decide NNO for various categories

Author: ScriptRaccoon

database/data/004_property-assignments/Delta.sql

@@ -127,5 +127,5 @@ VALUES
     'Delta',
     'pushouts',
     FALSE,
-    'Assume that the two inclusions $\{0 < 1\} \leftarrow \{0\} \rightarrow \{0 < 2\}$ have a pushout in $\mathbf{FinOrd} \setminus \{\emptyset\}$. This would be a universal non-empty finite ordered set $X$ with three elements $0,1,2$ satisfying $0 \leq 1$ and $0 \leq 2$. Assume w.l.o.g. $1 \leq 2$ (the case $2 \leq 1$ is similar). The universal property yields an order-preserving map $X \to \{a < b < c\}$ with $0 \mapsto a$, $1 \mapsto c$, $2 \mapsto b$. But then $c \leq b$, which is a contradiction.'
+    'Assume that the two inclusions $\{0 < 1\} \leftarrow \{0\} \rightarrow \{0 < 2\}$ have a pushout in $\mathbf{FinOrd} \setminus \{\varnothing\}$. This would be a universal non-empty finite ordered set $X$ with three elements $0,1,2$ satisfying $0 \leq 1$ and $0 \leq 2$. Assume w.l.o.g. $1 \leq 2$ (the case $2 \leq 1$ is similar). The universal property yields an order-preserving map $X \to \{a < b < c\}$ with $0 \mapsto a$, $1 \mapsto c$, $2 \mapsto b$. But then $c \leq b$, which is a contradiction.'
 );

database/data/004_property-assignments/FinSet.sql

@@ -82,4 +82,12 @@ VALUES
 	'countable',
 	FALSE,
 	'This is trivial.'
+),
+(
+	'FinSet',
+	'natural numbers object',
+	FALSE,
+	'If $(N,z,s)$ is a natural numbers object, then
+	<p>$1 \xrightarrow{z} N \xleftarrow{s} N$</p>
+	is a coproduct cocone by <a href="https://ncatlab.org/nlab/show/Sketches+of+an+Elephant" target="_blank">Johnstone</a>, Part A, Lemma 2.5.5. But there is no finite set $N$ with $N \cong 1 + N$.'
 );

database/data/004_property-assignments/Met.sql

@@ -136,4 +136,12 @@ VALUES
 	'Malcev',
 	FALSE,
 	'Consider the metric subspace $\{(a,b) \in \mathbb{R}^2 : a \leq b\}$ of $\mathbb{R}^2$.'
-);
+),
+(
+	'Met',
+	'natural numbers object',
+	FALSE,
+	'If $(N,z,s)$ is a natural numbers object in $\mathbf{Met}$, then
+	<p>$1 \xrightarrow{z} N \xleftarrow{s} N$</p>
+	is a coproduct cocone by <a href="https://ncatlab.org/nlab/show/Sketches+of+an+Elephant" target="_blank">Johnstone</a>, Part A, Lemma 2.5.5. Since there is a map $1 \to N$, we have $N \neq \varnothing$. However, the coproduct of two non-empty metric spaces does not exist, see <a href="https://math.stackexchange.com/questions/1778408" target="_blank">MSE/1778408</a>.'
+);

database/data/004_property-assignments/Setne.sql

@@ -102,6 +102,12 @@ VALUES
 	TRUE,
 	'Let $D$ be a diagram in $\mathbf{Set}_{\neq \varnothing}$, and let $L$ be a limit of $D$ in $\mathbf{Set}$. If $L$ is non-empty, it gives a limit in $\mathbf{Set}_{\neq \varnothing}$ as well. If $L$ is the empty set, there is no cone over $D$ in $\mathbf{Set}_{\neq \varnothing}$; hence the empty set of cones gives a multi-limit of $D$ in $\mathbf{Set}_{\neq \varnothing}$.'
 ),
+(
+	'Setne',
+	'natural numbers object',
+	TRUE,
+	'Any natural numbers object in $\mathbf{Set}$, such as $(\mathbb{N},0,n \mapsto n+1)$, is clearly also one in $\mathbf{Set}_{\neq \varnothing}$.'
+),
 (
 	'Setne',
 	'sequential limits',

database/data/004_property-assignments/Sp.sql

@@ -70,4 +70,12 @@ VALUES
 	'essentially countable',
 	FALSE,
 	'Any function $f\colon\mathbb{N} \to \mathbb{N}$ can be regarded as a combinatorial species with trivial actions, and distinct functions yield non-isomorphic species.'
-);
+),
+(
+	'Sp',
+	'natural numbers object',
+	FALSE,
+	'If $(N,z,s)$ is a natural numbers object, then
+	<p>$1 \xrightarrow{z} N \xleftarrow{s} N$</p>
+	is a coproduct cocone by <a href="https://ncatlab.org/nlab/show/Sketches+of+an+Elephant" target="_blank">Johnstone</a>, Part A, Lemma 2.5.5. But there is no combinatorial species $N$ with $N \cong 1 + N$, since evaluating this at, say, $\varnothing$, would yield a finite set $N$ with this property.'
+);