN { dimension: 1
      --                                  , cones: { index: 0
      --                                           , source: { forward: Rewrite0 { source: spaceG
      --                                                                         , target: wireG
      --                                                                         }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: wireG
      --                                                                          }
      --                                                     } : Nil
      --                                           , target: { forward: Rewrite0 { source: spaceG
      --                                                                         , target: comonoidG
      --                                                                         }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: comonoidG
      --                                                                          }
      --                                                     }
      --                                           , slices: (Rewrite0 { source: wireG
      --                                                               , target: comonoidG
      --                                                               })
      --                                                     : Nil
      --                                           } : Nil
      --                                  }
      --              , backward: RewriteN { dimension: 1
      --                                   , cones: { index: 0
      --                                            , source: { forward: Rewrite0 { source: spaceG
      --                                                                          , target: wireG
      --                                                                          }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: wireG
      --                                                                           }
      --                                                      } : { forward: Rewrite0 { source: spaceG
      --                                                                              , target: wireG
      --                                                                              }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: wireG
      --                                                                           }
      --                                                      } : Nil
      --                                            , target: { forward: Rewrite0 { source: spaceG
      --                                                                          , target: comonoidG
      --                                                                          }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: comonoidG
      --                                                                           }
      --                                                      }
      --                                            , slices: (Rewrite0 { source: wireG
      --                                                                , target: comonoidG
      --                                                                })
      --                                                      : (Rewrite0 { source: wireG
      --                                                                  , target: comonoidG
      --                                                                  })
      --                                                      : Nil
      --                                            } : Nil
      --                                   }
      --              } : Nil
      --   }
scalar = fromGenerator (DiagramN $ identity space) (DiagramN $ identity space) scalarG :: DiagramN
      -- InternalDiagram
      --   { source: (DiagramN $ identity space)
      --   , cospans: { forward: RewriteN { dimension: 1
      --                                  , cones: { index: 0
      --                                           , source: Nil
      --                                           , target: { forward: Rewrite0 { source: spaceG
      --                                                                         , target: scalarG
      --                                                                         }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: scalarG
      --                                                                          }
      --                                                     }
      --                                           , slices: Nil
      --                                           } : Nil
      --                                  }
      --              , backward: RewriteN { dimension: 1
      --                                   , cones: { index: 0
      --                                            , source: Nil
      --                                            , target: { forward: Rewrite0 { source: spaceG
      --                                                                          , target: scalarG
      --                                                                          }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: scalarG
      --                                                                           }
      --                                                      }
      --                                            , slices: Nil
      --                                            } : Nil
      --                                   }
      --              } : Nil
      --   }
endomorphism = fromGenerator (DiagramN wire) (DiagramN wire) endomorphismG :: DiagramN
      -- InternalDiagram
      --   { source: (DiagramN $ wire)
      --   , cospans: { forward: RewriteN { dimension: 1
      --                                  , cones: { index: 0
      --                                           , source: { forward: Rewrite0 { source: spaceG
      --                                                                         , target: wireG
      --                                                                         }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: wireG
      --                                                                          }
      --                                                     } : Nil
      --                                           , target: { forward: Rewrite0 { source: spaceG
      --                                                                         , target: endomorphismG
      --                                                                         }
      --                                                     , backward: Rewrite0 { source: spaceG
      --                                                                          , target: endomorphismG
      --                                                                          }
      --                                                     }
      --                                           , slices: (Rewrite0 { source: wireG
      --                                                               , target: endomorphismG
      --                                                               })
      --                                                     : Nil
      --                                           } : Nil
      --                                  }
      --              , backward: RewriteN { dimension: 1
      --                                   , cones: { index: 0
      --                                            , source: { forward: Rewrite0 { source: spaceG
      --                                                                          , target: wireG
      --                                                                          }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: wireG
      --                                                                           }
      --                                                      } : Nil
      --                                            , target: { forward: Rewrite0 { source: spaceG
      --                                                                          , target: endomorphismG
      --                                                                          }
      --                                                      , backward: Rewrite0 { source: spaceG
      --                                                                           , target: endomorphismG
      --                                                                           }
      --                                                      }
      --                                            , slices: (Rewrite0 { source: wireG
      --                                                                , target: endomorphismG
      --                                                                })
      --                                                      : Nil
      --                                            } : Nil
      --                                   }
      --              } : Nil
      --   }

With a couple of imports, this can be tested in the repl (spago repl), and the output can be examined to give something similar to the Pseudo-PureScript in the comments.

Let's unpack what is going on here.

• First, we defined a bunch of Generators, which are essentially names (with an additional dimension field) which will be bound to diagrams.

• The 0-diagram space is fully determined by giving it a generator of dimension 0: spaceG, with the Diagram0 constructor.

• The wire 1-diagram is constructed with the fromGenerator function, and has type $\text{Space} \xrightarrow{\text{Wire}} \text{Space}$ (domain is the first argument to fromGenerator, while codomain is the second argument; the third argument is the generator we wish to bind to the resulting diagram). Internally, this gets represented as a DiagramN; it specifies a source diagram and a list of cospans which rewrite from source to target. With respect to zigzags, the wire 1-diagram is represented as

  and this precisely corresponds to the singleton Cospan in wire's list of cospans, with the left arrow being the forward component and the right arrow the backward. Just as a Diagram0 is fairly trivial, a Rewrite0 is a fairly trivial (i.e.\ no information) relationship between Generators.

  The other 1-diagram we need to represent is wire⊗wire (as it is the source of the monoid, and target of the comonoid). With zigzags, it is represented as so

  The only difference is that its list of cospans is length two, instead of one.

• The monoid zigzag has one singular height, and two regular heights, looking like this:

  The source of this diagram is wire⊗wire, and it only contains a single Cospan (depicted vertically); however, in this case, the forward and backward components are not 0-dimensional rewrites. Higher dimensional rewrites are encoded by a list of Cones, which is a sparse encoding.

  The forward rewrite is depicted by the red arrows, and comprises of a single (red) cone. The index parameter of the cone identifies its starting position along the X-axis. The source parameter identifies what is replaced by target, by the rewrites in the slices parameter; the length of the source list should equal that of the slices list, and the $i$-th index of slices acts on the $i$-th index of source to encode the zigzag map --- observe that the first slice of the forward component is the left red arrow, and the second slice is the right one.

  The backward rewrite is similar (the only difference being it goes from a regular height to the preceding singular height), and is denoted in blue. Finally, in the zigzag formalism, there are identities between regular-regular heights, denoted in green; these are not represented explicitly in this encoding.

  For this particular example, no space has been saved by the sparse Cone encoding (but it will help significantly in larger diagrams).

  The comonoid diagram is similar, and the scalar zigzag looks like this:

  The endomorphism diagram looks like this:

Putting it together

We can identify the regular and singular levels of the big diagram:

This 2D-diagram should be captured by DiagramN, which has two constructors:

• source :: Diagram --- in this instance, a 1D-diagram, which is the zigzag corresponding to $[3]$ along the bottom of the figure;
• cospans :: List Cospan --- this encodes rewriting to the target, i.e.
  the forward and backward rewrites vertically along the right of the figure.