Add categories of operator algebras (WIP)

.vscode/settings.json

@@ -17,7 +17,8 @@
 		"Prost",
 		"SetxSet",
 		"hilberts",
-		"maxage"
+		"maxage",
+		"CVNA"
 	],
 	"cSpell.words": [
 		"abelian",
@@ -130,6 +131,7 @@
 		"Kashiwara",
 		"katex",
 		"Kolmogorov",
+		"Kornell",
 		"lextensive",
 		"libsql",
 		"Lindelöf",
@@ -145,6 +147,7 @@
 		"morphism",
 		"morphisms",
 		"naturality",
+		"Neumann",
 		"Niefield",
 		"nilradical",
 		"nlab",
@@ -155,6 +158,7 @@
 		"posets",
 		"preadditive",
 		"precomposed",
+		"predual",
 		"preimage",
 		"preimages",
 		"preordered",
@@ -174,6 +178,8 @@
 		"Rosicky",
 		"saft",
 		"Schapira",
+		"semigroup",
+		"semigroups",
 		"semisimple",
 		"setoid",
 		"Sheafifiable",

database/data/001_categories/002_analysis.sql

@@ -11,14 +11,54 @@ INSERT INTO categories (
 VALUES
 (
 	'Ban',
-	'category of Banach spaces with linear contractions',
+	'category of Banach spaces',
 	'$\mathbf{Ban}$',
 	'Banach spaces over $\mathbb{C}$',
 	'linear contractions, i.e. linear maps of norm $\leq 1$',
 	'The choice of morphisms is similar to that of $\mathbf{Met}$ which yields better categorical properties.',
 	'https://ncatlab.org/nlab/show/Banach+space',
 	NULL
 ),
+(
+	'BanAlg_nu',
+	'category of non-unital Banach algebras',
+	'$\mathbf{BanAlg}_{nu}$',
+	'non-unital Banach algebras over $\mathbb{C}$',
+	'non-unital algebra homomorphisms of norm $\leq 1$',
+	'With "non-unital" we mean "not necessarily unital". This category is the category of semigroup objects in the cocomplete monoidal category $(\mathbf{Ban},\otimes_p)$.',
+	'https://ncatlab.org/nlab/show/Banach+algebra',
+	NULL
+),
+(
+	'BanAlg_u',
+	'category of unital Banach algebras',
+	'$\mathbf{BanAlg}_u$',
+	'unital Banach algebras over $\mathbb{C}$',
+	'unital algebra homomorphisms of norm $\leq 1$',
+	'This category is the category of monoid objects in the cocomplete monoidal category $(\mathbf{Ban},\otimes_p)$.',
+	'https://ncatlab.org/nlab/show/Banach+algebra',
+	NULL
+),
+(
+	'C*Alg_nu',
+	'category of non-unital C<sup>*</sup>-algebras',
+	'$\mathbf{C^*Alg}_{nu}$',
+	'non-unital C<sup>*</sup>-algebras',
+	'non-unital $*$-algebra homomorphisms',
+	'With "non-unital" we mean "not necessarily unital".',
+	'https://ncatlab.org/nlab/show/C-star-algebra',
+	NULL
+),
+(
+	'C*Alg_u',
+	'category of unital C<sup>*</sup>-algebras',
+	'$\mathbf{C^*Alg}_{u}$',
+	'unital C<sup>*</sup>-algebras',
+	'unital $*$-algebra homomorphisms',
+	NULL,
+	'https://ncatlab.org/nlab/show/C-star-algebra',
+	NULL
+),
 (
 	'Meas',
 	'category of measurable spaces',
@@ -58,4 +98,14 @@ VALUES
 	'This category is equivalent to the subcategory of $\mathbf{Top}$ (or $\mathbf{Haus}$) that consists of metrizable topological spaces. Hence, the metrics only play a secondary role here.',
 	'https://ncatlab.org/nlab/show/metrisable+topological+space',
 	NULL
+),
+(
+	'CVNA',
+	'category of commutative von Neumann algebras',
+	'$\mathbf{CVNA}$',
+	'commutative von Neumann algebras, i.e. commutative unital $C^*$-algebras that have a predual',
+	'normal unital $C^*$-morphisms',
+	'Von Neumann algebras are also called $W^*$-algebras. In Kornell''s paper <a href="https://arxiv.org/abs/1202.2994v2" target="_blank">Quantum Collections</a> this category is denoted $\mathbf{cW}^*$.',
+	'https://ncatlab.org/nlab/show/von+Neumann+algebra',
+	NULL
 );

database/data/001_categories/100_related-categories.sql

@@ -11,8 +11,18 @@ VALUES
 ('Alg(R)', 'CAlg(R)'),
 ('Alg(R)', 'R-Mod'),
 ('Alg(R)', 'Ring'),
-('Ban','Met'),
 ('B', 'FI'),
+('Ban','Met'),
+('Ban','CVNA'),
+('Ban','BanAlg_nu'),
+('Ban','BanAlg_u'),
+('BanAlg_u', 'CVNA'),
+('BanAlg_u', 'BanAlg_nu'),
+('BanAlg_u', 'Alg(R)'),
+('BanAlg_u', 'C*Alg_u'),
+('BanAlg_nu', 'BanAlg_u'),
+('BanAlg_nu', 'Rng'),
+('BanAlg_nu', 'C*Alg_nu'),
 ('B', 'FS'),
 ('BG', 'BG_f'),
 ('BG', 'BN'),
@@ -21,6 +31,12 @@ VALUES
 ('BN', 'BG'),
 ('BN', 'BOn'),
 ('BOn', 'BN'),
+('C*Alg_u', 'C*Alg_nu'),
+('C*Alg_u', 'BanAlg_u'),
+('C*Alg_u', 'Alg(R)'),
+('C*Alg_nu', 'C*Alg_u'),
+('C*Alg_nu', 'BanAlg_nu'),
+('C*Alg_nu', 'Rng'),
 ('CAlg(R)', 'Alg(R)'),
 ('CAlg(R)', 'CRing'),
 ('CAlg(R)', 'R-Mod'),
@@ -32,6 +48,8 @@ VALUES
 ('CRing', 'CAlg(R)'),
 ('CRing', 'Ring'),
 ('CRing', 'Rng'),
+('CVNA','BanAlg_u'),
+('CVNA','CAlg(R)'),
 ('FI', 'B'),
 ('FI', 'FS'),
 ('FS', 'B'),

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

@@ -0,0 +1,37 @@
+INSERT INTO category_property_assignments (
+	category_id,
+	property_id,
+	is_satisfied,
+	reason
+)
+VALUES
+(
+	'BanAlg_nu',
+	'complete',
+	TRUE,
+	'This is because $\mathbf{Ban}$ is complete and then the algebra operations are defined point-wise.'
+),
+(
+	'BanAlg_nu',
+	'cocomplete',
+	TRUE,
+	'If $(\mathcal{C},\otimes)$ is a cocomplete monoidal category (which includes the requirement that $\otimes$ is cocontinuous in each variable), then its category of semigroup objects is cocomplete: this is a straight forward generalization of the construction of colimits of classical semigroups.'
+),
+(
+	'BanAlg_nu',
+	'zero morphisms',
+	TRUE,
+	'The zero maps provide zero morphisms because we do not assume units here.'
+),
+(
+	'BanAlg_nu',
+	'skeletal',
+	FALSE,
+	'This is trivial.'
+),
+(
+	'BanAlg_nu',
+	'balanced',
+	FALSE,
+	'We can take the same counterexample as in the <a href="/category/BanAlg_u">unital case</a>.'
+);

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

@@ -0,0 +1,37 @@
+INSERT INTO category_property_assignments (
+	category_id,
+	property_id,
+	is_satisfied,
+	reason
+)
+VALUES
+(
+	'BanAlg_u',
+	'complete',
+	TRUE,
+	'This is because $\mathbf{Ban}$ is complete and then the algebra operations are defined point-wise.'
+),
+(
+	'BanAlg_u',
+	'cocomplete',
+	TRUE,
+	'If $(\mathcal{C},\otimes)$ is a cocomplete monoidal category (which includes the requirement that $\otimes$ is cocontinuous in each variable), then its category of monoid objects is cocomplete: this is a straight forward generalization of the construction of colimits of classical monoids.'
+),
+(
+	'BanAlg_u',
+	'semi-strongly connected',
+	FALSE,
+	'There is no unital $\mathbb{C}$-algebra homomorphism between $M_2(\mathbb{C})$ and $M_3(\mathbb{C})$.'
+),
+(
+	'BanAlg_u',
+	'skeletal',
+	FALSE,
+	'This is trivial.'
+),
+(
+	'BanAlg_u',
+	'balanced',
+	FALSE,
+	'The inclusion $\alpha : C^1([0,1]) \hookrightarrow C([0,1])$ provides a counterexample, where $C^1([0,1])$ is equipped with the norm $||f||_1 := ||f|| + ||f''||$. Since $\alpha$ is injective, it is a monomorphism. It is also an epimorphism since it has dense image by the Weierstrass approximation theorem. But of course $\alpha$ is no isomorphism.'
+);

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

@@ -0,0 +1,37 @@
+INSERT INTO category_property_assignments (
+	category_id,
+	property_id,
+	is_satisfied,
+	reason
+)
+VALUES
+(
+	'CVNA',
+	'complete',
+	TRUE,
+	'See Prop. 5.3 and Prop. 6.6 in <a href="https://arxiv.org/pdf/1202.2994v2" target="_blank">Quantum Collections</a>.'
+),
+(
+	'CVNA',
+	'coproducts',
+	TRUE,
+	'See Prop. 6.1 in <a href="https://arxiv.org/pdf/1202.2994v2" target="_blank">Quantum Collections</a>.'
+),
+(
+	'CVNA',
+	'coequalizers',
+	TRUE,
+	'See Prop. 5.7 in <a href="https://arxiv.org/pdf/1202.2994v2" target="_blank">Quantum Collections</a>. Commutativity is preserved.'
+), 
+(
+	'CVNA',
+	'pointed',
+	FALSE,
+	'The terminal object $0$ is not the initial object $\mathbb{C}$.'
+),
+(
+	'CVNA',
+	'cocartesian coclosed',
+	FALSE,
+	'See Prop. 6.7 in <a href="https://arxiv.org/pdf/1202.2994v2" target="_blank">Quantum Collections</a>.'
+);

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

@@ -0,0 +1,20 @@
+INSERT INTO category_property_assignments (
+	category_id,
+	property_id,
+	is_satisfied,
+	reason
+)
+VALUES
+-- TODO: add properties here
+(
+	'C*Alg_nu',
+	'inhabited',
+	TRUE,
+	'This is clear.'
+),
+(
+	'C*Alg_nu',
+	'skeletal',
+	FALSE,
+	'This is trivial.'
+);

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

@@ -0,0 +1,20 @@
+INSERT INTO category_property_assignments (
+	category_id,
+	property_id,
+	is_satisfied,
+	reason
+)
+VALUES
+-- TODO: add properties here
+(
+	'C*Alg_u',
+	'inhabited',
+	TRUE,
+	'This is clear.'
+),
+(
+	'C*Alg_u',
+	'skeletal',
+	FALSE,
+	'This is trivial.'
+);

database/data/006_special-objects/002_initial_objects.sql

@@ -10,10 +10,13 @@ VALUES
 ('Ab_fg', 'trivial group'),
 ('Alg(R)', '$R$'),
 ('Ban', 'trivial Banach space'),
+('BanAlg_nu', 'zero algebra'),
+('BanAlg_u', '$\mathbb{C}$'),
 ('CAlg(R)', '$R$'),
 ('Cat', 'empty category'),
 ('CMon', 'trivial monoid'),
 ('CRing', 'ring of integers'),
+('CVNA', '$\mathbb{C}$'),
 ('FI', 'empty set'),
 ('FinAb', 'trivial group'),
 ('FinGrp', 'trivial group'),

database/data/006_special-objects/003_terminal_objects.sql

@@ -10,10 +10,13 @@ VALUES
 ('Ab_fg', 'trivial group'),
 ('Alg(R)', 'trivial algebra'),
 ('Ban', 'trivial Banach space'),
+('BanAlg_nu', 'zero algebra'),
+('BanAlg_u', 'zero algebra'),
 ('CAlg(R)', 'trivial algebra'),
 ('Cat', '<a href="/category/1">trivial category</a>'),
 ('CMon', 'trivial monoid'),
 ('CRing', 'zero ring'),
+('CVNA', 'zero algebra'),
 ('FinAb', 'trivial group'),
 ('FinGrp', 'trivial group'),
 ('FinOrd', 'singleton ordered set'),

database/data/006_special-objects/004_coproducts.sql

@@ -12,6 +12,8 @@ VALUES
 ('Ab', 'direct sums'),
 ('Alg(R)', 'see <a href="https://math.stackexchange.com/questions/625874" target="_blank">MSE/625874</a>'),
 ('Ban', 'cocompletion of the direct sum with the $1$-norm'),
+('BanAlg_u', 'The coproduct of $A,B$ is the cocompletion of the normed algebra $A \oplus B \oplus (A \otimes B) \oplus (B \otimes A) \oplus \cdots$ modulo the closed ideal generated by $1_A - 1_B$. The general case is similar.'),
+('BanAlg_nu', 'The coproduct of $A,B$ is the cocompletion of the normed algebra $A \oplus B \oplus (A \otimes B) \oplus (B \otimes A) \oplus \cdots$. The general case is similar.'),
 ('CAlg(R)', 'tensor products over $R$'),
 ('Cat', 'disjoint unions'),
 ('CMon', 'direct sums'),

database/data/006_special-objects/005_products.sql

@@ -12,6 +12,8 @@ VALUES
 ('Ab', 'direct products with pointwise operations'),
 ('Alg(R)', 'direct products with pointwise operations'),
 ('Ban', 'direct products with the $\sup$-norm'),
+('BanAlg_nu', 'direct products with the $\sup$-norm'),
+('BanAlg_u', 'direct products with the $\sup$-norm'),
 ('CAlg(R)', 'direct products with pointwise operations'),
 ('Cat', 'direct products with pointwise operations'),
 ('CMon', 'direct products with pointwise operations'),

database/data/007_special-morphisms/002_isomorphisms.sql

@@ -50,6 +50,16 @@ VALUES
 	'bijective linear isometries',
 	'This is easy.'
 ),
+(
+	'BanAlg_nu',
+	'isometric bijective algebra homomorphisms',
+	'This is easy.'
+),
+(
+	'BanAlg_u',
+	'isometric bijective algebra homomorphisms',
+	'This is easy.'
+),
 (
 	'BG',
 	'every morphism',
@@ -90,6 +100,11 @@ VALUES
 	'bijective ring homomorphisms',
 	'This characterization holds in every algebraic category.'
 ),
+(
+	'CVNA',
+	'bijective isometric $*$-homomorphisms',
+	'This is trivial.'
+),
 (
 	'FI',
 	'bijective maps',

src/lib/client/utils.ts

@@ -44,3 +44,7 @@ export const resize_textarea: Attachment = (textarea) => {
 		textarea.removeEventListener('input', adjust)
 	}
 }
+
+export function strip_html_tags(input: string): string {
+	return input.replace(/<[^>]*>/g, '')
+}