Cateq is an implementation of the procedure given by Makkai to decide whether two cells of an omega-category induced by a polygraph are equal. You can give it a try directly in your browser. Some explanations on the syntax follow.

Some explanations on the syntax:

% A comment start with '%' and stops at the end of the line % A 0-cell is defined like this: x := gen * % Several 0-cells can be defined using commas y,z := gen * % An (n+1)-cell between two n-cells is defined as follows f := gen x -> y % As before, we can define several such (n+1)-cells at the same time g,h := gen x -> y f',g',h' := gen y -> z % Cells can be composed with *0, *1, etc. % Identities are obtained by applying id1, id2, etc. % Variables containing a cell can be defined using ':=' u := f *0 f' v := f *0 g' v' := id1 x *0 f *0 f' % We ask whether two n-cells are equal with '=' u = u % prints: true u = v % prints: false u = v' % prints: true % Let's add 2-cells al := gen f -> g be := gen g -> h al' := gen f' -> g' be' := gen g' -> h' % Writing formulas with higher cells become quickly tiresome (f *0 al') *1 ((al *1 be) *0 g') *1 (h *0 be') = (al *1 be) *0 (al' *1 be') % cateq understands pasting diagrams % pasting diagrams represent cells by the set of the generators "involved" in them % cateq deduces how to compose the generators to build the cell % pasting diagrams are introduced with '{ ... }' { f, f' } = f *0 f' % prints: true % Of course, cateq checks that your pasting diagram is correct % It fails otherwise X := { f , g } % fails Y := { x , g' } % fails

A complete example:

% LICS19 Example x,y,z := gen * a,b,c := gen x -> y d,e,f := gen y -> z al,al' := gen a -> b be,be' := gen b -> c ga,ga' := gen d -> e de,de' := gen e -> f A := gen al *0 de -> al' *0 de' B := gen be *0 ga -> be' *0 ga' Aext := ( a *0 ga ) *1 A *1 ( be *0 f ) Aext' := ( a *0 ga' ) *1 A *1 ( be' *0 f ) Bext := ( al *0 d ) *1 B *1 ( c *0 de ) Bext' := ( al' *0 d ) *1 B *1 ( c *0 de' ) Aext *2 Bext' = Bext *2 Aext' % prints: false % The same request can be expressed with pasting diagrams { be, A, ga } *2 { al', B, de'} = { al, B, de } *2 { be', A, ga' } % prints: false