# 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 $\Upsilon$ 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):

1. $(a \sqcup b) X = a X \sqcup b X$
2. $(a \sqcap b) X \sqsubseteq a X \sqcap b X$
3. $(\lambda x \in \mathfrak{A}: x \sqcap c) \in \Upsilon (\mathfrak{A}; \mathfrak{A})$ for every $c \in \mathfrak{A}$
4. $a \bot = \bot$
5. $\top X = \top$ unless $X = \bot$

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.