Add some very large Grothendieck abelian categories

.cspell.json

@@ -266,6 +266,7 @@
 		"Neves",
 		"Niefield",
 		"nilradical",
+		"nimbers",
 		"nlab",
 		"Noetherian",
 		"Noncommutative",
@@ -323,6 +324,7 @@
 		"subconjugated",
 		"subcover",
 		"subfunctor",
+		"subfunctors",
 		"submanifold",
 		"submonoid",
 		"subobject",
@@ -360,6 +362,7 @@
 		"Universalis",
 		"Universitext",
 		"Urysohn",
+		"Vandermonde",
 		"vercel",
 		"Verlag",
 		"Vicari",

content/foundations.md

@@ -10,26 +10,26 @@ In _CatDat_, we work with the following convenient set-theoretic foundation for
 ## Sets, collections, and hypercollections
 
 We work with [ZFC](https://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory) and two [Grothendieck universes](https://en.wikipedia.org/wiki/Grothendieck_universe), which we denote by
-$$\mathrm{Set} \in \mathrm{Set}^+.$$
+$$\SetColl \in \SetColl^+.$$
 Thus, in principle everything is a set, but we rename them as follows to introduce three "levels of size":
 
-- The sets in $\mathrm{Set}$ are renamed to _sets_ (sometimes also _small sets_).
-- The sets in $\mathrm{Set}^+$ are renamed to _collections_ (sometimes also _large sets_).
-- All available sets are renamed to _hypercollections_ (which may or may not lie in $\mathrm{Set}^+$).
+- The sets in $\SetColl$ are renamed to _sets_ (sometimes also _small sets_).
+- The sets in $\SetColl^+$ are renamed to _collections_ (sometimes also _large sets_).
+- All available sets are renamed to _hypercollections_ (which may or may not lie in $\SetColl^+$).
 
-For example, $\IR$ is a set, $\mathrm{Set}$ is a collection, and $\mathrm{Set}^+$ is a hypercollection. The collection $\mathrm{Set}$ consists of all sets, and the hypercollection $\mathrm{Set}^+$ consists of all collections. Every set is also a collection, and every collection is also a hypercollection. There is a collection $\mathrm{Grp}$ that consists of all groups, a collection $\mathrm{Top}$ of all topological spaces, etc.
+For example, $\IR$ is a set, $\SetColl$ is a collection, and $\SetColl^+$ is a hypercollection. The collection $\SetColl$ consists of all sets, and the hypercollection $\SetColl^+$ consists of all collections. Every set is also a collection, and every collection is also a hypercollection. There is a collection $\GrpColl$ that consists of all groups, a collection $\TopColl$ of all topological spaces, etc.
 
 Note that sets, collections, and hypercollections all satisfy the ZFC axioms. In this sense, (hyper)collections behave in the same way as sets. This is crucial for category theory. For example, we can form the collection of all maps between two collections. This basic property is not satisfied by [classes](<https://en.wikipedia.org/wiki/Class_(set_theory)>), which are not adequate for category theory.
 
-For example, there is a collection $[\mathrm{Set},\mathrm{Set}]$ that consists of all maps $\mathrm{Set} \to \mathrm{Set}$.
+For example, there is a collection $[\SetColl,\SetColl]$ that consists of all maps $\SetColl \to \SetColl$.
 
 Just imagine three copies of ZFC embedded into each other, each representing a "level of size". Grothendieck universes are merely 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. Concrete mathematical objects such as numbers or functions can be thought of as living on level 0 (even though they are usually modeled as sets in ZFC).
 
 <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, $\{\mathrm{Set}\}$ is a collection with just one element, but it is not a set (since otherwise $\mathrm{Set}$ would be a set). In particular, not every finite collection is a set. However, every finite collection is isomorphic to a set.
+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 $\mathrm{Set}^{++}$, but such a grouping is usually not required.
+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.
 
@@ -56,9 +56,9 @@ We write $\Hom(X,Y)$ or $\Mor(X,Y)$ for the collection of such morphisms. This c
 
 When a morphism $f : X \to Y$ happens to be uniquely determined, it will be written as $!_{X,Y}$ or even just $!$.
 
-A _small category_ is defined as above, but using _sets_ $O$ and $M$ (instead of collections). A _hypercategory_ is defined similarly using _hypercollections_ $O$ and $M$. Every small category is a category, and every category is a hypercategory. Notice that there is a collection of all small categories $\mathrm{Cat}$, and likewise a hypercollection of all categories $\mathrm{Cat}^+$.
+A _small category_ is defined as above, but using _sets_ $O$ and $M$ (instead of collections). A _hypercategory_ is defined similarly using _hypercollections_ $O$ and $M$. Every small category is a category, and every category is a hypercategory. Notice that there is a collection of all small categories $\CatColl$, and likewise a hypercollection of all categories $\CatColl^+$.
 
-For example, the category of sets $\Set$ has $\Ob(\Set) = \mathrm{Set}$, the collection of all sets. The category of groups $\Grp$ has $\Ob(\Grp) = \mathrm{Grp}$, the collection of all groups. Other typical categories (topological spaces, graphs, metric spaces, etc.) are constructed as usual. All these examples are locally small.
+For example, the category of sets $\Set$ has $\Ob(\Set) = \SetColl$, the collection of all sets. The category of groups $\Grp$ has $\Ob(\Grp) = \GrpColl$, the collection of all groups. Other typical categories (topological spaces, graphs, metric spaces, etc.) are constructed as usual. All these examples are locally small.
 
 Collections are the objects of a hypercategory $\Set^+$.
 

database/data/categories/Ab.yaml

@@ -20,6 +20,8 @@ related:
   - TorsFreeAb
   - grAb
   - SeqAb
+  - Set_disc_Ab
+  - TransSeqAb
 
 satisfied_properties: []
 

database/data/categories/BOn.yaml

@@ -1,6 +1,6 @@
 id: BOn
 name: delooping of the additive monoid of ordinal numbers
-notation: $B\On$
+notation: $B\OnColl$
 objects: a single object
 morphisms: ordinal numbers, with addition as composition
 description: Every monoid $M$ induces a category $BM$ with a single object $*$. This also works when $M$ is large, in which case $BM$ is not locally small. In this example, we apply this construction to the large monoid of ordinal numbers with respect to addition, so composition is $\alpha \circ \beta = \alpha + \beta$.
@@ -32,7 +32,7 @@ satisfied_properties:
     proof: See <a href="https://math.stackexchange.com/questions/5129138/" target="_blank">MSE/5129138</a>.
 
   - property: locally cartesian closed
-    proof: The slice category $B\On / *$ is isomorphic to the poset $(\On,\geq)$ (not to $(\On,\leq)$). This category is thin and and semi-strongly connected, <a href="/category-implication/sequential_implies_lcc">hence</a> cartesian closed.
+    proof: The slice category $B\OnColl / *$ is isomorphic to the poset $(\OnColl,\geq)$ (not to $(\OnColl,\leq)$). This category is thin and and semi-strongly connected, <a href="/category-implication/sequential_implies_lcc">hence</a> cartesian closed.
 
   - property: ℵ₁-cofiltered
     proof: In fact, it is $\kappa$-cofiltered for every cardinal $\kappa$. By the dual of Theorem 2.2 at the <a href="https://ncatlab.org/nlab/show/filtered+category">nLab</a> it suffices to prove any set of objects has a cone (which is trivial in a one-object category) and that any set of parallel morphisms is equalized by some morphism. Here, this means that for every set of ordinals $A$ there is some ordinal $\beta$ such that $\alpha + \beta$ for $\alpha \in A$ does not depend on $\alpha$. Take $\beta$ to be any ordinal larger than $\sup(A)$ of the form $\omega^\gamma$. It is well-known that $\omega^\gamma$ has the property that $\alpha + \omega^\gamma = \omega^\gamma$ for all $\alpha < \omega^\gamma$ (Kunen's <i>Set Theory</i>, Exercise I.9.53), from which the claim follows.
@@ -41,29 +41,29 @@ satisfied_properties:
     references:
       - BN_aleph1-filtered_colimits
     proof: >-
-      The proof is similar to <a href="/category/BN">$B\IN$</a>. Let $\I$ be an $\aleph_1$-filtered small category and $D : \I \to B\On$ a diagram. A cocone $\lambda = (\lambda_i)_{i \in \I}$ for $D$ is a family of ordinals satisfying $\lambda_i = \lambda_j + D(f)$ for every morphism $f: i \to j$ in $\I$.
+      The proof is similar to <a href="/category/BN">$B\IN$</a>. Let $\I$ be an $\aleph_1$-filtered small category and $D : \I \to B\OnColl$ a diagram. A cocone $\lambda = (\lambda_i)_{i \in \I}$ for $D$ is a family of ordinals satisfying $\lambda_i = \lambda_j + D(f)$ for every morphism $f: i \to j$ in $\I$.
 
 
-      We first observe that $D$ factors uniquely through the preorder reflection of $\I$. Indeed, any two parallel morphisms in $\I$ are coequalized by some morphism, and $B\On$ is left cancellative. Thus, we may assume that $\I$ is a preordered set. Each inequality $i \leq j$ in $\I$ is mapped to an ordinal number $\alpha_{i,j} \coloneqq D(i \to j)$, and these numbers satisfy
+      We first observe that $D$ factors uniquely through the preorder reflection of $\I$. Indeed, any two parallel morphisms in $\I$ are coequalized by some morphism, and $B\OnColl$ is left cancellative. Thus, we may assume that $\I$ is a preordered set. Each inequality $i \leq j$ in $\I$ is mapped to an ordinal number $\alpha_{i,j} \coloneqq D(i \to j)$, and these numbers satisfy
       $$\alpha_{i,k} = \alpha_{j,k} + \alpha_{i,j}$$
       for all $i \leq j \leq k$. In particular, $\alpha_{i,j} \leq \alpha_{i,k}$.
 
 
-      For fixed $i \in \I$, the collection $\{\alpha_{i,j} : j \geq i\}$ is a set of ordinals because $\I$ is small, hence bounded above in $\On$. We claim that it has a maximum element. Otherwise, we can find a countable chain $i = j_0 \leq j_1 \leq j_2 \leq \dotsc$ in $\I$ such that $\alpha_{i,j_n} < \alpha_{i,j_{n+1}}$ for all $n \in \IN$. Since $\I$ is $\aleph_1$-filtered, there is an upper bound $j_\infty \in \I$ of $(j_n)_{n \in \IN}$. For each $n \in \IN$, the equation
+      For fixed $i \in \I$, the collection $\{\alpha_{i,j} : j \geq i\}$ is a set of ordinals because $\I$ is small, hence bounded above in $\OnColl$. We claim that it has a maximum element. Otherwise, we can find a countable chain $i = j_0 \leq j_1 \leq j_2 \leq \dotsc$ in $\I$ such that $\alpha_{i,j_n} < \alpha_{i,j_{n+1}}$ for all $n \in \IN$. Since $\I$ is $\aleph_1$-filtered, there is an upper bound $j_\infty \in \I$ of $(j_n)_{n \in \IN}$. For each $n \in \IN$, the equation
       $$\alpha_{i,j_{n+1}} = \alpha_{j_n,j_{n+1}} + \alpha_{i,j_n}$$
       implies that $\alpha_{j_n,j_{n+1}} > 0$. Hence,
       $$\alpha_{j_n,j_\infty} = \alpha_{j_{n+1},j_\infty} + \alpha_{j_n,j_{n+1}} > \alpha_{j_{n+1},j_\infty},$$
-      so $(\alpha_{j_n,j_\infty})_{n \in \IN}$ is a strictly decreasing infinite sequence of ordinals, contradicting the well-foundedness of $\On$. Thus, the maximum
+      so $(\alpha_{j_n,j_\infty})_{n \in \IN}$ is a strictly decreasing infinite sequence of ordinals, contradicting the well-foundedness of $\OnColl$. Thus, the maximum
       $$u_i \coloneqq \max \{ \alpha_{i,j} : j \geq i \}$$
-      is a well-defined ordinal number, which we regard as a morphism in $B\On$. The family $(u_i)_{i \in \I}$ forms a cocone for $D$, since for all $i \leq j$ we have
+      is a well-defined ordinal number, which we regard as a morphism in $B\OnColl$. The family $(u_i)_{i \in \I}$ forms a cocone for $D$, since for all $i \leq j$ we have
       $$\begin{align*}
       u_i & = \max \{ \alpha_{i,k} : k \geq i \} \\
       & = \max \{ \alpha_{i,k} : k \geq j \} \\
       & = \max \{ \alpha_{j,k} + \alpha_{i,j} : k \geq j \} \\
       & = \max \{ \alpha_{j,k} : k \geq j \} + \alpha_{i,j} \\
       & = u_j + \alpha_{i,j}.
       \end{align*}$$
-      To establish the universal property, let $(\lambda_i)_{i \in \I}$ be any cocone for $D$, so that $\lambda_i = \lambda_j + \alpha_{i,j}$ for all $i \leq j$. The cocone relation $u_i = u_j + \alpha_{i,j}$ implies that $u_i \geq u_j$ whenever $i \leq j$. By the well-foundedness of $\On$, there exists $i_0 \in \I$ such that $u_j = u_{i_0}$ for all $j \geq i_0$. For such $j$, the relation
+      To establish the universal property, let $(\lambda_i)_{i \in \I}$ be any cocone for $D$, so that $\lambda_i = \lambda_j + \alpha_{i,j}$ for all $i \leq j$. The cocone relation $u_i = u_j + \alpha_{i,j}$ implies that $u_i \geq u_j$ whenever $i \leq j$. By the well-foundedness of $\OnColl$, there exists $i_0 \in \I$ such that $u_j = u_{i_0}$ for all $j \geq i_0$. For such $j$, the relation
       $$u_{i_0} = u_j + \alpha_{i_0,j} = u_{i_0} + \alpha_{i_0,j}$$
       forces $\alpha_{i_0,j} = 0$. Consequently,
       $$u_{i_0} = \max \{ \alpha_{i_0,j} : j \geq i_0 \} = 0.$$
@@ -78,14 +78,14 @@ satisfied_properties:
 
       For uniqueness, suppose $\kappa'$ is any ordinal satisfying $\lambda_i = \kappa' + u_i$ for all $i \in \I$. Evaluating at $i_0$ yields
       $$\lambda_{i_0} = \kappa' + u_{i_0} = \kappa' + 0 = \kappa',$$
-      hence $\kappa' = \kappa$. Therefore, the cocone $(u_i)_{i \in \I}$ is the colimit of $D$ in $B\On$.
+      hence $\kappa' = \kappa$. Therefore, the cocone $(u_i)_{i \in \I}$ is the colimit of $D$ in $B\OnColl$.
 
 unsatisfied_properties:
   - property: one-way
     proof: This is trivial.
 
   - property: locally essentially small
-    proof: This is because $\On$ is large.
+    proof: This is because $\OnColl$ is a large collection.
 
   - property: balanced
     proof: Every finite ordinal is both a mono- and an epimorphism (see below), but only $0$ is an isomorphism.

database/data/categories/On.yaml

@@ -1,9 +1,9 @@
 id: On
 name: partially ordered collection of ordinal numbers
-notation: $(\On,\leq)$
+notation: $\On$
 objects: ordinal numbers
 morphisms: 'a unique morphism $(\alpha,\beta): \alpha \to \beta$ if $\alpha \leq \beta$'
-description: This category is a large variant of the thin <a href="/category/N">category of natural numbers</a>.
+description: This is the partially ordered collection $\On = (\OnColl,\leq)$, regarded as a thin category. This category is a large variant of the thin <a href="/category/N">category of natural numbers</a>.
 nlab_link: null
 
 tags:

database/data/categories/SeqAb.yaml

@@ -15,6 +15,7 @@ related:
   - grAb
   - FiltVect
   - Ch(Ab)
+  - TransSeqAb
 
 satisfied_properties: []
 

database/data/categories/Set_disc_Ab.yaml

@@ -0,0 +1,74 @@
+id: Set_disc_Ab
+name: category of set-indexed families of abelian groups
+notation: $[\Set_{\disc},\Ab]$
+objects: families of abelian groups $(A_X)_{X \in \SetColl}$ indexed by all sets
+morphisms: families of homomorphisms
+description: This functor category $[\Set_{\disc},\Ab] \cong \Ab^{\SetColl}$ is a larger variant of $\grAb = [\IZ_{\disc}, \Ab]$. Instead of $\Set_{\disc}$, we may take any other large discrete category. It does not appear in practice, but we have added it because of its interesting combinations of properties. For example, it shows that a Grothendieck abelian category is not necessarily locally small. For some background on why this functor category is well-defined, see <a href="/content/foundations">Foundations</a>.
+nlab_link: null
+
+tags:
+  - algebra
+
+related:
+  - Ab
+  - Z
+  - Set_disc
+  - grAb
+  - TransSeqAb
+  - Vect_large
+
+satisfied_properties:
+  - property: preadditive
+    proof: This property is immediately inherited from <a href="/category/Ab">$\Ab$</a>, because we may define the preadditive structure pointwise via $(f+g)_X \coloneqq f_X + g_X$. Note that for two families $A,B$, the collection $\Hom(A,B)$ is a <i>possibly large</i> abelian group, which is compatible with our definition of a preadditive category.
+
+  - property: cocomplete
+    proof: This property is immediately inherited from <a href="/category/Ab">$\Ab$</a>. Colimits are defined pointwise.
+
+  - property: complete
+    proof: This property is immediately inherited from <a href="/category/Ab">$\Ab$</a>. Limits are defined pointwise.
+
+  - property: exact filtered colimits
+    proof: This property is immediately inherited from <a href="/category/Ab">$\Ab$</a>.
+
+  - property: normal
+    proof: This property is immediately inherited from <a href="/category/Ab">$\Ab$</a>.
+
+  - property: conormal
+    proof: This property is immediately inherited from <a href="/category/Ab">$\Ab$</a>.
+
+  - property: generator
+    proof: We know that <a href="/category/Ab">$\Ab$</a> has a cogenerator $G$, for example $G = \IZ$. Then the constant family $(G)_{X \in \SetColl}$ is a generator of $[\Set_{\disc},\Ab]$.
+
+  - property: cogenerator
+    proof: We know that <a href="/category/Ab">$\Ab$</a> has a cogenerator $Q$, for example $Q = \IQ / \IZ$. Then the constant family $(Q)_{X \in \SetColl}$ is a cogenerator of $[\Set_{\disc},\Ab]$.
+
+unsatisfied_properties:
+  - property: skeletal
+    proof: This is trivial.
+
+  - property: split abelian
+    proof: Since there is an exact embedding $\Ab \to [\Set_{\disc},\Ab]$ which inserts an abelian group at some index, this follows from the fact that <a href="/category/Ab">$\Ab$</a> is not split abelian.
+
+  - property: well-powered
+    proof: The collection of subobjects of the constant family $(\IZ/2)_{X \in \SetColl}$ identifies with the collection $P(\SetColl)$, which is not isomorphic to a set.
+
+special_objects:
+  initial object:
+    description: family of trivial groups
+  terminal object:
+    description: family of trivial groups
+  coproducts:
+    description: pointwise defined direct sums
+  products:
+    description: pointwise defined direct products
+
+special_morphisms:
+  isomorphisms:
+    description: families of bijective homomorphisms
+    proof: This is trivial.
+  monomorphisms:
+    description: families of injective homomorphisms
+    proof: The category is abelian and hence has kernels, constructed pointwise. Thus, a homomorphism $f = (f_X)_{X \in \SetColl}$ is a monomorphism if and only if $\ker(f_X) = 0$ for all $X$, i.e. each $f_X$ is a monomorphism.
+  epimorphisms:
+    description: families of surjective homomorphisms
+    proof: The category is abelian and hence has cokernels, constructed pointwise. Thus, a homomorphism $f = (f_X)_{X \in \SetColl}$ is an epimorphism if and only if $\coker(f_X) = 0$ for all $X$, i.e. each $f_X$ is an epimorphism.