remove essentially small for now

Author: ScriptRaccoon

src/lib/content/foundations.md

@@ -16,7 +16,7 @@ Note that sets, collections, and hypercollections all satisfy the ZFC axioms. In
 
 Just imagine three copies of ZFC included into each other, each representing a "level of size". Grothendieck universes are just an implementation detail, which we can _and will_ drop from now on. Sets are on level 1, collections on level 2, and hypercollections on level 3. You can imagine concrete mathematical objects like numbers or functions as being on level 0 (even though they are usually modeled as sets in ZFC).
 
-The levels are not defined by cardinality alone, though. For example, $\\{ U \\}$ is a collection which has just one element, but it is not a set (since otherwise $U$ would be a set). In particular, not every finite collection is a set. But of course, every finite collection is isomorphic to (has a bijection to) a set. More generally, a collection is called _essentially small_ when it is isomorphic to a set. For the purpose of category theory, essentially small collections are just as good as sets.
+The levels are not defined by cardinality alone, though. For example, $\\{ U \\}$ is a collection which has just one element, but it is not a set (since otherwise $U$ would be a set). In particular, not every finite collection is a set. But of course, every finite collection is isomorphic to (has a bijection to) a set.
 
 In our framework, we have no way to group all hypercollections into a single mathematical object; for this we would need a third Grothendieck universe, but usually this grouping is not required anyway.
 
@@ -35,7 +35,7 @@ $\mathcal{C} = (O,M,i,s,t,c)$
 
 of collections (and hence a collection itself). We write $\mathrm{Ob}(\mathcal{C}) := O$ and $\mathrm{Mor}(\mathcal{C}) := M$. Instead of $X \in \mathrm{Ob}(\mathcal{C})$ we often write $X \in \mathcal{C}$.
 
-When $f \in \mathrm{Mor}(\mathcal{C})$ is a morphism and $s(f) = X$, $t(f) = Y$, we write $f : X \to Y$. We write $\mathrm{Hom}(X,Y)$ or $\mathrm{Mor}(X,Y)$ for the collection of such morphisms with source $X$ and target $Y$. It is not necessarily a set. If this collection is a set for all $X,Y$, the category is called _locally small_. If it is always isomorphic to a set (but not necessarily a set itself), the category is called _locally essentially small_. For example, every thin category (meaning that between two objects there is at most one morphism) is locally essentially small.
+When $f \in \mathrm{Mor}(\mathcal{C})$ is a morphism and $s(f) = X$, $t(f) = Y$, we write $f : X \to Y$. We write $\mathrm{Hom}(X,Y)$ or $\mathrm{Mor}(X,Y)$ for the collection of such morphisms with source $X$ and target $Y$. It is not necessarily a set. If this collection is a set for all $X,Y$, the category is called _locally small_.
 
 A _small category_ is defined as above, just by using _sets_ $O$ and $M$ (instead of collections). A _hypercategory_ is also defined in the same way, but by using _hypercollections_ $O$ and $M$ (instead of collections). Every small category is also a category, and every category is also a hypercategory.