I recall reading some time ago about some pattern/structure in category theory. Now I need to study some related properties and can't recall the proper name of it.
Let me describe it (pardon me if I am not 100% precise with my notation).
A be monoid over set
a, with identity element
a_id and binary assoc. operation:
(a op_a a) -> a
B is a monoid over set
b, with identity element
b_id and binary assoc. operation:
(b op_b b) -> a
What is the name of structure
S, that consists of:
a -> b
b -> a
... such that:
(a1 op_a a2) = m2(m1(a1) op_b m2(a1))
... for all
Example of this in math is:
exp(log(a1) + log(a2)) = a1 * a2
B being monoids over rational numbers with multiplication and addition operations,
S-like structure example is when you define mappings for
integers <-> strings (where strings are limited to be repetition of some symbol n times) and
concat forming monoids.
So, what is the proper name for this structure in category theory (or other branches of math)?