Page 151 - Textos de Matemática Vol. 46

MATHEMATICAL MORPHOLOGY WITH REGIMES IN ALGEBRAIC UNIVERSES 141
As a consequence, the monad P = (9, a, S) is “strong” i.e. with a strengthening (in the sense of Kock),
s t AE : E A ! P ( E ) P ( A ) , or equivalently an internal image calculus,
imA,E :PA⇥EA !PE. 1.2. The canonical universe of sets.
Proposition 1.1. The category Set of sets and maps is equipped with a canon- ical structure of universe as follows: F ⇥ E is the cartesian product of the two sets E and F, PE = P(E) is the powerset of E, with elements the subsets of E, the coupling cE : E2 ! P(E) is given by
cE ((y, x)) = {y, x}, the two maps and ⇡ are given by
E(A)={B;9x(x2A^x2B)}, ⇡E(A)={B;8x(x2B(x2A)}, and Cf is defined by inverse image,
Cf(B) = {x 2 E;f(x) 2 B}.
Proposition 1.2. In the algebraic universe of sets from Proposition 1.1, any structure (in the Bourbaki’s sense), algebraic or topological, can be defined equa- tionally. Especially we can study hyperspaces and continuous relations.
Remark 1.3. The proposition allows, in principle, to write any set theoretical construction or specification by an equation or a set of equations in the context of the set canonical universe, in such a way that, these equations can serve of specification of “the same thing” in an arbitrary algebraic universe. Concretely, this allows us to introduce operations and notions in arbitrary algebraic uni- verses by using the set theoretical notations. In the continuation of this paper at several moments we do that, mainly because for the standard reader it is clearly more intuitive. Nevertheless, if we can emphasize the part played by and ⇡, which is the core of the structure in the universe, we do not forget to do so. It will be the case notably for topological data and the construction of hyperspaces.
1.3. The case of an elementary topos.
Proposition 1.4. An elementary topos E is equipped with a canonical structure of algebraic universe, given, in the internal language, by the same formula as in the case of Set in Proposition 1.1.