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a/static/pdf/walking_parallel_pair_sifted_colimit.tex b/static/pdf/walking_parallel_pair_sifted_colimit.tex index 9bfb0fbc..af0feabe 100644 --- a/static/pdf/walking_parallel_pair_sifted_colimit.tex +++ b/static/pdf/walking_parallel_pair_sifted_colimit.tex @@ -100,8 +100,8 @@ \item $D(c)=0$ and there is a morphism from itself sent to $v$ by $D$. \end{enumerate} -Now, we have a cocone $(\alpha_c\colon D(c) \to 1)_{c\in\mathcal{C}}$ over $D$ by letting $\alpha_c\coloneqq \mathrm{id}_1$ if $c$ is classified into the first case, $\alpha_c\coloneqq u$ for the second case, and $\alpha_c\coloneqq v$ for the third case. -Moreover, this is a unique cocone over $D$: +Now, we have a cocone $(\alpha_c\colon D(c) \to 1)_{c\in\mathcal{C}}$ under $D$ by letting $\alpha_c\coloneqq \mathrm{id}_1$ if $c$ is classified into the first case, $\alpha_c\coloneqq u$ for the second case, and $\alpha_c\coloneqq v$ for the third case. +Moreover, this is a unique cocone under $D$: If $\beta$ is another cocone, its vertex should be $1$ by the existence of $c_0$. If $c\in\mathcal{C}$ is classified into the first case, $\beta_c$ should be the identity. For the second case, taking a morphism $f\colon c\to x$ such that $D(f)=u$, we can obtain $\beta_c = \beta_x \circ D(f) = D(f) = u$. From 41e34ca458e015334272ac0493d474aead4fbf12 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:41:53 +0200 Subject: [PATCH 02/13] add property countable, remove invalid implication finite => small --- .../data/001_categories/008_one-object.sql | 8 +++---- .../001_categories/100_related-categories.sql | 8 +++---- database/data/002_tags/002_category-tags.sql | 6 ++--- .../003_properties/004_size-constraints.sql | 16 +++++++++++++ database/data/004_property-assignments/0.sql | 6 +++++ database/data/004_property-assignments/1.sql | 6 +++++ database/data/004_property-assignments/2.sql | 6 +++++ .../data/004_property-assignments/Ab_fg.sql | 10 ++++++-- database/data/004_property-assignments/B.sql | 8 ++++++- .../{BG.sql => BG_c.sql} | 24 +++++++++---------- .../data/004_property-assignments/BG_f.sql | 6 +++++ database/data/004_property-assignments/BN.sql | 6 +++++ database/data/004_property-assignments/FI.sql | 8 ++++++- database/data/004_property-assignments/FS.sql | 8 ++++++- .../data/004_property-assignments/FinAb.sql | 8 ++++++- .../data/004_property-assignments/FinGrp.sql | 8 ++++++- .../data/004_property-assignments/FinOrd.sql | 8 ++++++- .../data/004_property-assignments/FinSet.sql | 8 ++++++- .../data/004_property-assignments/Man.sql | 6 +++++ database/data/004_property-assignments/N.sql | 6 +++++ .../data/004_property-assignments/N_oo.sql | 6 +++++ database/data/004_property-assignments/Sp.sql | 6 +++++ .../data/004_property-assignments/Z_div.sql | 6 +++++ .../real_interval.sql | 2 +- .../walking_commutative_square.sql | 6 +++++ .../walking_composable_pair.sql | 6 +++++ .../004_property-assignments/walking_fork.sql | 6 +++++ .../walking_idempotent.sql | 6 +++++ .../walking_isomorphism.sql | 6 +++++ .../walking_morphism.sql | 6 +++++ .../004_property-assignments/walking_pair.sql | 6 +++++ .../004_property-assignments/walking_span.sql | 6 +++++ .../003_size-constraints-implications.sql | 18 ++++++++++++-- .../006_trivial-categories-implications.sql | 7 ++++++ .../002_isomorphisms.sql | 2 +- .../003_monomorphisms.sql | 2 +- .../004_epimorphisms.sql | 2 +- .../005_regular-monomorphisms.sql | 2 +- .../006_regular-epimorphisms.sql | 2 +- 39 files changed, 233 insertions(+), 40 deletions(-) rename database/data/004_property-assignments/{BG.sql => BG_c.sql} (79%) diff --git a/database/data/001_categories/008_one-object.sql b/database/data/001_categories/008_one-object.sql index cfad9bca..74de146d 100644 --- a/database/data/001_categories/008_one-object.sql +++ b/database/data/001_categories/008_one-object.sql @@ -10,12 +10,12 @@ INSERT INTO categories ( ) VALUES ( - 'BG', - 'delooping of an infinite group', + 'BG_c', + 'delooping of an infinite countable group', '$BG$', 'a single object', - 'the elements of an infinite group $G$', - 'Every group $G$ yields a groupoid $BG$ with a single object $*$, morphisms given by the elements of $G$, and composition given by the group operation. In this example, we consider the case of an infinite group $G$.', + 'the elements of an infinite countable group $G$', + 'Every group $G$ yields a groupoid $BG$ with a single object $*$, morphisms given by the elements of $G$, and composition given by the group operation. In this example, we consider the case of an infinite countable group $G$ (such as $G = \mathbb{Z}$).', 'https://ncatlab.org/nlab/show/delooping', NULL ), diff --git a/database/data/001_categories/100_related-categories.sql b/database/data/001_categories/100_related-categories.sql index ff9aec8f..571d5705 100644 --- a/database/data/001_categories/100_related-categories.sql +++ b/database/data/001_categories/100_related-categories.sql @@ -14,11 +14,11 @@ VALUES ('Ban','Met'), ('B', 'FI'), ('B', 'FS'), -('BG', 'BG_f'), -('BG', 'BN'), -('BG_f', 'BG'), +('BG_c', 'BG_f'), +('BG_c', 'BN'), +('BG_f', 'BG_c'), ('BG_f', 'BN'), -('BN', 'BG'), +('BN', 'BG_c'), ('BN', 'BOn'), ('BOn', 'BN'), ('CAlg(R)', 'Alg(R)'), diff --git a/database/data/002_tags/002_category-tags.sql b/database/data/002_tags/002_category-tags.sql index 52c3011b..e71ce633 100644 --- a/database/data/002_tags/002_category-tags.sql +++ b/database/data/002_tags/002_category-tags.sql @@ -13,9 +13,9 @@ VALUES ('B', 'combinatorics'), ('B', 'set theory'), ('Ban', 'analysis'), -('BG', 'single object'), -('BG', 'algebra'), -('BG', 'category theory'), +('BG_c', 'single object'), +('BG_c', 'algebra'), +('BG_c', 'category theory'), ('BG_f', 'single object'), ('BG_f', 'algebra'), ('BG_f', 'finite'), diff --git a/database/data/003_properties/004_size-constraints.sql b/database/data/003_properties/004_size-constraints.sql index 3861f7db..7f64adff 100644 --- a/database/data/003_properties/004_size-constraints.sql +++ b/database/data/003_properties/004_size-constraints.sql @@ -63,6 +63,22 @@ VALUES 'essentially finite', TRUE ), +( + 'countable', + 'is', + 'A category is countable if it has countably many objects and morphisms.', + NULL, + 'countable', + FALSE +), +( + 'essentially countable', + 'is', + 'A category is essentially countable if it is equivalent to a countable category.', + NULL, + 'essentially countable', + TRUE +), ( 'well-powered', 'is', diff --git a/database/data/004_property-assignments/0.sql b/database/data/004_property-assignments/0.sql index a1bf48f3..2112a10a 100644 --- a/database/data/004_property-assignments/0.sql +++ b/database/data/004_property-assignments/0.sql @@ -29,6 +29,12 @@ VALUES TRUE, 'This is trivial.' ), +( + '0', + 'small', + TRUE, + 'This is trivial.' +), ( '0', 'inhabited', diff --git a/database/data/004_property-assignments/1.sql b/database/data/004_property-assignments/1.sql index dede0a97..050857b4 100644 --- a/database/data/004_property-assignments/1.sql +++ b/database/data/004_property-assignments/1.sql @@ -17,6 +17,12 @@ VALUES TRUE, 'This is trivial.' ), +( + '1', + 'small', + TRUE, + 'This is trivial.' +), ( '1', 'discrete', diff --git a/database/data/004_property-assignments/2.sql b/database/data/004_property-assignments/2.sql index d9886e05..84be63f1 100644 --- a/database/data/004_property-assignments/2.sql +++ b/database/data/004_property-assignments/2.sql @@ -17,6 +17,12 @@ VALUES TRUE, 'This is trivial.' ), +( + '2', + 'small', + TRUE, + 'This is trivial.' +), ( '2', 'inhabited', diff --git a/database/data/004_property-assignments/Ab_fg.sql b/database/data/004_property-assignments/Ab_fg.sql index dbbb9040..c0450466 100644 --- a/database/data/004_property-assignments/Ab_fg.sql +++ b/database/data/004_property-assignments/Ab_fg.sql @@ -13,9 +13,9 @@ VALUES ), ( 'Ab_fg', - 'essentially small', + 'essentially countable', TRUE, - 'Every finitely generated abelian group is isomorphic to a group of the form $\mathbb{Z}^n / U$, where $n \in \mathbb{N}$ and $U$ is a subgroup of $\mathbb{Z}^n$. And these constitute a set.' + 'Every finitely generated abelian group is isomorphic to a group of the form $\mathbb{Z}^n / U$, where $n \in \mathbb{N}$ and $U$ is a subgroup of $\mathbb{Z}^n$. Since $\mathbb{Z}^n$ is Noetherian as a $\mathbb{Z}$-module, $U$ is finitely generated, hence the category $\mathbf{Ab}_\mathrm{fg}$ has only countably many objects up to isomorphism. Furthermore, for any objects $A \cong \mathbb{Z}^n / U$ and $B \cong \mathbb{Z}^m / T$, the hom-set $\mathrm{Hom}(A,B)$ is countable. Indeed, precomposition with the quotient map yields an injection $\mathrm{Hom}(A,B) \hookrightarrow \mathrm{Hom}(\mathbb{Z}^n, B) \cong B^n$, and $B^n$ is countable.' ), ( 'Ab_fg', @@ -64,4 +64,10 @@ VALUES 'skeletal', FALSE, 'This is trivial.' +), +( + 'Ab_fg', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/B.sql b/database/data/004_property-assignments/B.sql index 7e98390f..8a94d1b1 100644 --- a/database/data/004_property-assignments/B.sql +++ b/database/data/004_property-assignments/B.sql @@ -13,7 +13,7 @@ VALUES ), ( 'B', - 'essentially small', + 'essentially countable', TRUE, 'Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.' ), @@ -58,4 +58,10 @@ VALUES 'skeletal', FALSE, 'This is trivial.' +), +( + 'B', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/BG.sql b/database/data/004_property-assignments/BG_c.sql similarity index 79% rename from database/data/004_property-assignments/BG.sql rename to database/data/004_property-assignments/BG_c.sql index dedf3ec9..61864e12 100644 --- a/database/data/004_property-assignments/BG.sql +++ b/database/data/004_property-assignments/BG_c.sql @@ -6,43 +6,43 @@ INSERT INTO category_property_assignments ( ) VALUES ( - 'BG', + 'BG_c', + 'countable', + TRUE, + 'This is because $G$ is countable.' +), +( + 'BG_c', 'small', TRUE, 'This is trivial.' ), ( - 'BG', + 'BG_c', 'groupoid', TRUE, 'This is trivial.' ), ( - 'BG', + 'BG_c', 'connected', TRUE, 'This is trivial.' ), ( - 'BG', + 'BG_c', 'generator', TRUE, 'The unique object is a generator for trivial reasons.' ), ( - 'BG', + 'BG_c', 'skeletal', TRUE, 'There is just one object.' ), ( - 'BG', - 'trivial', - FALSE, - 'This is trivial.' -), -( - 'BG', + 'BG_c', 'essentially finite', FALSE, 'This is because we choose $G$ to be infinite.' diff --git a/database/data/004_property-assignments/BG_f.sql b/database/data/004_property-assignments/BG_f.sql index e5cb2916..f2fd24a6 100644 --- a/database/data/004_property-assignments/BG_f.sql +++ b/database/data/004_property-assignments/BG_f.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'BG_f', + 'small', + TRUE, + 'This is trivial.' +), ( 'BG_f', 'groupoid', diff --git a/database/data/004_property-assignments/BN.sql b/database/data/004_property-assignments/BN.sql index 15027efe..1177063a 100644 --- a/database/data/004_property-assignments/BN.sql +++ b/database/data/004_property-assignments/BN.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'BN', + 'countable', + TRUE, + 'This is trivial.' +), ( 'BN', 'strongly connected', diff --git a/database/data/004_property-assignments/FI.sql b/database/data/004_property-assignments/FI.sql index bdcd897c..f79eae38 100644 --- a/database/data/004_property-assignments/FI.sql +++ b/database/data/004_property-assignments/FI.sql @@ -25,7 +25,7 @@ VALUES ), ( 'FI', - 'essentially small', + 'essentially countable', TRUE, 'Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.' ), @@ -118,4 +118,10 @@ VALUES 'strongly connected', FALSE, 'There is no map from a non-empty set to the empty set.' +), +( + 'FI', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/FS.sql b/database/data/004_property-assignments/FS.sql index 633e868b..0564e3be 100644 --- a/database/data/004_property-assignments/FS.sql +++ b/database/data/004_property-assignments/FS.sql @@ -13,7 +13,7 @@ VALUES ), ( 'FS', - 'essentially small', + 'essentially countable', TRUE, 'Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.' ), @@ -116,4 +116,10 @@ VALUES but $(E_1 \vee E_2) \wedge (E_1 \vee E_3) = \top \wedge \top = \top$.

*For thin categories, the properties codistributive and distributive are equivalent.' +), +( + 'FS', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/FinAb.sql b/database/data/004_property-assignments/FinAb.sql index ff7575ef..7f8d58c1 100644 --- a/database/data/004_property-assignments/FinAb.sql +++ b/database/data/004_property-assignments/FinAb.sql @@ -13,7 +13,7 @@ VALUES ), ( 'FinAb', - 'essentially small', + 'essentially countable', TRUE, 'The underlying set of a finite structure can be chosen to be a subset of $\mathbb{N}$.' ), @@ -58,4 +58,10 @@ VALUES 'skeletal', FALSE, 'There are many trivial and hence isomorphic groups which are not equal.' +), +( + 'FinAb', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/FinGrp.sql b/database/data/004_property-assignments/FinGrp.sql index 096cc4aa..babb0a4f 100644 --- a/database/data/004_property-assignments/FinGrp.sql +++ b/database/data/004_property-assignments/FinGrp.sql @@ -19,7 +19,7 @@ VALUES ), ( 'FinGrp', - 'essentially small', + 'essentially countable', TRUE, 'The underlying set of a finite structure can be chosen to be a subset of $\mathbb{N}$.' ), @@ -106,4 +106,10 @@ VALUES 'sequential colimits', FALSE, 'This follows from this lemma.' +), +( + 'FinGrp', + 'countable', + FALSE, + 'This is trivial.' ); \ No newline at end of file diff --git a/database/data/004_property-assignments/FinOrd.sql b/database/data/004_property-assignments/FinOrd.sql index a38c218f..8f145d89 100644 --- a/database/data/004_property-assignments/FinOrd.sql +++ b/database/data/004_property-assignments/FinOrd.sql @@ -61,7 +61,7 @@ VALUES ), ( 'FinOrd', - 'essentially small', + 'essentially countable', TRUE, 'Every finite ordered set is isomorphic to $\{0 < \cdots < n-1 \}$ for some $n \in \mathbb{N}$.' ), @@ -124,4 +124,10 @@ VALUES 'one-way', FALSE, 'There are three different order-preserving maps $\{0 < 1\} \to \{0 < 1\}$.' +), +( + 'FinOrd', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/FinSet.sql b/database/data/004_property-assignments/FinSet.sql index 0e308502..1adf99e2 100644 --- a/database/data/004_property-assignments/FinSet.sql +++ b/database/data/004_property-assignments/FinSet.sql @@ -13,7 +13,7 @@ VALUES ), ( 'FinSet', - 'essentially small', + 'essentially countable', TRUE, 'Every finite set is isomorphic to some $\{1,\dotsc,n\}$ for some $n \in \mathbb{N}$.' ), @@ -70,4 +70,10 @@ VALUES 'Malcev', FALSE, 'There are lots of non-symmetric reflexive relations.' +), +( + 'FinSet', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/Man.sql b/database/data/004_property-assignments/Man.sql index b3fc32c9..304c9f6c 100644 --- a/database/data/004_property-assignments/Man.sql +++ b/database/data/004_property-assignments/Man.sql @@ -119,4 +119,10 @@ VALUES 'sequential colimits', FALSE, 'If $\mathbf{Man}$ had sequential colimits, then by this lemma there would be a manifold $M$ that admits a split epimorphism $M \to \mathbb{R}^n$ for every $n$. But then $M$ will have an infinite-dimensional tangent space, which is a contradiction.' +), +( + 'Man', + 'countable', + FALSE, + 'This is trivial.' ); diff --git a/database/data/004_property-assignments/N.sql b/database/data/004_property-assignments/N.sql index 79a47446..7d367ccb 100644 --- a/database/data/004_property-assignments/N.sql +++ b/database/data/004_property-assignments/N.sql @@ -53,6 +53,12 @@ VALUES TRUE, 'This is because the natural numbers with respect to $<$ are well-founded.' ), +( + 'N', + 'countable', + TRUE, + 'This is trivial.' +), ( 'N', 'countable coproducts', diff --git a/database/data/004_property-assignments/N_oo.sql b/database/data/004_property-assignments/N_oo.sql index c301d62a..beba103d 100644 --- a/database/data/004_property-assignments/N_oo.sql +++ b/database/data/004_property-assignments/N_oo.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'N_oo', + 'countable', + TRUE, + 'This is trivial.' +), ( 'N_oo', 'coproducts', diff --git a/database/data/004_property-assignments/Sp.sql b/database/data/004_property-assignments/Sp.sql index 58181ab1..b75420d1 100644 --- a/database/data/004_property-assignments/Sp.sql +++ b/database/data/004_property-assignments/Sp.sql @@ -64,4 +64,10 @@ VALUES 'generator', FALSE, 'Assume that a generator $G$ exists. For $n \geq 0$ let $F_n$ be the combinatorial species of sets of cardinality $\neq n$: $F_n(A) = \varnothing$ when $A$ has cardinality $n$, otherwise $F_n(A) = \{A\}$. There are two different morphisms $F_n \rightrightarrows F_n \sqcup F_n$. Hence, there must be at least one morphism $G \to F_n$. If $A$ has cardinality $n$, this implies $G(A) = \varnothing$. Since this holds for all $n$, $G$ is the initial object. But this is clearly no generator (it would mean that the category is thin).' +), +( + 'Sp', + '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.' ); \ No newline at end of file diff --git a/database/data/004_property-assignments/Z_div.sql b/database/data/004_property-assignments/Z_div.sql index d18ec05b..4f38b97f 100644 --- a/database/data/004_property-assignments/Z_div.sql +++ b/database/data/004_property-assignments/Z_div.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'Z_div', + 'countable', + TRUE, + 'This is trivial.' +), ( 'Z_div', 'products', diff --git a/database/data/004_property-assignments/real_interval.sql b/database/data/004_property-assignments/real_interval.sql index 9f66736c..85cd3ccb 100644 --- a/database/data/004_property-assignments/real_interval.sql +++ b/database/data/004_property-assignments/real_interval.sql @@ -37,7 +37,7 @@ VALUES ), ( 'real_interval', - 'essentially finite', + 'essentially countable', FALSE, 'This is trivial.' ), diff --git a/database/data/004_property-assignments/walking_commutative_square.sql b/database/data/004_property-assignments/walking_commutative_square.sql index 86551e68..6f3f356a 100644 --- a/database/data/004_property-assignments/walking_commutative_square.sql +++ b/database/data/004_property-assignments/walking_commutative_square.sql @@ -17,6 +17,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_commutative_square', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_commutative_square', 'infinitary distributive', diff --git a/database/data/004_property-assignments/walking_composable_pair.sql b/database/data/004_property-assignments/walking_composable_pair.sql index bbc83274..d0476dba 100644 --- a/database/data/004_property-assignments/walking_composable_pair.sql +++ b/database/data/004_property-assignments/walking_composable_pair.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_composable_pair', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_composable_pair', 'semi-strongly connected', diff --git a/database/data/004_property-assignments/walking_fork.sql b/database/data/004_property-assignments/walking_fork.sql index 970f6844..9cb538b1 100644 --- a/database/data/004_property-assignments/walking_fork.sql +++ b/database/data/004_property-assignments/walking_fork.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_fork', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_fork', 'semi-strongly connected', diff --git a/database/data/004_property-assignments/walking_idempotent.sql b/database/data/004_property-assignments/walking_idempotent.sql index 9b13b5cf..37dcde65 100644 --- a/database/data/004_property-assignments/walking_idempotent.sql +++ b/database/data/004_property-assignments/walking_idempotent.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_idempotent', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_idempotent', 'generator', diff --git a/database/data/004_property-assignments/walking_isomorphism.sql b/database/data/004_property-assignments/walking_isomorphism.sql index 8bfc2e45..e11c8406 100644 --- a/database/data/004_property-assignments/walking_isomorphism.sql +++ b/database/data/004_property-assignments/walking_isomorphism.sql @@ -17,6 +17,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_isomorphism', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_isomorphism', 'skeletal', diff --git a/database/data/004_property-assignments/walking_morphism.sql b/database/data/004_property-assignments/walking_morphism.sql index 73c9607b..89328cd8 100644 --- a/database/data/004_property-assignments/walking_morphism.sql +++ b/database/data/004_property-assignments/walking_morphism.sql @@ -23,6 +23,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_morphism', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_morphism', 'infinitary distributive', diff --git a/database/data/004_property-assignments/walking_pair.sql b/database/data/004_property-assignments/walking_pair.sql index e1c0a20b..08e119aa 100644 --- a/database/data/004_property-assignments/walking_pair.sql +++ b/database/data/004_property-assignments/walking_pair.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_pair', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_pair', 'self-dual', diff --git a/database/data/004_property-assignments/walking_span.sql b/database/data/004_property-assignments/walking_span.sql index 7583e21c..c089d604 100644 --- a/database/data/004_property-assignments/walking_span.sql +++ b/database/data/004_property-assignments/walking_span.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'This is trivial.' ), +( + 'walking_span', + 'small', + TRUE, + 'This is trivial.' +), ( 'walking_span', 'skeletal', diff --git a/database/data/005_implications/003_size-constraints-implications.sql b/database/data/005_implications/003_size-constraints-implications.sql index 5b4ee2cf..2e209325 100644 --- a/database/data/005_implications/003_size-constraints-implications.sql +++ b/database/data/005_implications/003_size-constraints-implications.sql @@ -23,13 +23,27 @@ VALUES ( 'finite_consequence', '["finite"]', - '["small", "essentially finite"]', + '["countable", "essentially finite"]', 'This is trivial.', FALSE ), ( - 'essentially_finite_consequence', + 'essentially_finite_raise', '["essentially finite"]', + '["essentially countable"]', + 'This is trivial.', + FALSE +), +( + 'countable_consequence', + '["countable"]', + '["essentially countable"]', + 'This is trivial.', + FALSE +), +( + 'essentially_countable_consequence', + '["essentially countable"]', '["essentially small"]', 'This is trivial.', FALSE diff --git a/database/data/005_implications/006_trivial-categories-implications.sql b/database/data/005_implications/006_trivial-categories-implications.sql index fea25223..71222ad6 100644 --- a/database/data/005_implications/006_trivial-categories-implications.sql +++ b/database/data/005_implications/006_trivial-categories-implications.sql @@ -62,6 +62,13 @@ VALUES 'See Mac Lane, V.2, Prop. 3. The proof works for any category with products.', FALSE ), +( + 'freyd_countable', + '["essentially countable", "countable products"]', + '["thin", "products"]', + 'To see that the category is thin, use Mac Lane, V.2, Prop. 3. The proof can easily be adapted to this case. So the category is equivalent to a countable preordered set. But then products are just infima, so that repetitions of objects do not matter, and every product can be reduced to a countable one.', + FALSE +), ( 'freyd_finite', '["essentially finite", "finite products"]', diff --git a/database/data/007_special-morphisms/002_isomorphisms.sql b/database/data/007_special-morphisms/002_isomorphisms.sql index 0e5baf81..2824f322 100644 --- a/database/data/007_special-morphisms/002_isomorphisms.sql +++ b/database/data/007_special-morphisms/002_isomorphisms.sql @@ -51,7 +51,7 @@ VALUES 'This is easy.' ), ( - 'BG', + 'BG_c', 'every morphism', 'It is a groupoid.' ), diff --git a/database/data/007_special-morphisms/003_monomorphisms.sql b/database/data/007_special-morphisms/003_monomorphisms.sql index 465596e7..6e6a0586 100644 --- a/database/data/007_special-morphisms/003_monomorphisms.sql +++ b/database/data/007_special-morphisms/003_monomorphisms.sql @@ -51,7 +51,7 @@ VALUES 'The unit ball functor $U : \mathbf{Ban} \to \mathbf{Set}$ is faithful and representable (by $\mathbb{C}$), hence reflects and preserves monomorphisms.' ), ( - 'BG', + 'BG_c', 'every morphism', 'This is trivial.' ), diff --git a/database/data/007_special-morphisms/004_epimorphisms.sql b/database/data/007_special-morphisms/004_epimorphisms.sql index 24a0714d..fe1b3d5b 100644 --- a/database/data/007_special-morphisms/004_epimorphisms.sql +++ b/database/data/007_special-morphisms/004_epimorphisms.sql @@ -46,7 +46,7 @@ VALUES 'Let $f : X \to Y$ be an epimorphism of Banach spaces. The subspace $U := \overline{f(X)} \subseteq Y$ is closed. It is well-known that the quotient $Y/U$ is also a Banach space with a projection $p : Y \to Y/U$. Since $p \circ f = 0 = 0 \circ f$, we infer $p = 0$, so that $U = Y$.' ), ( - 'BG', + 'BG_c', 'every morphism', 'This holds because it is a groupoid.' ), diff --git a/database/data/007_special-morphisms/005_regular-monomorphisms.sql b/database/data/007_special-morphisms/005_regular-monomorphisms.sql index a8e93cf0..b5efdc48 100644 --- a/database/data/007_special-morphisms/005_regular-monomorphisms.sql +++ b/database/data/007_special-morphisms/005_regular-monomorphisms.sql @@ -41,7 +41,7 @@ VALUES 'This is because the category is mono-regular.' ), ( - 'BG', + 'BG_c', 'same as monomorphisms', 'This is because the category is mono-regular.' ), diff --git a/database/data/007_special-morphisms/006_regular-epimorphisms.sql b/database/data/007_special-morphisms/006_regular-epimorphisms.sql index 5e7ebdfa..08dfacb7 100644 --- a/database/data/007_special-morphisms/006_regular-epimorphisms.sql +++ b/database/data/007_special-morphisms/006_regular-epimorphisms.sql @@ -46,7 +46,7 @@ VALUES 'This is because the category is epi-regular.' ), ( - 'BG', + 'BG_c', 'same as isomorphisms', 'This is because the category is left cancellative.' ), From 094198466602ef1ee2617fac13c4d63dfd61a4f6 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:42:14 +0200 Subject: [PATCH 03/13] add multi-complete and variants --- .../002_limits-colimits-existence.sql | 32 +++++++++++++++++++ 1 file changed, 32 insertions(+) diff --git a/database/data/003_properties/002_limits-colimits-existence.sql b/database/data/003_properties/002_limits-colimits-existence.sql index 9c89b5fa..c4ae9af0 100644 --- a/database/data/003_properties/002_limits-colimits-existence.sql +++ b/database/data/003_properties/002_limits-colimits-existence.sql @@ -342,4 +342,36 @@ VALUES 'https://ncatlab.org/nlab/show/sifted+colimit', 'cosifted limits', TRUE +), +( + 'multi-complete', + 'is', + 'A multi-limit of a diagram $D\colon \mathcal{S} \to \mathcal{C}$ is a set $I$ of cones over $D$ such that every cone over $D$ uniquely factors through a unique cone belonging to $I$. This property refers to the existence of multi-limits of small diagrams. Note that any diagram with no cone admits a multi-limit, which is the empty set of cones.', + 'https://ncatlab.org/nlab/show/multilimit', + 'multi-cocomplete', + TRUE +), +( + 'multi-cocomplete', + 'is', + 'A multi-colimit of a diagram $D\colon \mathcal{S} \to \mathcal{C}$ is a set $I$ of cocones under $D$ such that every cocone under $D$ uniquely factors through a unique cocone belonging to $I$. This property refers to the existence of multi-colimits of small diagrams. Note that any diagram with no cocone admits a multi-colimit, which is the empty set of cocones.', + 'https://ncatlab.org/nlab/show/multilimit', + 'multi-complete', + TRUE +), +( + 'multi-terminal object', + 'has a', + 'This property refers to the existence of a multi-limit of the empty diagram. A category has a multi-terminal object if and only if each connected component has a terminal object.', + 'https://ncatlab.org/nlab/show/multilimit', + 'multi-initial object', + TRUE +), +( + 'multi-initial object', + 'has a', + 'This property refers to the existence of a multi-colimit of the empty diagram. A category has a multi-initial object if and only if each connected component has a initial object.', + 'https://ncatlab.org/nlab/show/multilimit', + 'multi-terminal object', + TRUE ); \ No newline at end of file From c535c50e250bf114f1d1db7bd8a7051ab0dc868e Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:43:08 +0200 Subject: [PATCH 04/13] add accessible and variants --- .../008_locally-presentable.sql | 125 +++++++++++++++++- 1 file changed, 121 insertions(+), 4 deletions(-) diff --git a/database/data/003_properties/008_locally-presentable.sql b/database/data/003_properties/008_locally-presentable.sql index 17e6ee20..99f4a353 100644 --- a/database/data/003_properties/008_locally-presentable.sql +++ b/database/data/003_properties/008_locally-presentable.sql @@ -10,7 +10,14 @@ VALUES ( 'locally finitely presentable', 'is', - 'A category is locally finitely presentable if it is cocomplete and there is a set $S$ of finitely presentable objects such that every object is a filtered colimit of objects in $S$. This is the same as being locally $\aleph_0$-presentable.', + 'A category is locally finitely presentable if it satisfies one of the following equivalent conditions: +

    +
  1. It is finitely accessible and cocomplete.
    2. +
  2. It is finitely accessible and complete.
    3. +
  3. It is equivalent to the category of finite-limit-preserving functors to $\mathbf{Set}$ from a small category with finite limits.
    4. +
  4. It is equivalent to the category of models of a small finite-limit sketch.
    5. +
+ For equivalence of conditions above, see Cor. 2.47, Thm. 1.46, and Cor. 1.52 in Adamek-Rosicky. This is the same as being locally $\aleph_0$-presentable.', 'https://ncatlab.org/nlab/show/locally+finitely+presentable+category', NULL, TRUE @@ -18,15 +25,31 @@ VALUES ( 'locally presentable', 'is', - 'Let $\kappa$ be a regular cardinal. A category is locally $\kappa$-presentable if it is cocomplete and there is a set of $\kappa$-presentable objects $S$ such that every object is a $\kappa$-filtered colimit of objects in $S$. A category is locally presentable if it is locally $\kappa$-presentable for some regular cardinal $\kappa$.', + 'Let $\kappa$ be a regular cardinal. A category is locally $\kappa$-presentable if it satisfies one of the following equivalent conditions: +
    +
  1. It is $\kappa$-accessible and cocomplete.
    2. +
  2. It is $\kappa$-accessible and complete.
    3. +
  3. It is equivalent to the category of $\kappa$-limit-preserving functors to $\mathbf{Set}$ from a small category with $\kappa$-limits.
    4. +
  4. It is equivalent to the category of models of a small $\kappa$-limit sketch.
    5. +
+ For equivalence of conditions above, see Cor. 2.47, Thm. 1.46, and Cor. 1.52 in Adamek-Rosicky. + A category is locally presentable if it is locally $\kappa$-presentable for some regular cardinal $\kappa$.', 'https://ncatlab.org/nlab/show/locally+presentable+category', + 'locally copresentable', + TRUE +), +( + 'locally copresentable', + 'is', + 'A category is locally copresentable if its opposite category is locally presentable.', NULL, + 'locally presentable', TRUE ), ( 'locally ℵ₁-presentable', 'is', - 'This is the special case of the notion of a locally $\kappa$-presentable, where $\kappa = \aleph_1$ is the first uncountable cardinal.', + 'This is the special case of the notion of locally $\kappa$-presentable categories, where $\kappa = \aleph_1$ is the first uncountable cardinal.', 'https://ncatlab.org/nlab/show/locally+presentable+category', NULL, TRUE @@ -34,11 +57,12 @@ VALUES ( 'locally strongly finitely presentable', 'is', - 'A category is a locally strongly finitely presentable if it is cocomplete and there is a set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$. + 'A category is locally strongly finitely presentable if it is cocomplete and there is a set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$. There are several equivalent conditions:
  1. It is equivalent to the category of models of a many-sorted finitary algebraic theory.
  2. It is equivalent to the category of finite-product-preserving functors to $\mathbf{Set}$ from a small category with finite products (=Lawvere theory).
    3. +
  3. It is equivalent to the category of models of a small finite-product sketch.
  4. It is equivalent to the Eilenberg–Moore category of a finitary (=filtered-colimit-preserving) monad on $\mathbf{Set}^S$ for some set $S$.
  5. It is equivalent to the Eilenberg–Moore category of a sifted-colimit-preserving monad on $\mathbf{Set}^S$ for some set $S$. (cf. [KR12, Proposition 3.3])
@@ -46,4 +70,97 @@ VALUES 'https://ncatlab.org/nlab/show/locally+strongly+finitely+presentable+category', NULL, TRUE +), +( + 'accessible', + 'is', + 'Let $\kappa$ be a regular cardinal. A category is $\kappa$-accessible if it has $\kappa$-filtered colimits and there is a (small) set $G$ of $\kappa$-presentable objects such that every object is a $\kappa$-filtered colimit of objects in $G$. A category is accessible if it is $\kappa$-accessible for some regular cardinal $\kappa$.', + 'https://ncatlab.org/nlab/show/accessible+category', + 'coaccessible', + TRUE +), +( + 'coaccessible', + 'is', + 'A category is coaccessible if its opposite category is accessible.', + NULL, + 'accessible', + TRUE +), +( + 'finitely accessible', + 'is', + 'A category is finitely accessible if it has filtered colimits and there is a (small) set $G$ of finitely presentable objects such that every object is a filtered colimit of objects in $S$.', + 'https://ncatlab.org/nlab/show/accessible+category', + NULL, + TRUE +), +( + 'ℵ₁-accessible', + 'is', + 'This is the special case of the notion of $\kappa$-accessible categories, where $\kappa = \aleph_1$ is the first uncountable cardinal.', + 'https://ncatlab.org/nlab/show/accessible+category', + NULL, + TRUE +), +( + 'generalized variety', + 'is a', + 'A category is a generalized variety if it has sifted colimits and there is a (small) set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$. Generalized varieties are like locally strongly finitely presentable categories but without colimits. The relation is similar as between finitely accessible and locally finitely presentable categories. This notion is defined in [AR01, Def. 3.6].', + NULL, + NULL, + TRUE +), +( + 'multi-algebraic', + 'is', + 'A category is multi-algebraic if it satisfies one of the following equivalent conditions: +
    +
  1. It is a multi-cocomplete generalized variety, that is, it has multi-colimits and sifted colimits of all small diagrams, and there is a (small) set $G$ of strongly finitely presentable objects such that every object is a sifted colimit of objects from $G$.
    2. +
  2. It is equivalent to the category of models of a small (finite product, coproduct)-sketch, shortly small FPC-sketch.
    3. +
  3. It is equivalent to the category of multi-finite-product-preserving functors to $\mathbf{Set}$ from a small category with multi-finite-products (multi-algebraic theory). Here, multi-finite-products means multi-limits of finite discrete diagrams.
    4. +
  4. It is equivalent to the category of models of a small multi-finite-product sketch.
    5. +
+ Multi-algebraic categories are like locally strongly finitely presentable categories but only with multi-colimits. The relation is similar as between locally finitely multi-presentable and locally finitely presentable categories. + For equivalence of conditions above, see [AR01a, Lem. 1] and [AR01b, Thm. 4.4]. + This notion was originally introduced by Diers.', + NULL, + NULL, + TRUE +), +( + 'locally multi-presentable', + 'is', + 'Let $\kappa$ be a regular cardinal. A category is locally $\kappa$-multi-presentable if it satisfies one of the following equivalent conditions: +
    +
  1. It is $\kappa$-accessible and multi-cocomplete.
    2. +
  2. It is $\kappa$-accessible and has connected limits.
    3. +
  3. It is equivalent to the category of models of a small ($\kappa$-limit, coproduct)-sketch.
    4. +
+ For equivalence of conditions above, see Thm. 4.30, Thm. 4.32, and the remark below in Adamek-Rosicky. A category is locally multi-presentable if it is locally $\kappa$-multi-presentable for some $\kappa$.', + 'https://ncatlab.org/nlab/show/locally+multipresentable+category', + NULL, + TRUE +), +( + 'locally finitely multi-presentable', + 'is', + 'A category is locally finitely multi-presentable if it satisfies one of the following equivalent conditions: +
    +
  1. It is finitely accessible and multi-cocomplete.
    2. +
  2. It is finitely accessible and has connected limits.
    3. +
  3. It is equivalent to the category of models of a small (finite limit, coproduct)-sketch.
    4. +
+ For equivalence of conditions above, see Thm. 4.30, Thm. 4.32, and the remark below in Adamek-Rosicky.', + 'https://ncatlab.org/nlab/show/locally+multipresentable+category', + NULL, + TRUE +), +( + 'locally poly-presentable', + 'is', + 'A category is locally poly-presentable if it is accessible and has wide pullbacks.', + 'https://ncatlab.org/nlab/show/locally+polypresentable+category', + NULL, + TRUE ); \ No newline at end of file From 63bf67e4fc3a5e274ef62f2045eafa845329a632 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:43:20 +0200 Subject: [PATCH 05/13] add property relationships --- .../003_properties/100_related-properties.sql | 65 ++++++++++++++++++- 1 file changed, 64 insertions(+), 1 deletion(-) diff --git a/database/data/003_properties/100_related-properties.sql b/database/data/003_properties/100_related-properties.sql index a9d28f8e..5dfc8c8a 100644 --- a/database/data/003_properties/100_related-properties.sql +++ b/database/data/003_properties/100_related-properties.sql @@ -40,22 +40,33 @@ VALUES ('locally finitely presentable', 'locally presentable'), ('locally finitely presentable', 'locally strongly finitely presentable'), ('locally finitely presentable', 'locally ℵ₁-presentable'), +('locally finitely presentable', 'finitely accessible'), +('locally finitely presentable', 'locally finitely multi-presentable'), ('locally strongly finitely presentable', 'locally finitely presentable'), +('locally strongly finitely presentable', 'multi-algebraic'), +('locally strongly finitely presentable', 'generalized variety'), ('locally presentable', 'cocomplete'), ('locally presentable', 'locally finitely presentable'), ('locally presentable', 'locally ℵ₁-presentable'), +('locally presentable', 'accessible'), +('locally presentable', 'locally multi-presentable'), +('locally presentable', 'locally poly-presentable'), ('locally ℵ₁-presentable', 'cocomplete'), ('locally ℵ₁-presentable', 'locally finitely presentable'), ('locally ℵ₁-presentable', 'locally presentable'), +('locally ℵ₁-presentable', 'ℵ₁-accessible'), ('elementary topos', 'Grothendieck topos'), ('elementary topos', 'cartesian closed'), ('elementary topos', 'finitely complete'), ('elementary topos', 'subobject classifier'), ('Grothendieck topos', 'elementary topos'), ('initial object', 'finite coproducts'), +('initial object', 'multi-initial object'), ('terminal object', 'finite products'), +('terminal object', 'multi-terminal object'), ('complete', 'equalizers'), ('complete', 'products'), +('complete', 'multi-complete'), ('equalizers', 'finitely complete'), ('equalizers', 'coreflexive equalizers'), ('equalizers', 'kernels'), @@ -70,6 +81,7 @@ VALUES ('cokernels', 'conormal'), ('cocomplete', 'coequalizers'), ('cocomplete', 'coproducts'), +('cocomplete', 'multi-cocomplete'), ('products', 'complete'), ('products', 'finite products'), ('products', 'powers'), @@ -122,8 +134,14 @@ VALUES ('essentially discrete', 'discrete'), ('finite', 'essentially finite'), ('finite', 'small'), +('finite', 'countable'), ('essentially finite', 'finite'), ('essentially finite', 'essentially small'), +('essentially finite', 'essentially countable'), +('countable', 'essentially countable'), +('countable', 'finite'), +('essentially countable', 'countable'), +('essentially countable', 'essentially finite'), ('pullbacks', 'wide pullbacks'), ('pullbacks', 'binary products'), ('pushouts', 'wide pushouts'), @@ -278,4 +296,49 @@ VALUES ('normal', 'kernels'), ('conormal', 'zero morphisms'), ('conormal', 'epi-regular'), -('conormal', 'cokernels'); \ No newline at end of file +('conormal', 'cokernels'), +('multi-complete', 'complete'), +('multi-complete', 'multi-terminal object'), +('multi-terminal object', 'multi-complete'), +('multi-terminal object', 'terminal object'), +('multi-cocomplete', 'cocomplete'), +('multi-cocomplete', 'multi-initial object'), +('multi-initial object', 'multi-cocomplete'), +('multi-initial object', 'initial object'), +('accessible', 'finitely accessible'), +('accessible', 'ℵ₁-accessible'), +('accessible', 'locally presentable'), +('accessible', 'locally multi-presentable'), +('accessible', 'locally poly-presentable'), +('ℵ₁-accessible', 'accessible'), +('ℵ₁-accessible', 'finitely accessible'), +('ℵ₁-accessible', 'locally ℵ₁-presentable'), +('finitely accessible', 'accessible'), +('finitely accessible', 'ℵ₁-accessible'), +('finitely accessible', 'locally finitely presentable'), +('finitely accessible', 'locally finitely multi-presentable'), +('finitely accessible', 'filtered colimits'), +('locally multi-presentable', 'locally finitely multi-presentable'), +('locally multi-presentable', 'locally presentable'), +('locally multi-presentable', 'locally poly-presentable'), +('locally multi-presentable', 'accessible'), +('locally multi-presentable', 'multi-cocomplete'), +('locally multi-presentable', 'connected limits'), +('locally finitely multi-presentable', 'locally multi-presentable'), +('locally finitely multi-presentable', 'finitely accessible'), +('locally finitely multi-presentable', 'filtered colimits'), +('locally finitely multi-presentable', 'multi-cocomplete'), +('locally finitely multi-presentable', 'connected limits'), +('locally finitely multi-presentable', 'multi-algebraic'), +('locally poly-presentable', 'locally presentable'), +('locally poly-presentable', 'locally multi-presentable'), +('locally poly-presentable', 'accessible'), +('locally poly-presentable', 'wide pullbacks'), +('multi-algebraic', 'locally finitely multi-presentable'), +('multi-algebraic', 'locally strongly finitely presentable'), +('multi-algebraic', 'generalized variety'), +('multi-algebraic', 'sifted colimits'), +('multi-algebraic', 'multi-cocomplete'), +('generalized variety', 'multi-algebraic'), +('generalized variety', 'locally strongly finitely presentable'), +('generalized variety', 'sifted colimits'); From f04899751c8a99f860a7cefc07300e90c61cc0c2 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:44:21 +0200 Subject: [PATCH 06/13] add results about multi-complete categories --- ...limits-colimits-existence-implications.sql | 28 +++++++++++++++++++ .../006_trivial-categories-implications.sql | 7 +++++ 2 files changed, 35 insertions(+) diff --git a/database/data/005_implications/001_limits-colimits-existence-implications.sql b/database/data/005_implications/001_limits-colimits-existence-implications.sql index 15eaa5a3..311cbcb5 100644 --- a/database/data/005_implications/001_limits-colimits-existence-implications.sql +++ b/database/data/005_implications/001_limits-colimits-existence-implications.sql @@ -292,4 +292,32 @@ VALUES '["powers"]', 'The product $X^I$ is the cofiltered limit of the finite powers $X^E$, where $E$ ranges over the finite subsets of $I$.', FALSE +), +( + 'multi-complete_generalize_limits', + '["complete"]', + '["multi-complete"]', + 'Limits are precisely multi-limits such that the set of cones is singleton.', + FALSE +), +( + 'multi-terminal_special_case', + '["multi-complete"]', + '["multi-terminal object"]', + 'This is trivial.', + FALSE +), +( + 'multi-terminal_with_connected', + '["connected","multi-terminal object"]', + '["terminal object"]', + 'Let $(T_i)_{i\in I}$ be a multi-terminal object in a connected category $\mathcal{C}$. By definition of multi-terminal objects, for each object $C$, there are a unique index $i_C\in I$ and a unique morphism $C \to T_{i_C}$. Since the index $i_C$ is invariant under connected components, $I$ must be a singleton. The converse is trivial.', + TRUE +), +( + 'multi-complete_with_finite_coproducts', + '["multi-complete", "finite coproducts"]', + '["complete"]', + 'Let $D\colon \mathcal{S} \to \mathcal{C}$ be a small diagram in a category $\mathcal{C}$. Since $\mathcal{C}$ has finite coproducts, the category $\mathbf{Cone}(D)$ of cones over $D$ has finite coproducts. In particular, $\mathbf{Cone}(D)$ is connected, hence a multi-terminal object in it automatically becomes a terminal object.', + FALSE ); \ No newline at end of file diff --git a/database/data/005_implications/006_trivial-categories-implications.sql b/database/data/005_implications/006_trivial-categories-implications.sql index 71222ad6..8d23cdcf 100644 --- a/database/data/005_implications/006_trivial-categories-implications.sql +++ b/database/data/005_implications/006_trivial-categories-implications.sql @@ -34,6 +34,13 @@ VALUES 'This is trivial.', FALSE ), +( + 'groupoid_with_multi-terminal', + '["groupoid", "multi-terminal object"]', + '["thin"]', + 'Let $f,g\colon A \rightrightarrows B$ be a parallel pair of morphisms. Since the category has a multi-terminal object, the connected component containing $A$ and $B$ has a terminal object. But since the category is a groupoid, both $A$ and $B$ are terminal objects in the connected component, hence $f=g$.', + FALSE +), ( 'trivial_consequence', '["trivial"]', From ea45544426699b3b63bc2249d52a126890735d27 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:44:52 +0200 Subject: [PATCH 07/13] add results about accessible and related categories --- ...limits-colimits-existence-implications.sql | 6 +- .../007_locally-presentable-implications.sql | 182 ++++++++++++++++-- 2 files changed, 171 insertions(+), 17 deletions(-) diff --git a/database/data/005_implications/001_limits-colimits-existence-implications.sql b/database/data/005_implications/001_limits-colimits-existence-implications.sql index 311cbcb5..8ef2e6c2 100644 --- a/database/data/005_implications/001_limits-colimits-existence-implications.sql +++ b/database/data/005_implications/001_limits-colimits-existence-implications.sql @@ -169,9 +169,9 @@ VALUES ), ( 'finite_filtered_colimits', - '["finite", "Cauchy complete"]', - '["filtered colimits"]', - 'See MO/509853.', + '["essentially finite", "Cauchy complete"]', + '["finitely accessible"]', + 'See MO/509853, where it is in fact shown that the ind-completion of any finite Cauchy-complete category becomes itself.', FALSE ), ( diff --git a/database/data/005_implications/007_locally-presentable-implications.sql b/database/data/005_implications/007_locally-presentable-implications.sql index 059756e1..30a87204 100644 --- a/database/data/005_implications/007_locally-presentable-implications.sql +++ b/database/data/005_implications/007_locally-presentable-implications.sql @@ -6,13 +6,6 @@ INSERT INTO implication_input ( is_equivalence ) VALUES -( - 'locally_finitely_presentable_condition', - '["locally finitely presentable"]', - '["locally presentable"]', - 'Locally finitely presentable categories are by definition the locally $\aleph_0$-presentable categories.', - FALSE -), ( 'locally_finitely_presentable_raise', '["locally finitely presentable"]', @@ -28,10 +21,24 @@ VALUES FALSE ), ( - 'locally_presentable_consequence', - '["locally presentable"]', - '["locally essentially small", "well-powered", "well-copowered", "complete", "cocomplete", "generating set"]', - 'For locally essential smallness, see the proof of Prop. 2.1.5 in Makkai-Pare. For the other non-trivial conclusions see Adamek-Rosicky, Thm. 1.20, Cor. 1.28, Rem. 1.56, Thm. 1.58.', + 'accessible_trivial_consequence', + '["accessible"]', + '["well-powered", "generating set"]', + 'For well-poweredness, see nLab. For a $\kappa$-accessible category, the set $G$ appearing in the definition gives a small dense full subcategory, which is in particular a generating set.', + FALSE +), +( + 'accessible_locally_small', + '["accessible"]', + '["locally essentially small"]', + 'See the proof of Prop. 2.1.5 in Makkai-Pare.', + FALSE +), +( + 'accessible_wellcopowered', + '["accessible", "pushouts"]', + '["well-copowered"]', + 'See Thm. 2.49 in Adamek-Rosicky or Prop. 6.1.3 in Makkai-Pare.', FALSE ), ( @@ -42,9 +49,9 @@ VALUES FALSE ), ( - 'locally_presentable_thin', - '["locally presentable", "self-dual"]', - '["thin"]', + 'locally_presentable_essentially_small', + '["locally presentable", "locally copresentable"]', + '["essentially small"]', 'This follows from Adamek-Rosicky, Thm. 1.64.', FALSE ), @@ -61,4 +68,151 @@ VALUES '["locally finitely presentable"]', 'See Adamek-Rosicky, Cor. 3.7.', FALSE +), +( + 'locally_presentable_definition_finite', + '["locally finitely presentable"]', + '["finitely accessible", "cocomplete"]', + 'This follows from one of equivalent formulations of locally finitely presentable categories.', + TRUE +), +( + 'locally_presentable_definition_countable', + '["locally ℵ₁-presentable"]', + '["ℵ₁-accessible", "cocomplete"]', + 'This follows from one of equivalent formulations of locally ℵ₁-presentable categories.', + TRUE +), +( + 'locally_presentable_definition', + '["locally presentable"]', + '["accessible", "cocomplete"]', + 'This follows from one of equivalent formulations of locally presentable categories.', + TRUE +), +( + 'finitely_accessible_raise', + '["finitely accessible"]', + '["ℵ₁-accessible"]', + 'This is because any regular cardinal is strictly smaller than its successor cardinal. See nLab.', + FALSE +), +( + 'countably_accessible_special_case', + '["ℵ₁-accessible"]', + '["accessible"]', + 'This is trivial.', + FALSE +), +( + 'accessible_require_filtered_colimit', + '["finitely accessible"]', + '["filtered colimits"]', + 'This holds by definition.', + FALSE +), +( + 'accessible_require_Cauchy_complete', + '["accessible"]', + '["Cauchy complete"]', + 'This is because the walking idempotent is $\kappa$-filtered for any regular cardinal $\kappa$.', + FALSE +), +( + 'small_accessible_characterization', + '["essentially small", "Cauchy complete"]', + '["accessible"]', + 'See Makkai-Pare, Thm. 2.2.2.', + FALSE +), +( + 'countably_accessible_thin', + '["essentially countable", "thin"]', + '["ℵ₁-accessible"]', + 'In general, every $\kappa$-filtered diagram in a poset whose elements are less than $\kappa$ admits the greatest element. Therefore, all the elements are $\kappa$-presentable, and the poset is $\kappa$-accessible.', + FALSE +), +( + 'locally_presentable_another_definition', + '["accessible", "complete"]', + '["locally presentable"]', + 'This follows from one of equivalent formulations of locally presentable categories.', + FALSE +), +( + 'locally_strongly_finitely_presentable_definition', + '["locally strongly finitely presentable"]', + '["generalized variety", "cocomplete"]', + 'This is trivial.', + TRUE +), +( + 'generalized_variety_require_sifted_colimit', + '["generalized variety"]', + '["sifted colimits"]', + 'This holds by definition.', + FALSE +), +( + 'generalized_variety_implies_accessible', + '["generalized variety"]', + '["ℵ₁-accessible"]', + 'See [AR01, Remark 4.8(2)].', + FALSE +), +( + 'locally_multi-presentable_definition', + '["locally multi-presentable"]', + '["accessible", "connected limits"]', + 'This follows from one of equivalent formulations of locally multi-presentable categories.', + TRUE +), +( + 'locally_multi-presentable_another_definition', + '["locally multi-presentable"]', + '["accessible", "multi-cocomplete"]', + 'This follows from one of equivalent formulations of locally multi-presentable categories.', + TRUE +), +( + 'locally_finitely_multi-presentable_definition', + '["locally finitely multi-presentable"]', + '["finitely accessible", "connected limits"]', + 'This follows from one of equivalent formulations of locally finitely multi-presentable categories.', + TRUE +), +( + 'locally_finitely_multi-presentable_another_definition', + '["locally finitely multi-presentable"]', + '["finitely accessible", "multi-cocomplete"]', + 'This follows from one of equivalent formulations of locally finitely multi-presentable categories.', + TRUE +), +( + 'locally_poly-presentable_definition', + '["locally poly-presentable"]', + '["accessible", "wide pullbacks"]', + 'This holds by definition.', + TRUE +), +( + 'multi-algebraic_implies_locally_finitely_multi-presentable', + '["multi-algebraic"]', + '["locally finitely multi-presentable"]', + 'This is because that every (finite product, coproduct)-sketch is clearly a (finite limit, coproduct)-sketch.', + FALSE +), +( + 'varieties_are_multi-algebraic', + '["locally strongly finitely presentable"]', + '["multi-algebraic"]', + 'This is because that every finite-product-sketch is clearly a (finite product, coproduct)-sketch.', + FALSE +), +( + 'multi-algebraic_another_definition', + '["multi-algebraic"]', + '["generalized variety", "multi-cocomplete"]', + 'This follows from one of equivalent formulations of multi-algebraic categories.', + TRUE ); \ No newline at end of file From f1a86e3e75329c2cd059bd4f3afeda6b3e7cc539 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:46:48 +0200 Subject: [PATCH 08/13] decide (co)accessibility for various categories --- database/data/004_property-assignments/FinAb.sql | 6 ++++++ database/data/004_property-assignments/FinGrp.sql | 6 ++++++ database/data/004_property-assignments/FinSet.sql | 6 ++++++ database/data/004_property-assignments/Setne.sql | 6 ++++++ database/data/004_property-assignments/Top.sql | 6 ++++++ database/data/004_property-assignments/Top_pointed.sql | 6 ++++++ 6 files changed, 36 insertions(+) diff --git a/database/data/004_property-assignments/FinAb.sql b/database/data/004_property-assignments/FinAb.sql index 7f8d58c1..ce7ec296 100644 --- a/database/data/004_property-assignments/FinAb.sql +++ b/database/data/004_property-assignments/FinAb.sql @@ -29,6 +29,12 @@ VALUES TRUE, 'This is a simple special case of Pontryagin duality: The functor $\hom(-,\mathbb{Q}/\mathbb{Z})$ provides the equivalence.' ), +( + 'FinAb', + 'ℵ₁-accessible', + TRUE, + 'The proof works exactly as for the category of finite sets.' +), ( 'FinAb', 'small', diff --git a/database/data/004_property-assignments/FinGrp.sql b/database/data/004_property-assignments/FinGrp.sql index babb0a4f..bf80a30b 100644 --- a/database/data/004_property-assignments/FinGrp.sql +++ b/database/data/004_property-assignments/FinGrp.sql @@ -59,6 +59,12 @@ VALUES TRUE, 'Since epimorphisms are surjective (see below), this is the first isomorphism theorem for finite groups.' ), +( + 'FinGrp', + 'ℵ₁-accessible', + TRUE, + 'The proof works exactly as for the category of finite sets.' +), ( 'FinGrp', 'normal', diff --git a/database/data/004_property-assignments/FinSet.sql b/database/data/004_property-assignments/FinSet.sql index 1adf99e2..c389e46b 100644 --- a/database/data/004_property-assignments/FinSet.sql +++ b/database/data/004_property-assignments/FinSet.sql @@ -41,6 +41,12 @@ VALUES TRUE, 'The two-element set is a cogenerator.' ), +( + 'FinSet', + 'ℵ₁-accessible', + TRUE, + 'The inclusion $\mathbf{FinSet} \hookrightarrow \mathbf{Set}$ is closed under ℵ₁-filtered colimits, that is, any ℵ₁-filtered colimit of finite sets is again finite. Since every finite set is ℵ₁-presentable in $\mathbf{Set}$, it is still ℵ₁-presentable in $\mathbf{FinSet}$. Therefore, $\mathbf{FinSet}$ is ℵ₁-accessible, where every object is ℵ₁-presentable.' +), ( 'FinSet', 'small', diff --git a/database/data/004_property-assignments/Setne.sql b/database/data/004_property-assignments/Setne.sql index 90f13269..d07468d5 100644 --- a/database/data/004_property-assignments/Setne.sql +++ b/database/data/004_property-assignments/Setne.sql @@ -83,6 +83,12 @@ VALUES TRUE, 'Use constant maps.' ), +( + 'Setne', + 'finitely accessible', + TRUE, + 'Since the inclusion $\mathbf{Set}_{\neq \varnothing} \hookrightarrow \mathbf{Set}$ is closed under non-empty colimits, it is also closed under filtered colimits. Therefore, non-empty finite sets are still finitely presentable in $\mathbf{Set}_{\neq \varnothing}$, and every non-empty set is written as a filtered colimit of them.' +), ( 'Setne', 'sequential limits', diff --git a/database/data/004_property-assignments/Top.sql b/database/data/004_property-assignments/Top.sql index a9e940b9..7b8cbdf8 100644 --- a/database/data/004_property-assignments/Top.sql +++ b/database/data/004_property-assignments/Top.sql @@ -112,4 +112,10 @@ VALUES 'co-Malcev', FALSE, 'See MO/509548. We can also phrase the proof as follows: Consider the forgetful functor $U : \mathbf{Top} \to \mathbf{Set}$ and the relation $R \subseteq U^2$ defined by $R(X) := \{(x,y) \in U(X)^2 : x \in \overline{\{y\}} \}$. Both are representable: $U$ by the singleton and $R$ by the Sierpinski space. It is clear that $R$ is reflexive, but not symmetric.' +), +( + 'Top', + 'coaccessible', + FALSE, + 'Assume $\mathbf{Top}$ is coaccessible. Let $p\colon S \to I$ be the identity map from the Sierpinski space to the two-element indiscrete space. Then, a topological space is discrete if and only if it is projective to the morphism $p$. This implies that the full subcategory spanned by all discrete spaces, which is equivalent to $\mathbf{Set}$, is coaccessible by Prop. 4.7 in Adamek-Rosicky. However, since $\mathbf{Set}$ is not coaccessible, this is a contradiction.' ); diff --git a/database/data/004_property-assignments/Top_pointed.sql b/database/data/004_property-assignments/Top_pointed.sql index fc5e500f..fa4e679d 100644 --- a/database/data/004_property-assignments/Top_pointed.sql +++ b/database/data/004_property-assignments/Top_pointed.sql @@ -137,6 +137,12 @@ VALUES 'regular quotient object classifier', FALSE, 'We can recycle the proof for the category of pointed sets using discrete topological spaces.' +), +( + 'Top*', + 'coaccessible', + FALSE, + 'We can adjust the proof for $\mathbf{Top}$ as follows: Assume $\mathbf{Top}_*$ is coaccessible. Let $S_0=\{x,*\}$ be the pointed topological space such that $\{*\}$ is the only non-trivial open set, and let $S_1=\{x,*\}$ be the pointed space such that $\{x\}$ is the only non-trivial open set. Let $p_i\colon S_i \to \{x,*\}$ be the identity function to the two-element indiscrete pointed space. Then, a pointed topological space is discrete if and only if it is projective to the morphisms $p_0$ and $p_1$. This implies that the full subcategory spanned by all discrete pointed spaces, which is equivalent to $\mathbf{Set}_*$, is coaccessible by Prop. 4.7 in Adamek-Rosicky. However, since $\mathbf{Set}_*$ is not coaccessible, this is a contradiction.' ); From 7fa6f9aa00831c926531b7f1d5e4f2c613016c32 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:47:40 +0200 Subject: [PATCH 09/13] identify examples of multi-algebraic categories --- database/data/004_property-assignments/0.sql | 6 ++++ database/data/004_property-assignments/2.sql | 6 ++++ .../data/004_property-assignments/Fld.sql | 34 +++++-------------- .../004_property-assignments/walking_span.sql | 6 ++++ 4 files changed, 26 insertions(+), 26 deletions(-) diff --git a/database/data/004_property-assignments/0.sql b/database/data/004_property-assignments/0.sql index 2112a10a..e15b2183 100644 --- a/database/data/004_property-assignments/0.sql +++ b/database/data/004_property-assignments/0.sql @@ -35,6 +35,12 @@ VALUES TRUE, 'This is trivial.' ), +( + '0', + 'multi-algebraic', + TRUE, + 'The terminal category $\mathbf{1}$ becomes an FPC-sketch by selecting the unique empty cone and cocone. Then, a $\mathbf{Set}$-valued model of this sketch is a functor $\mathbf{1} \to \mathbf{Set}$ sending the unique object to a terminal and initial object, which never exists. Hence, $\mathbf{0}$ is the category of models of this FPC-sketch.' +), ( '0', 'inhabited', diff --git a/database/data/004_property-assignments/2.sql b/database/data/004_property-assignments/2.sql index 84be63f1..74bf9fd8 100644 --- a/database/data/004_property-assignments/2.sql +++ b/database/data/004_property-assignments/2.sql @@ -29,6 +29,12 @@ VALUES TRUE, 'This is trivial.' ), +( + '2', + 'multi-algebraic', + TRUE, + 'There is an FPC-sketch whose $\mathbf{Set}$-model is precisely a pair $(X,Y)$ of sets such that the coproduct $X+Y$ is a singleton. Any $\mathbf{Set}$-model of such a sketch is isomorphic to either $(\varnothing, 1)$ or $(1, \varnothing)$, hence the category of models is equivalent to $\mathbf{2}$.' +), ( '2', 'connected', diff --git a/database/data/004_property-assignments/Fld.sql b/database/data/004_property-assignments/Fld.sql index 51a01ab8..4337fce6 100644 --- a/database/data/004_property-assignments/Fld.sql +++ b/database/data/004_property-assignments/Fld.sql @@ -11,18 +11,6 @@ VALUES TRUE, 'There is a forgetful functor $\mathbf{Fld} \to \mathbf{Set}$ and $\mathbf{Set}$ is locally small.' ), -( - 'Fld', - 'directed colimits', - TRUE, - 'Consider a directed diagram $(F_i)$ of fields and take the colimit $F$ in the category of commutative rings. Now one checks that $F$ is a field as well, and the universal property remains true for fields.' -), -( - 'Fld', - 'connected limits', - TRUE, - 'Consider a connected diagram $(F_i)$ of fields and take the limit $F$ in the category of commutative rings. Now one checks that $F$ is a field as well, and the universal property remains true for fields. Namely, $1 = 0$ in $F$ would imply that $1 = 0$ in each $F_i$ and hence, since the diagram is connected, in some $F_i$, which is a contradiction. And if $x \in F$ is non-zero, then all components $x_i$ are non-zero and hence invertible: Choose some $j$ such that $x_j$ is non-zero. Since there is a zig zag path of morphisms between $i$ and $j$ in the index category, which get mapped to field homomorphisms which are injective, it follows that $x_i$ is non-zero.' -), ( 'Fld', 'inhabited', @@ -35,12 +23,6 @@ VALUES TRUE, 'It is well-known that every field homomorphism is injective and hence a monomorphism.' ), -( - 'Fld', - 'well-powered', - TRUE, - 'The subfields of a given field form a set.' -), ( 'Fld', 'well-copowered', @@ -49,9 +31,9 @@ VALUES ), ( 'Fld', - 'generating set', + 'multi-algebraic', TRUE, - 'The fields $Q(\mathbb{Z}[X]/\mathfrak{p})$, where $\mathfrak{p}$ runs through all prime ideals of the polynomial ring, provide a generating set. This is because for every element $a \in K$ of a field $K$ there is a prime ideal $\mathfrak{p}$ with a homomorphism $Q(\mathbb{Z}[X]/\mathfrak{p}) \to K$ mapping $[X]$ to $a$: There is a homomorphism $\mathbb{Z}[X] \to K$ mapping $X \mapsto a$. Let $\mathfrak{p}$ be its kernel. Then it extends to a field homomorphism as desired.' + 'See Eg. 4.3(1) in [AR01].' ), ( 'Fld', @@ -83,12 +65,6 @@ VALUES FALSE, 'We apply this lemma to the collection of fields: Any homomorphism of fields is injective. For every infinite cardinal $\kappa$ the field of rational functions in $\kappa$ variables has cardinality $\geq \kappa$ and a non-trivial automorphism (swap two variables).' ), -( - 'Fld', - 'essentially small', - FALSE, - 'Consider function fields in any number of variables.' -), ( 'Fld', 'skeletal', @@ -112,4 +88,10 @@ VALUES 'binary powers', FALSE, 'Assume that the product $P := \mathbb{Q}(\sqrt{2}) \times \mathbb{Q}(\sqrt{2})$ exists. This field is isomorphic to a subfield of $\mathbb{Q}(\sqrt{2})$, hence $P \cong \mathbb{Q}$ or $P \cong \mathbb{Q}(\sqrt{2})$. In the first case, the two projections $P \rightrightarrows \mathbb{Q}(\sqrt{2})$ must be equal, which means that every two homomorphisms $K \rightrightarrows \mathbb{Q}(\sqrt{2})$ are equal, which is absurd (take $K = \mathbb{Q}(\sqrt{2})$ and its two automorphisms). In the second case, the projections induce for every field $K$ a bijection $\mathrm{Hom}(K,\mathbb{Q}(\sqrt{2})) \cong \mathrm{Hom}(K,\mathbb{Q}(\sqrt{2}))^2$, which however fails for $K = \mathbb{Q}(\sqrt{2})$: the left hand side has $2$ elements, the right hand side has $4$ elements. A more general result about products in $\mathbf{Fld}$ can be found at MSE/359352.' +), +( + 'Fld', + 'multi-terminal object', + FALSE, + 'Every field has a non-trivial extension, for instance, the rational function field over itself in one variable. Hence, a multi-terminal object never exists.' ); diff --git a/database/data/004_property-assignments/walking_span.sql b/database/data/004_property-assignments/walking_span.sql index c089d604..90a18a94 100644 --- a/database/data/004_property-assignments/walking_span.sql +++ b/database/data/004_property-assignments/walking_span.sql @@ -47,6 +47,12 @@ VALUES TRUE, 'The slice category over $0$ is the trivial category, and the slice category over $1$ is the interval category, which is cartesian closed (see there). The same holds for $2$ by symmetry.' ), +( + 'walking_span', + 'multi-algebraic', + TRUE, + 'We first remark that for a set $X$, the identity span $(\mathrm{id},\mathrm{id})\colon X \leftarrow X \rightarrow X$ exhibits a product if and only if $X$ is either a singleton or the empty set. Therefore, there is a (finite product, coproduct)-sketch whose $\mathbf{Set}$-model is precisely a pair $(X,Y)$ of sets such that each of $X$ and $Y$ is either a singleton or the empty set and the product $X \times Y$ is the empty set. Any $\mathbf{Set}$-model of such a sketch is isomorphic to either $(\varnothing, \varnothing)$, $(\varnothing, 1)$, or $(1, \varnothing)$; hence the category of models is equivalent to the walking span.' +), ( 'walking_span', 'sifted', From 84e5b6cfb43e8ec311c7607b7a0e7c1cb757d966 Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:47:55 +0200 Subject: [PATCH 10/13] remove redundant property of Set^op --- database/data/004_property-assignments/Set_op.sql | 13 ------------- 1 file changed, 13 deletions(-) delete mode 100644 database/data/004_property-assignments/Set_op.sql diff --git a/database/data/004_property-assignments/Set_op.sql b/database/data/004_property-assignments/Set_op.sql deleted file mode 100644 index 0e6c6982..00000000 --- a/database/data/004_property-assignments/Set_op.sql +++ /dev/null @@ -1,13 +0,0 @@ -INSERT INTO category_property_assignments ( - category_id, - property_id, - is_satisfied, - reason -) -VALUES -( - 'Set_op', - 'locally presentable', - FALSE, - 'This follows from the fact that the opposite category of a complete accessible large category is never accessible. See Makkai-Pare, Thm. 3.4.3.' -); From eaa764e5b4f1122f5346b067bd2411d5cc4c012a Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:48:31 +0200 Subject: [PATCH 11/13] assign multi-completeness --- database/data/004_property-assignments/FS.sql | 12 ++++++++++++ database/data/004_property-assignments/Setne.sql | 13 +++++++++++++ 2 files changed, 25 insertions(+) diff --git a/database/data/004_property-assignments/FS.sql b/database/data/004_property-assignments/FS.sql index 0564e3be..b8b8d8b2 100644 --- a/database/data/004_property-assignments/FS.sql +++ b/database/data/004_property-assignments/FS.sql @@ -53,6 +53,12 @@ VALUES TRUE, 'If $f : X \to Y$ is a surjective map of finite sets, it is the coequalizer of the two projections $p_1, p_2 : X \times_Y X \rightrightarrows X$ in $\mathbf{FinSet}$, but also in $\mathbf{FS}$. Notice that $p_1,p_2$ are surjective. Even though $X \times_Y X$ is not a pullback in $\mathbf{FS}$, we can use this finite set here.' ), +( + 'FS', + 'multi-terminal object', + TRUE, + 'The empty set and a singleton give a multi-terminal object.' +), ( 'FS', 'small', @@ -117,6 +123,12 @@ VALUES $(E_1 \vee E_2) \wedge (E_1 \vee E_3) = \top \wedge \top = \top$.

*For thin categories, the properties codistributive and distributive are equivalent.' ), +( + 'FS', + 'multi-initial object', + FALSE, + 'If a multi-initial object exists, then the connected component consisting of non-empty finite sets has an initial object $X$. Then, any non-empty finite set cannot have a cardinality strictly greater than $X$, which is a contradiction.' +), ( 'FS', 'countable', diff --git a/database/data/004_property-assignments/Setne.sql b/database/data/004_property-assignments/Setne.sql index d07468d5..fc969618 100644 --- a/database/data/004_property-assignments/Setne.sql +++ b/database/data/004_property-assignments/Setne.sql @@ -89,6 +89,19 @@ VALUES TRUE, 'Since the inclusion $\mathbf{Set}_{\neq \varnothing} \hookrightarrow \mathbf{Set}$ is closed under non-empty colimits, it is also closed under filtered colimits. Therefore, non-empty finite sets are still finitely presentable in $\mathbf{Set}_{\neq \varnothing}$, and every non-empty set is written as a filtered colimit of them.' ), +( + 'Setne', + 'generalized variety', + TRUE, + 'Since the inclusion $\mathbf{Set}_{\neq \varnothing} \hookrightarrow \mathbf{Set}$ is closed under non-empty colimits, it is also closed under sifted colimits. Therefore, non-empty finite sets are still strongly finitely presentable in $\mathbf{Set}_{\neq \varnothing}$, and every non-empty set is written as a sifted colimit of them.' +), + +( + 'Setne', + 'multi-complete', + 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', 'sequential limits', From 43607db7e4aaf1c43b7ea7ac2812c6eb2a655d7f Mon Sep 17 00:00:00 2001 From: ykawase5048 <114643337+ykawase5048@users.noreply.github.com> Date: Thu, 16 Apr 2026 22:49:09 +0200 Subject: [PATCH 12/13] adjust test files --- scripts/expected-data/Ab.json | 18 +++++++++++++++++- scripts/expected-data/Set.json | 18 +++++++++++++++++- scripts/expected-data/Top.json | 16 ++++++++++++++++ 3 files changed, 50 insertions(+), 2 deletions(-) diff --git a/scripts/expected-data/Ab.json b/scripts/expected-data/Ab.json index 350904ca..3b0ab31d 100644 --- a/scripts/expected-data/Ab.json +++ b/scripts/expected-data/Ab.json @@ -86,6 +86,18 @@ "cokernels": true, "normal": true, "conormal": true, + "accessible": true, + "finitely accessible": true, + "ℵ₁-accessible": true, + "generalized variety": true, + "locally finitely multi-presentable": true, + "locally multi-presentable": true, + "locally poly-presentable": true, + "multi-algebraic": true, + "multi-complete": true, + "multi-cocomplete": true, + "multi-terminal object": true, + "multi-initial object": true, "cartesian filtered colimits": true, "cocartesian cofiltered limits": true, @@ -126,5 +138,9 @@ "cocartesian coclosed": false, "locally cocartesian coclosed": false, "quotient object classifier": false, - "regular quotient object classifier": false + "regular quotient object classifier": false, + "locally copresentable": false, + "coaccessible": false, + "countable": false, + "essentially countable": false } diff --git a/scripts/expected-data/Set.json b/scripts/expected-data/Set.json index c11a16dd..899b6a95 100644 --- a/scripts/expected-data/Set.json +++ b/scripts/expected-data/Set.json @@ -81,6 +81,18 @@ "cofiltered": true, "sifted": true, "cosifted": true, + "accessible": true, + "finitely accessible": true, + "ℵ₁-accessible": true, + "generalized variety": true, + "locally finitely multi-presentable": true, + "locally multi-presentable": true, + "locally poly-presentable": true, + "multi-algebraic": true, + "multi-complete": true, + "multi-cocomplete": true, + "multi-terminal object": true, + "multi-initial object": true, "cartesian filtered colimits": true, "cocartesian cofiltered limits": true, @@ -126,5 +138,9 @@ "kernels": false, "cokernels": false, "normal": false, - "conormal": false + "conormal": false, + "locally copresentable": false, + "coaccessible": false, + "countable": false, + "essentially countable": false } diff --git a/scripts/expected-data/Top.json b/scripts/expected-data/Top.json index 94b5cc11..821cfbf5 100644 --- a/scripts/expected-data/Top.json +++ b/scripts/expected-data/Top.json @@ -65,6 +65,10 @@ "cofiltered": true, "sifted": true, "cosifted": true, + "multi-complete": true, + "multi-cocomplete": true, + "multi-terminal object": true, + "multi-initial object": true, "cocartesian cofiltered limits": true, "abelian": false, @@ -126,5 +130,17 @@ "cokernels": false, "normal": false, "conormal": false, + "accessible": false, + "finitely accessible": false, + "ℵ₁-accessible": false, + "generalized variety": false, + "locally finitely multi-presentable": false, + "locally multi-presentable": false, + "locally poly-presentable": false, + "multi-algebraic": false, + "locally copresentable": false, + "coaccessible": false, + "countable": false, + "essentially countable": false, "cartesian filtered colimits": false } From af4592ee94fd0d81e5fcd3949579b22f1a2c8f87 Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Thu, 16 Apr 2026 22:57:24 +0200 Subject: [PATCH 13/13] Delta is countable --- database/data/004_property-assignments/Delta.sql | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/database/data/004_property-assignments/Delta.sql b/database/data/004_property-assignments/Delta.sql index 0cad9358..ec5cb288 100644 --- a/database/data/004_property-assignments/Delta.sql +++ b/database/data/004_property-assignments/Delta.sql @@ -11,6 +11,12 @@ VALUES TRUE, 'If $[n] \to [m]$ is a bijection, then $n+1 = m+1$ by comparing the sizes, hence $n=m$.' ), +( + 'Delta', + 'countable', + TRUE, + 'This is obvious.' +), ( 'Delta', 'small',