small fixes

Author: ScriptRaccoon

database/data/003_properties/009_topos-theory.sql

@@ -101,7 +101,7 @@ VALUES
 ),
 (
 	'natural numbers object',
-	'has',
+	'has a',
 	'A <i>natural numbers object</i> (NNO) in a category with finite products is a triple
 	<br>$(N, z : 1 \to N, s : N \to N)$<br>
 	satisfying the following universal property: for all morphisms $f : A \to X$,  $g : X \to X$ there is a unique morphism $\Phi : A \times N \to X$ such that $\Phi(a,z)=f(a)$ and $\Phi(a,s(n)) = g(\Phi(a,n))$ in element notation. Intuitively, this allows recursive definitions of morphisms. This concept is an abstraction of the set of natural numbers, which indeed provide a NNO for the category of sets.',

database/data/005_implications/008_topos-theory-implications.sql

@@ -203,7 +203,7 @@ VALUES
 	FALSE
 ),
 (
-	'nno_additive_case',	
+	'nno_pointed_case',	
 	'["natural numbers object", "pointed"]',
 	'["trivial"]',
 	'Let $(N,z,s)$ be a natural numbers object in a category with a zero object, denoted $0$. The morphism $z : 0 \to N$ must be zero. The universal property applied to $A=1$ implies that $s : N \to N$ is an initial object in the category of endomorphisms. This exists, it is given by the identity $0 \to 0$. Therefore, $N = 0$. The general universal property now becomes: For all $f : A \to X$, $g : X \to X$ there is a unique $\Phi : A \to X$ such that $\Phi(a) = f(a)$ and $\Phi(a)=g(\Phi(a))$. Apply this to $g = 0$ to conclude $f = 0$.',