# A new math abstraction, categories of sides

I introduce a new math abstraction, *categories of sides*, in order to generalize two theorems into one.

Category of *sides* is an ordered category whose objects are (small) bounded lattices and whose morphisms are maps between lattices such that every Hom-set is a bounded lattice and (for all relevant variables):

- for every
- unless

I call morphisms of such categories *sides*.

The category of pointfree funcoids between boolean lattices is a category of sides. Also it seems (not checked yet) that the category of Galois connections between boolean lattices is a category of sides.

This way, it seems that I’ve found a common generalization of two theorems:

**Theorem** For category of pointfree funcoids, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

**Theorem** For category of Galois connections, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

The last theorem is a slight reformulation of theorem 3.8 in “Zahava Shmuely. The tensor product of distributive lattices. algebra universalis, 9(1):281–296.” (I borrowed the proof idea from that Zahava’s article.)

Common generalization:

**Theorem** For every category of sides, the set of morphisms between a non-atomic boolean lattice and itself is not a boolean lattice.

It is also conceivable to define *pointfree reloids* as filers on a (fixed) category of sides.

Note that the definition of “categories of sides” is preliminary, I may probably add more axioms in the future, if found convenient.

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