Consistent usage of essentially small collections

content/comphaus_copresentable.md

@@ -98,7 +98,7 @@ followed by
 $$\Hom_{\CompHaus}({-}, [0,1]) : \CompHaus^{\op} \to \Set.$$
 The first automatically preserves $\aleph_1$-filtered colimits (and in fact all colimits) since it has a right adjoint. The second one preserves $\aleph_1$-filtered colimits by the previous lemma.
 
-Alternately, applying the general framework of Lawvere theories shows that $\CompHaus^{\op}$ is equivalent to the category of functors $\T \to \Set$ preserving countable products, where $\T$ is the full subcategory of $\CompHaus$ of all spaces $[0,1]^A$ where $A$ is countable. Note that $\T$ is essentially small. We thus reproduce a result from [Isb82](#references) which also provides a nice description of a small set of generators of the operations of the $\aleph_0$-ary algebraic theory. A more recent treatment in [MR17](#references) refines this by providing a nice axiomatization of the relations of that theory.
+Alternately, applying the general framework of Lawvere theories shows that $\CompHaus^{\op}$ is equivalent to the category of functors $\T \to \Set$ preserving countable products, where $\T$ is the full subcategory of $\CompHaus$ of all spaces $[0,1]^A$ where $A$ is countable. Note that $\T$ is essentially small. We thus reproduce a result from [Isb82](#references) which also provides a nice description of a set of generators of the operations of the $\aleph_0$-ary algebraic theory. A more recent treatment in [MR17](#references) refines this by providing a nice axiomatization of the relations of that theory.
 :::
 
 ## References

content/foundations.md

@@ -27,13 +27,21 @@ Just imagine three copies of ZFC embedded into each other, each representing a "
 
 <img class="small" alt="visualization of three levels of size" src="/img/three-levels-of-size.webp" />
 
-The levels are not defined by cardinality alone. For example, $\{\SetColl\}$ is a collection with just one element, but it is not a set (since otherwise $\SetColl$ would be a set). In particular, not every finite collection is a set. However, every finite collection is isomorphic to a set.
-
 In our framework, there is no way to group all hypercollections into a single mathematical object; for this, one would need a third Grothendieck universe $\SetColl^{++}$, but such a grouping is usually not required.
 
-A family $(X_i)_{i \in I}$ of collections is called _small_ when its index collection is a set.
+## Essentially small, finite, and countable collections
+
+Let us call a collection $X$ _essentially small_ if it is isomorphic to a set $S$, i.e., if there is a bijection between $X$ and $S$. (An alternative terminology suggested [here](https://ncatlab.org/nlab/show/small+category) is _structurally small_.) For most parts of category theory, $X$ can then simply be replaced with $S$ and assumed to be small itself, i.e., a set.
+
+For example, the collection $\{\SetColl\}$ is not small, but essentially small, since it is isomorphic to the set $\{0\}$. This example shows that the three levels are not defined by cardinality alone. It also shows that the elements of a collection are not necessarily sets, which is yet another fundamental difference to classes.
+
+If a collection $X$ admits a surjective map from a set $S$, then it is also essentially small, since by the axiom of choice $X$ is isomorphic to a subset of $S$.
+
+A family of collections $(X_i)_{i \in I}$ is called _small_ when $I$ is a set. In this case, the collection of its elements $\{X_i : i \in I\}$ is essentially small. Every essentially small collection has this form.
 
-A collection is called _countable_ if it admits a surjective map from $\IN$. In particular, every finite collection is countable.
+A collection is called _finite_ if it is isomorphic to $\{1,\dotsc,n\}$ for some $n \in \IN$. In particular, every finite collection is essentially small. However, a finite collection is not necessarily small, as the example $\{\SetColl\}$ shows.
+
+A collection is called _countable_ if it admits a surjective map from the set of natural numbers $\IN$. In particular, every finite collection is countable, and every countable collection is essentially small.
 
 ## Categories
 
@@ -52,7 +60,9 @@ of collections (and hence a collection itself). We write $\Ob(\C) \coloneqq O$ a
 
 When $f \in \Mor(\C)$ is a morphism with $s(f) = X$ and $t(f) = Y$, we write
 $$f : X \to Y.$$
-We write $\Hom(X,Y)$ or $\Mor(X,Y)$ for the collection of such morphisms. This collection need not be a set. If it is a set for all $X,Y$, the category is called _locally small_.
+We write $\Hom(X,Y)$ or $\Mor(X,Y)$ for the collection of such morphisms.
+
+The collection $\Hom(X,Y)$ need not be a set. If it is a set for all $X,Y$, the category is called _locally small_. If $\Hom(X,Y)$ is essentially small (isomorphic to a set) for all $X,Y$, the category is called _locally essentially small_.
 
 When a morphism $f : X \to Y$ happens to be uniquely determined, it will be written as $!_{X,Y}$ or even just $!$.
 
@@ -77,6 +87,8 @@ If $\C, \D$ are categories, we can construct the functor category $[\C, \D]$ as
 
 It is better to state explicitly when the assumption of being locally small is needed.
 
+Equivalences of categories are defined [as usual](https://en.wikipedia.org/wiki/Equivalence_of_categories). A category is _essentially small_ if it is equivalent to a small category. A collection $X$ is essentially small if and only if the associated discrete category $X_{\disc}$ (which has only identity morphisms) is essentially small. In this sense, the two notions are compatible.
+
 ## Representable Functors
 
 If $\C$ is any category and $A \in \C$, we have the Hom-functor
@@ -97,10 +109,6 @@ Unless stated otherwise, we consider only small diagrams and hence small limits
 
 There are special types of limits, such as equalizers, products, and cofiltered limits, and their duals, such as coequalizers, coproducts, and filtered colimits. By convention, products and coproducts are indexed by a set, not a collection (unless stated otherwise). Filtered colimits are indexed by a small filtered category (unless stated otherwise).
 
-## Well-powered categories
-
-If $A$ is an object of a category, the collection of all monomorphisms $B \to A$ need not be a set. If, for every $A$, there exists a small family of such monomorphisms such that every monomorphism $B \to A$ is isomorphic over $A$ to one in the family, then the category is called _well-powered_. The dual notion of being _well-copowered_ is defined using epimorphisms $A \to B$. Every small category is well-powered, but there are many well-powered categories that are not small and not even equivalent to a small category.
-
 ## Conclusion
 
 There is much more to say about set-theoretic foundations for category theory (in fact, many papers have been written on the subject, and the approach developed above is just _one_ of [many](https://xkcd.com/927/) approaches), but this suffices for the purposes of _CatDat_.

content/free-cocompletion.md

@@ -11,7 +11,7 @@ that are _small_. This condition can be described in many equivalent ways:
 
 1. $F$ is a small colimit of representable functors.
 2. There is a small category $\I$ such that $F$ is the left Kan extension of a presheaf on $\I$ along a functor $\I \to \C$.
-3. There is small subcategory $\I \subseteq \C$ such that $F$ is the left Kan extension of its restriction to $\I$.
+3. There is an essentially small subcategory $\I \subseteq \C$ such that $F$ is the left Kan extension of its restriction to $\I$.
 4. The category of elements $\int F$ is [finally small](https://ncatlab.org/nlab/show/finally+small).
 
 Here, the objects of $\int F$ are pairs $(X,a)$, where $X \in \C$ and $a \in F(X)$, and a morphism $(X,a) \to (Y,b)$ is a morphism $f : X \to Y$ with $F(f)(b) = a$. The equivalence of the conditions (1), (2), (3) is proven as Proposition 4.83 in Kelly's book [Basic Concepts of Enriched Category Theory](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.html). The implication (1) $\implies$ (4) is proven as Proposition 3.7 in <a href="https://doi.org/10.1007/s10485-021-09671-9">Kan Extensions are Partial Colimits</a> by Perrone-Tholen (but there must be earlier references). The implication (4) $\implies$ (1) follows from the [co-Yoneda Lemma](https://ncatlab.org/nlab/show/co-Yoneda+lemma)
@@ -175,9 +175,9 @@ where $E \coloneqq \eq(F \rightrightarrows G)$ is the objectwise defined equaliz
 
 We view the pushout $P \coloneqq F \sqcup_E F$ as the union of two copies $F_1,F_2$ of $F$ with $F_1 \cap F_2 = E$. In particular, we regard $E,F_1,F_2$ as sub-presheaves of $P$. For a morphism $f$ in $\C$, we write $f^*$ instead of $P(f)$.
 
-Since $P \cong G$ is small, its category of elements $\int P$ has a finally small subcategory $\K$. Let $K \subseteq \Ob(\C)$ be the set of objects that appear in $\K$. We claim that
+Since $P \cong G$ is small, its category of elements $\int P$ has an essentially small final subcategory $\K$. Let $K \subseteq \Ob(\C)$ be the collection of objects that appear in $\K$. We claim that
 $$\{(A,a) : A \in K, \, a \in E(A)\}$$
-is a weakly terminal set in $\int E$, which is equivalent to saying that the canonical morphism
+is a weakly terminal essentially small collection in $\int E$, which is equivalent to saying that the canonical morphism
 $$\textstyle \coprod_{A \in K,\, a \in E(A)} \Hom(-,A) \to E$$
 is an epimorphism of presheaves, as required.
 

content/functors_on_discrete_categories.md

@@ -24,7 +24,7 @@ Let $\S$ be a non-trivial inhabited discrete category. Then a functor $F : \S \t
 ::: Proof
 Assume first that $F$ is continuous. An object $X$ is subterminal if and only if $X \times X$ exists and the diagonal $X \to X \times X$ is an isomorphism. Thus, every functor preserving binary products preserves subterminal objects. Since every object in a discrete category is subterminal, it follows that each $F(s) \in \C$ is subterminal.
 
-Conversely, assume that each $F(s) \in \C$ is subterminal. To show that $F$ is continuous, let $D : \I \to \S$ be a (small) diagram admitting a universal cone $(s \to D(i))_{i \in \I}$. Then $D(i) = s$ for all $i \in \I$, and each morphism $s \to D(i)$ is the identity. Since $\S$ has no terminal object (otherwise, $\S$ would be trivial), $\I$ is inhabited. We need to show that $(\id_{F(s)} : F(s) \to F(s))_{i \in \I}$ is a universal cone in $\C$. This follows immediately from $F(s)$ being subterminal: for a family of morphisms $X \to F(s)$ indexed by $\I$, all morphisms must be equal, and there is one such morphism since $\I$ is inhabited.
+Conversely, assume that each $F(s) \in \C$ is subterminal. To show that $F$ is continuous, let $D : \I \to \S$ be a diagram admitting a universal cone $(s \to D(i))_{i \in \I}$. Then $D(i) = s$ for all $i \in \I$, and each morphism $s \to D(i)$ is the identity. Since $\S$ has no terminal object (otherwise, $\S$ would be trivial), $\I$ is inhabited. We need to show that $(\id_{F(s)} : F(s) \to F(s))_{i \in \I}$ is a universal cone in $\C$. This follows immediately from $F(s)$ being subterminal: for a family of morphisms $X \to F(s)$ indexed by $\I$, all morphisms must be equal, and there is one such morphism since $\I$ is inhabited.
 :::
 
 Remark that in a thin category, every object is subterminal. Of course, Lemma 2 can also be dualized: A functor on a non-trivial inhabited discrete category is cocontinuous if and only if each object in its image is "co-subterminal". Here, an object $X$ is co-subterminal if any two morphisms with domain $X$ are equal (see [MSE/1092122](https://math.stackexchange.com/questions/1092122) for a discussion of the terminology).

content/generator_construction.md

@@ -1,18 +1,18 @@
 ---
 title: Construction of generators
-description: How to construct a generator from a generating set
+description: How to construct a generator from a generating collection
 ---
 
 # Construction of generators
 
 ::: Lemma
-In a category let $S$ be a generating set which is [strongly connected](/category-property/strongly_connected), i.e. between any two objects $G,G' \in S$ there is a morphism $G \to G'$. If the coproduct $U \coloneqq \coprod_{G \in S} G$ exists, then it is a generator. Moreover, if $S$ is an extremal generating set, then $U$ is an extremal generator.
+In a category let $S$ be a generating collection which is [strongly connected](/category-property/strongly_connected), i.e. between any two objects $G,G' \in S$ there is a morphism $G \to G'$. If the coproduct $U \coloneqq \coprod_{G \in S} G$ exists, then it is a generator. Moreover, if $S$ is an extremal generating collection, then $U$ is an extremal generator.
 :::
 
 ::: Proof
-We remark that the assumption on $S$ implies that each coprojection $i_G : G \to U$ has a left inverse. Now let $f,g : A \rightrightarrows B$ be two morphisms with $f \circ \bar a = g \circ \bar a$ for all $\bar a : U \to A$. If $G \in S$, any morphism $G \to A$ extends to $U$ by our preliminary remark. Thus, $f \circ a = g \circ a$ holds for all morphisms $a : G \to A$ with $G \in S$. Since $S$ is a generating set, this implies $f = g$.
+We remark that the assumption on $S$ implies that each coprojection $i_G : G \to U$ has a left inverse. Now let $f,g : A \rightrightarrows B$ be two morphisms with $f \circ \bar a = g \circ \bar a$ for all $\bar a : U \to A$. If $G \in S$, any morphism $G \to A$ extends to $U$ by our preliminary remark. Thus, $f \circ a = g \circ a$ holds for all morphisms $a : G \to A$ with $G \in S$. Since $S$ is a generating collection, this implies $f = g$.
 
-Similarly, for the case where $S$ is an extremal generating set, suppose we have a morphism $f : A \to B$ such that $f \circ {-} : \Hom(U, A) \to \Hom(U, B)$ is a bijection. In particular, because it is injective and $U$ is a generator, we can conclude that $f$ is a monomorphism, so $f \circ {-} : \Hom(G, A) \to \Hom(G, B)$ is injective for each $G \in S$. Now suppose $b \in \Hom(G, B)$ for $G \in S$. Then $b$ extends to a morphism $\bar b : U \to B$. By assumption, there exists $\bar a : U \to A$ such that $f \circ \bar a = \bar b$. Composing with the coprojection $i_G : G \to U$, we see
+Similarly, for the case where $S$ is an extremal generating collection, suppose we have a morphism $f : A \to B$ such that $f \circ {-} : \Hom(U, A) \to \Hom(U, B)$ is a bijection. In particular, because it is injective and $U$ is a generator, we can conclude that $f$ is a monomorphism, so $f \circ {-} : \Hom(G, A) \to \Hom(G, B)$ is injective for each $G \in S$. Now suppose $b \in \Hom(G, B)$ for $G \in S$. Then $b$ extends to a morphism $\bar b : U \to B$. By assumption, there exists $\bar a : U \to A$ such that $f \circ \bar a = \bar b$. Composing with the coprojection $i_G : G \to U$, we see
 $$f \circ \bar a \circ i_G = \bar b \circ i_G = b.$$
-This shows that $f \circ {-} : \Hom(G, A) \to \Hom(G, B)$ is also surjective for each $G \in S$. Since $S$ is an extremal generating set, this implies $f$ is an isomorphism.
+This shows that $f \circ {-} : \Hom(G, A) \to \Hom(G, B)$ is also surjective for each $G \in S$. Since $S$ is an extremal generating collection, this implies $f$ is an isomorphism.
 :::

content/isbell_concreteness.md

@@ -19,7 +19,7 @@ $$
 
 commutes. We say that two spans over $(A,B)$ are _equivalent_ if they commute with exactly the same cospans.
 
-The _Isbell condition_ says that, for each pair of objects $A,B$, the collection of equivalence classes of spans over $(A,B)$ is isomorphic to a set. Equivalently, there must be a set of selected spans over $(A,B)$ such that every span over $(A,B)$ is equivalent to one of the selected spans.
+The _Isbell condition_ says that, for each pair of objects $A,B$, the collection of equivalence classes of spans over $(A,B)$ is essentially small. Equivalently, there must be a set of selected spans over $(A,B)$ such that every span over $(A,B)$ is equivalent to one of the selected spans.
 
 ::: Lemma
 Every concretizable category satisfies the Isbell condition.
@@ -38,7 +38,7 @@ for $x \in U(X)$. Then the span commutes with the cospan if and only if
 $$U(a')(u) = U(b')(v)$$
 for every $(u,v) \in P_{a,b}$. Thus, whether a span commutes with a given cospan depends only on the subset $P_{a,b}$.
 
-Call a subset of $U(A) \times U(B)$ _realizable_ if it is of the form $P_{a,b}$ for some span $(a,b)$. Since $U(A) \times U(B)$ is a set, there is a set of realizable subsets. As we have just seen, the map that sends a realizable subset $P_{a,b}$ to the equivalence class of $(a,b)$ is well-defined, and it is clearly surjective. Hence, the collection of equivalence classes of spans over $(A,B)$ is isomorphic to a set.
+Call a subset of $U(A) \times U(B)$ _realizable_ if it is of the form $P_{a,b}$ for some span $(a,b)$. Since $U(A) \times U(B)$ is a set, there is a set of realizable subsets. As we have just seen, the map that sends a realizable subset $P_{a,b}$ to the equivalence class of $(a,b)$ is well-defined, and it is clearly surjective. Hence, the collection of equivalence classes of spans over $(A,B)$ is essentially small.
 :::
 
 ## References

content/missing_cogenerating_collections.md

@@ -0,0 +1,19 @@
+---
+title: Missing cogenerating collections
+description: A generalization of the proof that the category of commutative rings has no cogenerating collection.
+---
+
+# Missing cogenerating collections
+
+::: Lemma
+Let $\C$ be a category with a faithful functor $U: \C \to \Set$. Assume there exists a collection of objects $\F \subseteq \Ob(\C)$ satisfying the following conditions:
+
+1. For any $X \in \F$ and any non-terminal $Y \in \C$, for every morphism $f: X \to Y$ its underlying map $U(f) : U(X) \to U(Y)$ is injective.
+2. For every infinite cardinal number $\kappa$, there exists an object $X \in \F$ such that $\card(U(X)) \geq \kappa$ and such that $X$ has a non-identity endomorphism.
+
+Then $\C$ does not have a cogenerating collection.
+:::
+
+::: Proof
+Assume that there is a cogenerating collection $S$. By assumption (2) there is an object $X \in \F$ such that $U(X)$ is larger than all the $U(Y)$ with $Y \in S$ (w.r.t. cardinalities) and which has a non-identity endomorphism $\sigma : X \to X$. Since $S$ cogenerates, there is a morphism $f : X \to Y$ with $Y \in S$ and $f \sigma \neq f$. For this, $Y$ must be non-terminal. By (1) the map $U(f) : U(X) \to U(Y)$ is injective. This is a contradiction.
+:::