Mazzola, Guerino / Noll, Thomas / Lluis-Puebla, Emilio: Perspectives in Mathematical and Computational Music Theory

Here is the definition of the critical map $fs$ . Whenever no confusion is likely, we omit the indexes $F,D,E$ of $f$ . On the topoi $E1,E2$ , it is the given logical morphism. On diagrams, we have this construction: Let $X : S --> E1$ be diagram in $Dias(F1/E1)$ . We define a diagram $fs(X) = X': S'--> E2$ . Its quiver $S'$ has these data: The vertex set is the image $S'= f(S)$ . For a given vertex couple $A, B (- S'$ the arrow are all arrows $(U,V,i)$ in $S$ such that $f(U ) = A,f(V ) = B$ . We enumerate these arrows by natural indexes $j = 1,2,3,...$ and denote these indexes⁴

⁴: This indexing function is however only defined up to permutations, but in this theory, quivers are only considered modulo permutations of the arrow numbers for given vertex couples, as limits and colimits are invariant under these permutations.

by $j = j(U,V,i)$ . With this, the new diagram $' X$ sends an arrow $j = j(U,V,i)$ to the morphism $f(X(i)) : space(A) --> space(B)$ . This follows from the axiom $s Id2 o fF = f o Id1$ and therefore, form spaces and frames commute with $f$ , i.e., for any form $F$ , we have $s f (Id(F) : space(F) --> frame(F )) = Id(f(F )) : space(f(F )) --> frame(f (F ))$ .

Now, if $F$ is simple, the diagram and frame maps coincide, and diagram commutation with $s f$ means commutation of frames. If $F$ is of limit type, the conservation of the limit $frame(F ) = lim D(F )$ under the logical $f$ means precisely $s frame(f (F )) = lim f (D(F))$ ; analogously for colimit and powerset type. Of course, a number of conditions upon a morphism $f$ are intertwined, but this is not the place to discuss a minimal set of conditions. More important is the following fact:

Sorite 1 Given three form semiotics $Sem1,Sem2, Sem3$ , and two morphisms

f = (fF,fD,fE) : Sem1 --> Sem2,g = (gF ,gD, gE) : Sem2 --> Sem3,

the factorwise composition $go f : Sem1 --> Sem3$ is a morphism of form semiotics. This composition is associative, and the identity is such a morphism. Call $F orSem$ the category of form semiotics with these morphism data.

In (Mazzola, 2002, G.5.3.2), the category $F orSem$ is used to propose global form semiotics by the usual gluing procedure. Here, we are rather interested in the local extension problem.

4 Galois Correspondence of Form Semiotics

Given a topos $E$ , together with an address category $R$ , we denote by $Ø(E,R)$ the empty form semiotic with $F = D = Ø$ . An automorphism of $Ø(E,R)$ is just an (automatically logical) automorphism of $E$ which preserves addresses, i.e, induces an automorphism on the subcategory of addresses, denote by $Aut(E,R)$ the group of these automorphisms. Let $P$ be a subgroup of $Aut(E,R)$ . Given a form semiotic $S$ over $E,R$ , call $AutP(S)$ the group of automorphisms of a semiotic $S$ over $P$ , i.e., the automorphisms which have elements of $P$ as underlying topos morphisms. In particular, for the trivial group $P = IdE$ , we write $AutE(S)$ and call this the group of automorphisms of $S$ over $E$ . For any subsemiotic $R$ of $S$ over $E$ , we consider the subgroup $GalE(S/R) < AutE(S)$ consisting of those automorphisms which leave