Parent structures

database/data/categories/Ab.yaml

@@ -3,8 +3,9 @@ name: category of abelian groups
 notation: $\Ab$
 objects: abelian groups
 morphisms: group homomorphisms
-description: This is the prototype of an abelian category.
+description: This category is the prototype of an abelian category. It is the special case of <a href="/category/R-Mod_non_ss">$R{-}\Mod$</a> where $R = \IZ$.
 nlab_link: https://ncatlab.org/nlab/show/Ab
+parent: R-Mod_non_ss
 
 tags:
   - algebra
@@ -15,31 +16,15 @@ related:
   - FinAb
   - FreeAb
   - Grp
-  - R-Mod
   - TorsAb
   - TorsFreeAb
 
-satisfied_properties:
-  - property: locally small
-    proof: There is a forgetful functor $\Ab \to \Set$ and $\Set$ is locally small.
-
-  - property: abelian
-    proof: This is standard, see <a href="https://ncatlab.org/nlab/show/Categories+for+the+Working+Mathematician" target="_blank">Mac Lane</a>, Ch. VIII.
-    label: ab_abelian
-
-  - property: finitary algebraic
-    proof: Take the algebraic theory of a commutative group.
+satisfied_properties: []
 
 unsatisfied_properties:
-  - property: skeletal
-    proof: This is trivial.
-
   - property: split abelian
     proof: The short exact sequence $0 \xrightarrow{} \IZ \xrightarrow{p} \IZ \xrightarrow{} \IZ/p \xrightarrow{} 0$ does not split.
 
-  - property: CSP
-    proof: The canonical homomorphism $\bigoplus_{n \geq 0} \IZ \to \prod_{n \geq 0} \IZ$ is not surjective, hence no epimorphism.
-
 special_objects:
   initial object:
     description: trivial group
@@ -53,4 +38,4 @@ special_objects:
 special_morphisms:
   epimorphisms:
     description: surjective morphisms
-    proof: 'For the non-trivial direction, if $f : A \to B$ is an epimorphism, then $p \circ f = 0$ for the projection  $p : B \to B/f(A)$ implies that $p = 0$, so that $B = f(A)$.'
+    proof: 'For the non-trivial direction, if $f : A \to B$ is an epimorphism, then $p \circ f = 0$ for the projection $p : B \to B/f(A)$ implies that $p = 0$, so that $B = f(A)$.'

database/data/categories/Ab_fg.yaml

@@ -3,7 +3,7 @@ name: category of finitely generated abelian groups
 notation: $\Ab_{\fg}$
 objects: finitely generated abelian groups
 morphisms: group homomorphisms
-description: null
+description: This is the full subcategory of $\Ab$ that consists of the finitely generated abelian groups.
 nlab_link: https://ncatlab.org/nlab/show/finitely+generated+module
 
 tags:

database/data/categories/Alg(R).yaml

@@ -3,7 +3,7 @@ name: category of algebras
 notation: $\Alg(R)$
 objects: algebras over a commutative ring $R \neq 0$
 morphisms: maps preserving the ring and module structure
-description: This is a generalization of the category of rings, which we get for $R = \IZ$. We assume our rings (and algebras) to be unital. For $R = 0$ we would get the trivial category, which is why we exclude this here.
+description: This category is a generalization of the category of rings, which we get for $R = \IZ$. We assume our rings (and algebras) to be unital. For $R = 0$ we would get the trivial category, which is why we exclude this here.
 nlab_link: https://ncatlab.org/nlab/show/Alg
 
 tags:
@@ -12,7 +12,6 @@ tags:
 related:
   - CAlg(R)
   - R-Mod
-  - Ring
 
 satisfied_properties:
   - property: locally small
@@ -25,9 +24,7 @@ satisfied_properties:
     proof: 'If $f : 0 \to A$ is an algebra homomorphism, then $A$ satisfies $1=f(1)=f(0)=0$, so that $A=0$.'
 
   - property: disjoint finite products
-    proof: One can take the same proof as for <a href="/category/Ring">$\Ring$</a>.
-    references:
-      - ring_disjoint_finite_products
+    proof: 'Let $A,B$ be two $R$-algebras. To show that $A \sqcup_{A \times B} B$ is trivial, let $T$ be an $R$-algebra which admits homomorphisms $f : A \to T$, $g : B \to T$ with $f(p_1(a,b))=g(p_2(a,b))$ for all $(a,b) \in A \times B$, i.e. $f(a)=g(b)$. Applying this to $a=1$, $b=0$ yields $1=0$ in $T$. Hence, $T = 0$.'
 
   - property: Malcev
     proof: This follows in the same way as for <a href="/category/Grp">$\Grp$</a>, see also Example 2.2.5 in <a href="https://ncatlab.org/nlab/show/Malcev,+protomodular,+homological+and+semi-abelian+categories" target="_blank">Malcev, protomodular, homological and semi-abelian categories</a>.
@@ -54,9 +51,8 @@ unsatisfied_properties:
     proof: 'See <a href="https://mathoverflow.net/questions/509552">MO/509552</a>: Consider the forgetful functor $U : \Alg(R) \to \Set$ and the relation $S \subseteq U^2$ defined by $S(A) \coloneqq \{(a,b) \in U(A)^2 : ab = a^2\}$. Both are representable: $U$ by $R[X]$ and $S$ by $R \langle X,Y \rangle / \langle XY-X^2 \rangle$. It is clear that $S$ is reflexive, but not symmetric.'
 
   - property: coregular
-    proof: 'We just need to tweak the proof for <a href="/category/Ring">$\Ring$</a>. Since $R \neq 0$, there is an infinite field $K$ with a homomorphism $R \to K$. Since $K$ is infinite, we may choose some $\lambda \in K \setminus \{0,1\}$. Let $B \coloneqq M_2(K)$ and $A \coloneqq K \times K$. Then $A \to B$, $(x,y) \mapsto \diag(x,y)$ is a regular monomorphism: A direct calculation shows that a matrix is diagonal iff it commutes with $M \coloneqq \bigl(\begin{smallmatrix} 1 & 0 \\ 0 & \lambda \end{smallmatrix}\bigr)$, so that $A \to B$ is the equalizer of the identity $B \to B$ and the conjugation $B \to B$, $X \mapsto M X M^{-1}$. Consider the homomorphism $A \to K$, $(a,b) \mapsto a$. We claim that $K \to K \sqcup_A B$ is not a monomorphism, because in fact, the pushout $K \sqcup_A B$ is zero: Since $A \to K$ is surjective with kernel $0 \times K$, the pushout is $B/\langle 0 \times K \rangle$, which is $0$ because $B$ is simple (<a href="https://math.stackexchange.com/questions/22629" target="_blank">proof</a>) or via a direct calculation with elementary matrices.'
-    references:
-      - ring_not_coregular
+    proof: 'Since $R \neq 0$, there is an infinite field $K$ with a homomorphism $R \to K$. Since $K$ is infinite, we may choose some $\lambda \in K \setminus \{0,1\}$. Let $B \coloneqq M_2(K)$ and $A \coloneqq K \times K$. Then $A \to B$, $(x,y) \mapsto \diag(x,y)$ is a regular monomorphism: A direct calculation shows that a matrix is diagonal iff it commutes with $M \coloneqq \bigl(\begin{smallmatrix} 1 & 0 \\ 0 & \lambda \end{smallmatrix}\bigr)$, so that $A \to B$ is the equalizer of the identity $B \to B$ and the conjugation $B \to B$, $X \mapsto M X M^{-1}$. Consider the homomorphism $A \to K$, $(a,b) \mapsto a$. We claim that $K \to K \sqcup_A B$ is not a monomorphism, because in fact, the pushout $K \sqcup_A B$ is zero: Since $A \to K$ is surjective with kernel $0 \times K$, the pushout is $B/\langle 0 \times K \rangle$, which is $0$ because $B$ is simple (<a href="https://math.stackexchange.com/questions/22629" target="_blank">proof</a>) or via a direct calculation with elementary matrices.'
+    label: alg_not_coregular
 
   - property: regular quotient object classifier
     proof: We may copy the proof for <a href="/category/CAlg(R)">$\CAlg(R)$</a> (since the proof there did not use that $P$ is commutative). Alternatively, any regular quotient object classifier in $\Alg(R)$ would produce one in the reflective subcategory $\CAlg(R)$ by Lemma 1 <a href="/content/subcategories">here</a> (dualized).
@@ -65,23 +61,21 @@ unsatisfied_properties:
 
   - property: cocartesian cofiltered limits
     proof: >-
-      Consider the ring $A = R[X]$ and the sequence of rings $B_n = R[Y]/(Y^{n+1})$ with projections $B_{n+1} \to B_n$, whose limit is $R[[Y]]$. Every element in the coproduct of rings $R[X] \sqcup R[[Y]]$ has a finite "free product" length. Now consider the elements 
+      Consider the algebra $A = R[X]$ and the sequence of algebras $B_n = R[Y]/(Y^{n+1})$ with projections $B_{n+1} \to B_n$, whose limit is $R[[Y]]$. Every element in the coproduct of algebras $R[X] \sqcup R[[Y]]$ has a finite "free product" length. Now consider the elements 
       $$w_n = (1 + XY) (1+XY^2) \cdots (1+X Y^n) \in A \sqcup B_n.$$
       Because of $w_n \equiv w_{n-1} \bmod Y^n$ these form an element $w \in \lim_n (A \sqcup B_n)$. Expanding $w_n$, the longest term is $XY XY^2 \cdots X Y^n$ of "free product" length $2n$, which is unbounded.
 
   - property: cofiltered-limit-stable epimorphisms
     proof: We already know that <a href="/category/CRing">$\CAlg(R)$</a> does not have this property. Now apply the contrapositive of the dual of Lemma 2 <a href="/content/subcategories">here</a> to the forgetful functor $\CAlg(R) \to \Alg(R)$. It preserves epimorphisms by <a href="https://math.stackexchange.com/questions/5133488" target="_blank">MSE/5133488</a>.
 
   - property: effective cocongruences
-    proof: 'The counterexample is similar to the one for <a href="/category/Ring">$\Ring$</a>: Let $X \coloneqq R[p] / (p^2-p)$ with cocongruence $E \coloneqq R \langle p, q \rangle / (p^2-p, q^2-q, pq-q, qp-p)$.'
-    references:
-      - ring_no_effective_cocongruences
+    proof: '<a href="https://mathoverflow.net/a/510809" target="_blank">MO/510744</a> presents a counterexample for $\Ring$, and it can be easily generalized to $\Alg(R)$: Let $X \coloneqq R[p] / (p^2-p)$ with cocongruence $E \coloneqq R \langle p, q \rangle / (p^2-p, q^2-q, pq-q, qp-p)$.'
 
 special_objects:
   initial object:
     description: $R$
   terminal object:
-    description: trivial algebra
+    description: zero algebra
   coproducts:
     description: see <a href="https://math.stackexchange.com/questions/625874" target="_blank">MSE/625874</a>
   products:

database/data/categories/BG.yaml

@@ -0,0 +1,51 @@
+id: BG
+name: delooping of a group
+notation: $BG$
+objects: a single object $*$
+morphisms: the elements of $G$
+description: Every group $G$ yields a groupoid $BG$ with a single object $*$, morphisms given by the elements of $G$, and composition given by the group operation. We assume that $G$ is non-trivial, since otherwise we get the trivial category.
+nlab_link: https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid
+
+tags:
+  - algebra
+  - category theory
+
+related:
+  - BN
+
+satisfied_properties:
+  - property: small
+    proof: This is trivial.
+
+  - property: groupoid
+    proof: This is trivial.
+
+  - property: core-connected
+    proof: The category has exactly one object.
+
+  - property: skeletal
+    proof: The category has exactly one object.
+
+unsatisfied_properties:
+  - property: thin
+    proof: This is because $G$ is not trivial, so there are at least two morphisms $* \rightrightarrows *$.
+
+undecidable_properties:
+  - property: essentially countable
+    proof: This holds if and only if $G$ is countable.
+
+  - property: countable
+    proof: This holds if and only if $G$ is countable.
+
+  - property: finite
+    proof: This holds if and only if $G$ is finite.
+
+  - property: locally finite
+    proof: This holds if and only if $G$ is finite.
+
+  - property: essentially finite
+    proof: This holds if and only if $G$ is finite.
+
+special_objects: {}
+
+special_morphisms: {}

database/data/categories/BG_c.yaml

@@ -1,39 +1,26 @@
 id: BG_c
 name: delooping of an infinite countable group
 notation: $BG$
-objects: a single object
+objects: a single object $*$
 morphisms: the elements of an infinite countable group $G$
-description: Every group $G$ yields a groupoid $BG$ with a single object $*$, morphisms given by the elements of $G$, and composition given by the group operation. In this example, we consider the case of an infinite countable group $G$ (such as $G = \IZ$).
-nlab_link: https://ncatlab.org/nlab/show/delooping
+description: This is the special case of <a href="/category/BG">$BG$</a> where $G$ is an infinite countable group $G$ (such as $G = \IZ$).
+nlab_link: https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid
+parent: BG
 
 tags:
   - algebra
   - category theory
 
 related:
-  - BG_f
-  - BG_u
   - BN
 
 satisfied_properties:
-  - property: small
-    proof: This is trivial.
-
-  - property: groupoid
-    proof: This is trivial.
-
-  - property: core-connected
-    proof: The category has exactly one object.
-
-  - property: skeletal
-    proof: The category has exactly one object.
-
   - property: countable
-    proof: This is because $G$ is countable.
+    proof: This is because $G$ is countable by assumption.
 
 unsatisfied_properties:
   - property: locally finite
-    proof: This is because we choose $G$ to be infinite.
+    proof: This is because $G$ is infinite by assumption.
 
 special_objects: {}
 

database/data/categories/BG_f.yaml

@@ -1,39 +1,24 @@
 id: BG_f
 name: delooping of a non-trivial finite group
 notation: $BG$
-objects: a single object
+objects: a single object $*$
 morphisms: the elements of a non-trivial finite group $G$
-description: Every group $G$ yields a groupoid $BG$ with a single object $*$, morphisms given by the elements of $G$, and composition given by the group operation. In this example, we consider the case of a non-trivial finite group $G$ (such as $G = C_2$).
-nlab_link: https://ncatlab.org/nlab/show/delooping
+description: This is the special case of <a href="/category/BG">$BG$</a> where $G$ is a non-trivial finite group $G$ (such as $G = C_2$).
+nlab_link: https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid
+parent: BG
 
 tags:
   - algebra
   - category theory
 
 related:
-  - BG_c
-  - BG_u
   - BN
 
 satisfied_properties:
   - property: finite
-    proof: This is trivial.
+    proof: This is because $G$ is finite by assumption.
 
-  - property: small
-    proof: This is trivial.
-
-  - property: groupoid
-    proof: This is trivial.
-
-  - property: core-connected
-    proof: The category has exactly one object.
-
-  - property: skeletal
-    proof: The category has exactly one object.
-
-unsatisfied_properties:
-  - property: trivial
-    proof: This is trivial.
+unsatisfied_properties: []
 
 special_objects: {}
 

database/data/categories/BG_u.yaml

@@ -1,39 +1,27 @@
 id: BG_u
 name: delooping of an infinite uncountable group
 notation: $BG$
-objects: a single object
+objects: a single object $*$
 morphisms: the elements of an infinite uncountable group $G$
-description: Every group $G$ yields a groupoid $BG$ with a single object $*$, morphisms given by the elements of $G$, and composition given by the group operation. In this example, we consider the case of an uncountable group $G$ (such as $G = \IR$).
-nlab_link: https://ncatlab.org/nlab/show/delooping
+description: This is the special case of <a href="/category/BG">$BG$</a> where $G$ is an infinite uncountable group $G$ (such as $G = \IR$).
+nlab_link: https://ncatlab.org/nlab/show/delooping#delooping_of_a_group_to_a_groupoid
+parent: BG
 
 tags:
   - algebra
   - category theory
 
 related:
-  - BG_f
-  - BG_c
   - BN
 
-satisfied_properties:
-  - property: small
-    proof: This is trivial.
-
-  - property: groupoid
-    proof: This is trivial.
-
-  - property: core-connected
-    proof: The category has exactly one object.
-
-  - property: skeletal
-    proof: The category has exactly one object.
+satisfied_properties: []
 
 unsatisfied_properties:
   - property: locally finite
-    proof: This is because we choose $G$ to be infinite.
+    proof: This is because $G$ is infinite by assumption.
 
   - property: essentially countable
-    proof: This is because we choose $G$ to be uncountable.
+    proof: This is because $G$ is uncountable by assumption.
 
 special_objects: {}