Add special objects and morphisms for CompHaus

Author: dschepler

databases/catdat/data/005_special-objects/002_initial_objects.sql

@@ -13,6 +13,7 @@ VALUES
 ('CAlg(R)', '$R$'),
 ('Cat', 'empty category'),
 ('CMon', 'trivial monoid'),
+('CompHaus', 'empty space'),
 ('CRing', 'ring of integers'),
 ('FI', 'empty set'),
 ('FinAb', 'trivial group'),

databases/catdat/data/005_special-objects/003_terminal_objects.sql

@@ -13,6 +13,7 @@ VALUES
 ('CAlg(R)', 'trivial algebra'),
 ('Cat', '<a href="/category/1">trivial category</a>'),
 ('CMon', 'trivial monoid'),
+('CompHaus', 'singleton space'),
 ('CRing', 'zero ring'),
 ('FinAb', 'trivial group'),
 ('FinGrp', 'trivial group'),

databases/catdat/data/005_special-objects/004_coproducts.sql

@@ -15,6 +15,7 @@ VALUES
 ('CAlg(R)', 'tensor products over $R$'),
 ('Cat', 'disjoint unions'),
 ('CMon', 'direct sums'),
+('CompHaus', 'Stone-Čech compactification of the disjoint union with the disjoint union topology (in the finite case, the disjoint union is already compact Hausdorff so Stone-Čech compactification is not necessary)'),
 ('CRing', 'tensor products over $\IZ$'),
 ('FreeAb', 'direct sums'),
 ('Grp', 'free products'),

databases/catdat/data/005_special-objects/005_products.sql

@@ -15,6 +15,7 @@ VALUES
 ('CAlg(R)', 'direct products with pointwise operations'),
 ('Cat', 'direct products with pointwise operations'),
 ('CMon', 'direct products with pointwise operations'),
+('CompHaus', 'direct product with the <a href="https://en.wikipedia.org/wiki/Product_topology" target="_blank">product topology</a> (which is compact by the Tychonoff product theorem)'),
 ('CRing', 'direct products with pointwise operations'),
 ('Grp', 'direct products with pointwise operations'),
 ('Haus', 'direct product with the <a href="https://en.wikipedia.org/wiki/Product_topology" target="_blank">product topology</a>'),

databases/catdat/data/006_special-morphisms/002_isomorphisms.sql

@@ -85,6 +85,11 @@ VALUES
 	'bijective homomorphisms',
 	'This characterization holds in every algebraic category.'
 ),
+(
+	'CompHaus',
+	'homeomorphisms',
+	'This is easy.'
+),
 (
 	'CRing',
 	'bijective ring homomorphisms',

databases/catdat/data/006_special-morphisms/003_monomorphisms.sql

@@ -80,6 +80,11 @@ VALUES
 	'injective homomorphisms',
 	'This holds in every finitary algebraic category: the forgetful functor to $\Set$ is faithful, hence reflects monomorphisms, and it is continuous (even representable), hence preserves monomorphisms.'
 ),
+(
+	'CompHaus',
+	'injective continuous maps (which are automatically closed embeddings)',
+	'For the non-trivial direction, the forgetful functor to $\Set$ is representable (by the terminal object), hence preserves monomorphisms. To prove the parenthetical remark, given an injective continuous function $f : X \to Y$ between compact Hausdorff spaces, the image of $f$ is a closed subset. Also, the induced map from $X$ to $\im(f)$ with the subspace topology is a bijective continuous map between compact Hausdorff spaces, so it is a homeomorphism.'
+),
 (
 	'CRing',
 	'injective ring homomorphisms',

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

@@ -81,6 +81,11 @@ VALUES
 	'a homomorphism of algebras which is an epimorphism of commutative rings',
 	'The forgetful functor $\CAlg(R) \to \Ring$ is faithful and hence reflects epimorphisms, but it also preserves epimorphisms since it preserves pushouts (since $\CAlg(R) \cong R / \Ring$). For epimorphisms of commutative rings see their <a href="/category/CRing">detail page</a>.'
 ),
+(
+	'CompHaus',
+	'surjective continuous maps (which are automatically quotient maps)',
+	'For the non-trivial direction, and for a proof of the parenthetical remark, see the proof above that $\CompHaus$ is epi-regular.'
+),
 (
 	'CRing',
 	'A ring map $f : R \to S$ is an epimorphism iff $S$ equals the <i>dominion</i> of $f(R) \subseteq S$, meaning that for every $s \in S$ there is some matrix factorization $(s) = Y X Z$ with $X \in M_{n \times n}(R)$, $Y \in M_{1 \times n}(S)$, and $Z \in M_{n \times 1}(S)$.',
@@ -378,4 +383,4 @@ SELECT
 	category_id, description, reason, 'epimorphisms'
 FROM epimorphisms;
 
-DROP TABLE epimorphisms;
+DROP TABLE epimorphisms;

databases/catdat/data/006_special-morphisms/005_regular-monomorphisms.sql

@@ -65,6 +65,11 @@ VALUES
     'closed embeddings',
     'The non-trivial direction follows from the <a href="https://math.stackexchange.com/questions/319867" target="_blank">well-known fact</a> that for every closed subspace of a Banach space its quotient space is again a Banach space.'
 ),
+(
+    'CompHaus',
+    'same as monomorphisms',
+    'This is because the category is mono-regular.'
+),
 (
     'Delta',
     'same as monomorphisms',
@@ -317,4 +322,4 @@ SELECT
 	category_id, description, reason, 'regular monomorphisms'
 FROM regular_monomorphisms;
 
-DROP TABLE regular_monomorphisms;
+DROP TABLE regular_monomorphisms;

databases/catdat/data/006_special-morphisms/006_regular-epimorphisms.sql

@@ -74,7 +74,12 @@ VALUES
     'CMon',
     'surjective homomorphisms',
     'This holds in every finitary algebraic category.'
-),    
+),
+(
+    'CompHaus',
+    'same as epimorphisms',
+    'This is because the category is epi-regular.'
+),
 (
     'CRing',
     'surjective homomorphisms',
@@ -327,4 +332,4 @@ SELECT
 	category_id, description, reason, 'regular epimorphisms'
 FROM regular_epimorphisms;
 
-DROP TABLE regular_epimorphisms;
+DROP TABLE regular_epimorphisms;