describe epis for SemiNormVect

Author: ScriptRaccoon

databases/catdat/data/001_categories/003_analysis.sql

@@ -25,7 +25,8 @@ VALUES
 	'$\mathbf{SemiNormVect}$',
 	'semi-normed vector spaces over $\mathbb{C}$',
 	'linear contractions, i.e. linear maps $f$ with $|f(x)| \leq |x|$',
-	'In contrast to a norm, a semi-norm does not necessarily satisfy $|x|=0 \implies x=0$. In particular, every vector space $V$ yields a trivial semi-normed vector space $(V,0)$. The choice of morphisms is similar to that of $\mathbf{PMet}$ which yields better categorical properties than continuous linear maps.',
+	'In contrast to a norm, a semi-norm does not necessarily satisfy $|x|=0 \implies x=0$. In particular, every vector space $V$ yields a trivial semi-normed vector space $(V,0)$; and this construction yields a functor which is right adjoint to the forgetful functor $\mathbf{SemiNormVect} \to \mathbf{Vect}$.
+	<br>The choice of morphisms is similar to that of $\mathbf{PMet}$ which yields better categorical properties than continuous linear maps.',
 	NULL,
 	NULL
 ),

databases/catdat/data/006_special-morphisms/004_epimorphisms.sql

@@ -226,6 +226,11 @@ VALUES
 	'A relation $R : A \to B$ is an epimorphism iff the map $R^* : P(B) \to P(A)$ defined by $S \mapsto \{a \in A : \exists \, b \in S: (a,b) \in R \}$ is injective.',
 	'See <a href="https://math.stackexchange.com/questions/350716/" target="_blank">MSE/350716</a>.'
 ),
+(
+	'SemiNormVect',
+	'surjective linear contractions',
+	'For the non-trivial direction, use that the forgetful functor $\mathbf{SemiNormVect} \to \mathbf{Vect}$ has a right adjoint, hence preserves all colimits, and therefore preserves epimorphisms.'
+),
 (
 	'Set_c',
 	'surjective maps',