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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).

Let A be monoid over set a, with identity element a_id and binary assoc. operation:

(a op_a a) -> a

Likewise 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
  • mapping m1: a -> b
  • mapping m2: b -> a

... such that:

(a1 op_a a2) = m2(m1(a1) op_b m2(a1))

... for all a1, a2 in a?

Example of this in math is:

exp(log(a1) + log(a2)) = a1 * a2

... with A and B being monoids over rational numbers with multiplication and addition operations, m1 being log and m2 being exp.

Another 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 sum and concat forming monoids.

So, what is the proper name for this structure in category theory (or other branches of math)?

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    $\begingroup$ It seems that you are trying to describe the situation where $A,B$ are monoids and $m_1: A \to B$ is an isomorphism or at least an injective monomorphism. However, the requirement you have set down does not quite imply that. Do you more generally want that $m_1(\mathrm{id}_A) = \mathrm{id}_B$ and $m_1(a_1 \circ_A a_2) = m_1(a_1) \circ_B m_1(a_2)$? $\endgroup$ – Mees de Vries Nov 1 '16 at 11:15
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    $\begingroup$ If $m_1$ and $m_2$ are not supposed to be homomorphisms a priori, then this situation reminds me to having a structure (now monoid) $A$, a set $b$ and mappings $m_1:a\to b,\ m_2:b\to a$ that are inverses to each others (=> bijections). Then it implies a sturcture $B$ of the same kind on $b$. -- Even if this is what you think of, I don't know if it has any specific name.. $\endgroup$ – Berci Nov 1 '16 at 22:00
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I was looking for group isomorphism. Kudos to commenters for pointing me in right direction.

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