@acset_colim variants: exposing name mapping, and morphisms

[kris-brown]

This draft introduces two new macros related to @acset_colim. The first, @named_acset_colim, takes in an extra parameter which will be assigned a NamedTuple relating the names of the generators to their respective elements (Pair{Symbol, Int}).

acs = @named_acset_colim y_Graph MyDomain begin
  v::V; e::E; src(e)==v
end

@test acs[MyDomain.e[2],:src] == MyDomain.v[2]

The second @acset_transformation_colim, accepts two @acset_colim-style expressions as well as a sequence of pairs (which initializing a hom search) as well as keyword arguments for hom-search.

# sending path graph •→•→• to •→•↺↺. 
f = @acset_transformation_colim y_Graph #=domain=# begin
  (vstart,vend)::V; (e1,e2)::E; 
  src(e1)==vstart; tgt(e1)==src(e2); tgt(e2)==vend
end #=codomain =# begin
  v::V; (ea,eb,ec)::E; 
  src(ea)==v; tgt(ea)==src(eb) == tgt(eb) == src(ec) == tgt(ec)
end #=mapping=# begin vstart=>v; end #=kw=# (any=true, monic=[:E])

In order for this to truly work, the data of yoneda is actually not enough. We are missing the data of, in a representable Foo, which element is the representing element. For example, the representable edge in a reflexive graph has three edges in it. Only one of them is the representing element. Presently, for simplicity so that the same data can be plugged into @acset_colim and these macros, we throw an error if there is any ambiguity in what the representing element is.

[wwaites]

Moving some discussion from the Zulip here. Having some experience with these new macros now, I think that it might be unwise to have @named_acset_colimit implemented the way it is because it is not referentially transparent -- the names appear effectively by side-effect. This more functional (and simpler!) implementation might be better :

macro named_acset_colim(yon, body)
  quote 
    colimit_representables($(parse_diagram_data(body, __module__)), $(esc(yon)))
  end
end

[wwaites]

names is not yet defined here

[wwaites]

Comment on the PR moved here: #986 (comment)

Having some experience with these new macros now, I think that it might be unwise to have @named_acset_colimit implemented the way it is because it is not referentially transparent -- the names appear effectively by side-effect. This more functional (and simpler!) implementation might be better:

macro named_acset_colim(yon, body)
  quote 
    colimit_representables($(parse_diagram_data(body, __module__)), $(esc(yon)))
  end
end

[wwaites]

Probably ought to change the comment too:

This would behave like `@acset_colim` and assign to `Dom` the walking edge 
ACSet with src=1 and tgt=2. However, unlike `@acset_colim`, this returns the pair
`(names, colimit)` where `names =  (v=(:V, 1),e=(:E, 1))`.