From e740174da53627159da5d3aca301573537457b18 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Wed, 22 Apr 2026 22:22:28 -0400 Subject: [PATCH 01/18] Initial trial of 'has effective (co)congruences properties' --- .../002_limits-colimits-existence.sql | 16 ++++++++++++++++ .../003_properties/100_related-properties.sql | 8 ++++++++ ..._limits-colimits-existence-implications.sql | 7 +++++++ .../005_additional-structure-implications.sql | 9 ++++++++- .../007_locally-presentable-implications.sql | 9 ++++++++- .../008_topos-theory-implications.sql | 18 ++++++++++++++++-- 6 files changed, 63 insertions(+), 4 deletions(-) diff --git a/databases/catdat/data/003_properties/002_limits-colimits-existence.sql b/databases/catdat/data/003_properties/002_limits-colimits-existence.sql index 9d395181..4d269ec3 100644 --- a/databases/catdat/data/003_properties/002_limits-colimits-existence.sql +++ b/databases/catdat/data/003_properties/002_limits-colimits-existence.sql @@ -343,6 +343,22 @@ VALUES 'quotients of congruences', TRUE ), +( + 'effective congruences', + 'has', + 'A congruence $f, g : E \rightrightarrows X$ (see definition here) is effective if it is the kernel pair of some morphism, i.e. if there is a morphism $h : X \to Y$ such that we have a cartesian square [put diagram here]. This property asserts that every congruence in the category is effective.', + 'https://ncatlab.org/nlab/show/congruence', + 'effective cocongruences', + TRUE +), +( + 'effective cocongruences', + 'has', + 'A cocongruence $f, g : X \rightrightarrows E$ (see definition here) is effective if it is the cokernel pair of some morphism, i.e. if there is a morphism $h : Y \to X$ such that we have a cocartesian square [put diagram here]. This property asserts that every cocongruence in the category is effective.', + NULL, + 'effective congruences', + TRUE +), ( 'cosifted limits', 'has', diff --git a/databases/catdat/data/003_properties/100_related-properties.sql b/databases/catdat/data/003_properties/100_related-properties.sql index 100866aa..3979549e 100644 --- a/databases/catdat/data/003_properties/100_related-properties.sql +++ b/databases/catdat/data/003_properties/100_related-properties.sql @@ -285,9 +285,15 @@ VALUES ('quotients of congruences', 'coequalizers'), ('quotients of congruences', 'cokernels'), ('quotients of congruences', 'regular'), +('quotients of congruences', 'effective congruences'), ('coquotients of cocongruences', 'equalizers'), ('coquotients of cocongruences', 'kernels'), ('coquotients of cocongruences', 'coregular'), +('coquotients of cocongruences', 'effective cocongruences'), +('effective congruences', 'normal'), +('effective congruences', 'quotients of congruences'), +('effective cocongruences', 'conormal'), +('effective cocongruences', 'coquotients of cocongruences'), ('direct', 'one-way'), ('direct', 'skeletal'), ('inverse', 'one-way'), @@ -327,9 +333,11 @@ VALUES ('normal', 'zero morphisms'), ('normal', 'mono-regular'), ('normal', 'kernels'), +('normal', 'effective congruences'), ('conormal', 'zero morphisms'), ('conormal', 'epi-regular'), ('conormal', 'cokernels'), +('conormal', 'effective cocongruences'), ('multi-complete', 'complete'), ('multi-complete', 'multi-terminal object'), ('multi-terminal object', 'multi-complete'), diff --git a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql index 5864b899..592ddc66 100644 --- a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql +++ b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql @@ -69,6 +69,13 @@ VALUES 'For any congruence $E$ on an object $X$ of a preadditive category, let $E_0$ be the kernel of $p_2 : E \to X$. The restriction of $p_1$ to $E_0$ is a monomorphism. We can then see that $E$ must be the pullback of $p_1 - p_2 : E \to X$ and $E_0 \hookrightarrow X$. Then the cokernel of $E_0 \hookrightarrow X$ is a quotient of $E$.', FALSE ), +( + 'thin_effective_congruences', + '["thin"]', + '["effective congruences"]', + 'In a thin category, any congruence pair is a reflexive fork, and thus consists of equal isomorphisms. Therefore, the congruence is the kernel pair of the identity morphism on the target.', + FALSE +), ( 'products_consequences', '["products"]', diff --git a/databases/catdat/data/005_implications/005_additional-structure-implications.sql b/databases/catdat/data/005_implications/005_additional-structure-implications.sql index 97962495..5f6d8d0f 100644 --- a/databases/catdat/data/005_implications/005_additional-structure-implications.sql +++ b/databases/catdat/data/005_implications/005_additional-structure-implications.sql @@ -27,6 +27,13 @@ VALUES 'This is trivial.', FALSE ), +( + 'preadditive_kernels_normal_imply_effective_congruences', + '["preadditive", "kernels", "normal"]', + '["effective congruences"]', + 'Let $f, g : E \rightrightarrows X$ be a congruence. Then let $E_0$ be the kernel of $g$. We see that $f$ restricted to $E_0$ is a monomorphism $E_0 \hookrightarrow X$. If this is the kernel of a morphism $h : X \to Y$, then $E$ is the kernel pair of $h$.', + FALSE +), ( 'additive_definition', '["additive"]', @@ -110,4 +117,4 @@ VALUES '["trivial"]', 'This follows since the dual of a non-trivial Grothendieck abelian category cannot be Grothendieck abelian. See Peter Freyd, Abelian categories, p. 116.', FALSE -); \ No newline at end of file +); diff --git a/databases/catdat/data/005_implications/007_locally-presentable-implications.sql b/databases/catdat/data/005_implications/007_locally-presentable-implications.sql index 3d52bab4..db0c6273 100644 --- a/databases/catdat/data/005_implications/007_locally-presentable-implications.sql +++ b/databases/catdat/data/005_implications/007_locally-presentable-implications.sql @@ -153,6 +153,13 @@ VALUES 'This is trivial.', TRUE ), +( + 'locally_strongly_finitely_presentable_exact', + '["locally strongly finitely presentable"]', + '["effective congruences"]', + 'By one of the equivalent conditions, the category is monadic over a small power of $\mathbf{Set}$, and therefore is Barr-exact.', + FALSE +), ( 'generalized_variety_require_sifted_colimit', '["generalized variety"]', @@ -229,4 +236,4 @@ VALUES '["filtered-colimit-stable monomorphisms"]', 'Every locally finitely multi-presentable category is a multi-reflective full subcategory of a presheaf category closed under filtered colimits (Adamek-Rosicky, 4.30). Since multi-reflective full subcategories are in general closed under connected limits (Adamek-Rosicky, Thm. 4.26), in particular, we can calculate not only filtered colimits but also kernel pairs as well as in a presheaf category.', FALSE -); \ No newline at end of file +); diff --git a/databases/catdat/data/005_implications/008_topos-theory-implications.sql b/databases/catdat/data/005_implications/008_topos-theory-implications.sql index 60bf7679..27c88e42 100644 --- a/databases/catdat/data/005_implications/008_topos-theory-implications.sql +++ b/databases/catdat/data/005_implications/008_topos-theory-implications.sql @@ -121,7 +121,7 @@ VALUES ( 'topos_consequence', '["elementary topos"]', - '["finitely cocomplete", "disjoint finite coproducts", "epi-regular"]', + '["finitely cocomplete", "disjoint finite coproducts", "epi-regular", "effective congruences"]', 'See Mac Lane & Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. 4.7.8.', FALSE ), @@ -167,6 +167,20 @@ VALUES 'This is proven in Johnstone, A2.6.3 (for every quasitopos).', FALSE ), +( + 'regular_effective_congruences_implies_quotients', + '["regular", "effective congruences"]', + '["quotients of congruences"]', + 'We assume that every congruence is effective, and the regularity condition implies that every effective congruence has a quotient.', + FALSE +), +( + 'pretopos_balanced', + '["regular", "effective congruences", "extensive"]', + '["balanced"]', + 'This is the theorem that a pretopos is balanced.', + FALSE +), ( 'lcc_implies_extensive', '["locally cartesian closed", "disjoint finite coproducts"]', @@ -222,4 +236,4 @@ VALUES '["natural numbers object"]', 'The triple $(1, \mathrm{id}_1, \mathrm{id}_1)$ is clearly a NNO.', FALSE -); \ No newline at end of file +); From 6be294101915b7f1fddbe192c263c9dd71bbd35d Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Wed, 22 Apr 2026 23:08:33 -0400 Subject: [PATCH 02/18] Found a proof of effective congruences + extensive -> balanced so can drop the regularity hypothesis --- .../data/005_implications/008_topos-theory-implications.sql | 4 ++-- 1 file changed, 2 insertions(+), 2 deletions(-) diff --git a/databases/catdat/data/005_implications/008_topos-theory-implications.sql b/databases/catdat/data/005_implications/008_topos-theory-implications.sql index 0d4c5276..2a694e13 100644 --- a/databases/catdat/data/005_implications/008_topos-theory-implications.sql +++ b/databases/catdat/data/005_implications/008_topos-theory-implications.sql @@ -176,9 +176,9 @@ VALUES ), ( 'pretopos_balanced', - '["regular", "effective congruences", "extensive"]', + '["effective congruences", "extensive"]', '["balanced"]', - 'This is the theorem that a pretopos is balanced.', + 'Let $i : Y \to X$ be both a monomorphism and an epimorphism. Now define a congruence $f, g : X + Y + Y + X \rightrightarrows X+X$ acting as $i_1,i_1$ on the first copy of $X$; $i_1\circ i, i_2\circ i$ on the first copy of $Y$; $i_2\circ i, i_1\circ i$ on the second copy of $Y$; and $i_2\circ i_2$ on the second copy of $X$. We use extensitivity in showing that $f, g$ are jointly monomorphic, and again in proving transitivity. Now suppose this is the kernel pair of a morphism $h : X \to Z$. Then consider the map pair $i_2, i_1 : X \to X+X$. We must have $h \circ i_2 \circ i = h \circ i_1 \circ i$ since the pair of maps $i_2\circ i, i_1\circ i$ factor through $E$. Since $i$ is an epimorphism, that implies $h\circ i_2 = h\circ i_1$, so $i_2, i_1$ factor through $E$ as well. By disjointness of coproducts, we can conclude that $i_2, i_1$ factor uniquely through the second copy of $Y$. We thus get a morphism $X \to Y$ which is a left inverse of $i$, showing that $i$ must in fact be an isomorphism.', FALSE ), ( From ffeb572edddcfdc79de723a0711f405437045d5b Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 00:36:43 -0400 Subject: [PATCH 03/18] Core-thin categories also have effective congruences --- .../005_implications/004_morphism-behavior-implications.sql | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql b/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql index e406791e..b4c70755 100644 --- a/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql +++ b/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql @@ -177,8 +177,8 @@ VALUES ( 'core-hin_quotients', '["core-thin"]', - '["quotients of congruences"]', - 'If $p_1, p_2 : E \rightrightarrows X$ is a congruence, the symmetry morphism $s : E \to E$ is an automorphism of $E$, hence equal to $\mathrm{id}_E$ by assumption. But then $p_1 = p_2 \circ s = p_2$, and simply $\mathrm{id}_X$ is a coequalizer.', + '["quotients of congruences", "effective congruences"]', + 'If $p_1, p_2 : E \rightrightarrows X$ is a congruence, the symmetry morphism $s : E \to E$ is an automorphism of $E$, hence equal to $\mathrm{id}_E$ by assumption. But then $p_1 = p_2 \circ s = p_2$, and simply $\mathrm{id}_X$ is a coequalizer. Also, for the reflexivity morphism $r : X \to E$, we have $p_1 \circ r = \mathrm{id}$. For the reverse composition, $p_1 \circ r \circ p_1 = p_1 \circ \mathrm{id}$ and $p_2 \circ r \circ p_1 = p_2 \circ \mathrm{id}$, so since $p_1, p_2$ are jointly monomorphic, we get $r \circ p_1 = \mathrm{id}$. Therefore, $p_1 = p_2$ is an isomorphism, so $E$ is the kernel pair of $\mathrm{id}_X$.', FALSE ), ( @@ -187,4 +187,4 @@ VALUES '["skeletal", "core-thin"]', 'This is trivial.', TRUE -); \ No newline at end of file +); From 5601649a1bbbda5e35127f787bb3b62a7282e820 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 09:35:13 -0400 Subject: [PATCH 04/18] Make a few easy assignments --- .../002_limits-colimits-existence.sql | 4 ++-- .../catdat/data/004_property-assignments/FS.sql | 6 ++++++ .../data/004_property-assignments/FinGrp.sql | 8 +++++++- .../data/004_property-assignments/FinSet.sql | 6 ++++++ .../data/004_property-assignments/FreeAb.sql | 8 +++++++- .../catdat/data/004_property-assignments/Met.sql | 8 +++++++- .../catdat/data/004_property-assignments/Rel.sql | 6 ++++++ .../catdat/data/004_property-assignments/Set.sql | 6 ++++++ .../data/004_property-assignments/Set_c.sql | 12 ++++++++++++ .../data/004_property-assignments/Set_f.sql | 8 +++++++- .../data/004_property-assignments/Set_pointed.sql | 6 ++++++ .../data/004_property-assignments/Setne.sql | 15 ++++++++++++++- .../data/004_property-assignments/SetxSet.sql | 6 ++++++ .../catdat/data/004_property-assignments/Z.sql | 14 +++++++++++++- ...001_limits-colimits-existence-implications.sql | 7 +++++++ 15 files changed, 112 insertions(+), 8 deletions(-) diff --git a/databases/catdat/data/003_properties/002_limits-colimits-existence.sql b/databases/catdat/data/003_properties/002_limits-colimits-existence.sql index 4d269ec3..30b9d1bc 100644 --- a/databases/catdat/data/003_properties/002_limits-colimits-existence.sql +++ b/databases/catdat/data/003_properties/002_limits-colimits-existence.sql @@ -346,7 +346,7 @@ VALUES ( 'effective congruences', 'has', - 'A congruence $f, g : E \rightrightarrows X$ (see definition here) is effective if it is the kernel pair of some morphism, i.e. if there is a morphism $h : X \to Y$ such that we have a cartesian square [put diagram here]. This property asserts that every congruence in the category is effective.', + 'A congruence $f, g : E \rightrightarrows X$ (see definition here) is effective if it is the kernel pair of some morphism, i.e. if there is a morphism $h : X \to Y$ such that we have a cartesian square [put diagram here]. A category has effective congruences if every congruence in the category is effective.', 'https://ncatlab.org/nlab/show/congruence', 'effective cocongruences', TRUE @@ -354,7 +354,7 @@ VALUES ( 'effective cocongruences', 'has', - 'A cocongruence $f, g : X \rightrightarrows E$ (see definition here) is effective if it is the cokernel pair of some morphism, i.e. if there is a morphism $h : Y \to X$ such that we have a cocartesian square [put diagram here]. This property asserts that every cocongruence in the category is effective.', + 'A cocongruence $f, g : X \rightrightarrows E$ (see definition here) is effective if it is the cokernel pair of some morphism, i.e. if there is a morphism $h : Y \to X$ such that we have a cocartesian square [put diagram here]. A category has effective cocongruences if every cocongruence in the category is effective.', NULL, 'effective congruences', TRUE diff --git a/databases/catdat/data/004_property-assignments/FS.sql b/databases/catdat/data/004_property-assignments/FS.sql index 3a10f8fd..9ed8b0e5 100644 --- a/databases/catdat/data/004_property-assignments/FS.sql +++ b/databases/catdat/data/004_property-assignments/FS.sql @@ -65,6 +65,12 @@ VALUES TRUE, 'The empty set and a singleton give a multi-terminal object.' ), +( + 'FS', + 'effective congruences', + TRUE, + 'In fact, for any congruence $E \rightrightarrows X$ in $\mathbf{FS}$, we must have the reflexivity morphism $X \to E$ surjective. That implies $E$ is the kernel pair of $\mathrm{id}_X$.' +), ( 'FS', 'small', diff --git a/databases/catdat/data/004_property-assignments/FinGrp.sql b/databases/catdat/data/004_property-assignments/FinGrp.sql index bf80a30b..a3e49cb4 100644 --- a/databases/catdat/data/004_property-assignments/FinGrp.sql +++ b/databases/catdat/data/004_property-assignments/FinGrp.sql @@ -65,6 +65,12 @@ VALUES TRUE, 'The proof works exactly as for the category of finite sets.' ), +( + 'FinGrp', + 'effective congruences', + TRUE, + 'For a finite kernel pair $E$ of $h : X \to Z$ where $X$ is a finite group, replacing $Z$ with the image of $h$ gives the same kernel pair.' +), ( 'FinGrp', 'normal', @@ -118,4 +124,4 @@ VALUES 'countable', FALSE, 'This is trivial.' -); \ No newline at end of file +); diff --git a/databases/catdat/data/004_property-assignments/FinSet.sql b/databases/catdat/data/004_property-assignments/FinSet.sql index 3b91f482..4e532c7a 100644 --- a/databases/catdat/data/004_property-assignments/FinSet.sql +++ b/databases/catdat/data/004_property-assignments/FinSet.sql @@ -47,6 +47,12 @@ VALUES 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', + 'effective cocongruences', + TRUE, + 'For a finite cokernel pair $E$ of $h : Z \to X$ where $X$ is a finite set, replacing $h$ with the inclusion map of its image gives the same cokernel pair.' +), ( 'FinSet', 'small', diff --git a/databases/catdat/data/004_property-assignments/FreeAb.sql b/databases/catdat/data/004_property-assignments/FreeAb.sql index 3be5d2a7..b8f8d8cb 100644 --- a/databases/catdat/data/004_property-assignments/FreeAb.sql +++ b/databases/catdat/data/004_property-assignments/FreeAb.sql @@ -61,6 +61,12 @@ VALUES (1) If $f : A \to B$ is surjective, it is the coequalizer of $A \times_B A \rightrightarrows A$ in $\mathbf{Ab}$. Since $A \times_B A$ is free abelian, $f$ is also an coequalizer in $\mathbf{FreeAb}$.
(2) If $f : A \to B$ is a regular epimorphism in $\mathbf{FreeAb}$, consider the factorization $f = i \circ p$ as above. Since $f$ is an extremal epimorphism, $i$ must be an isomorphism, so that $f$ is surjective.' ), +( + 'FreeAb', + 'effective cocongruences', + TRUE, + 'Since $\mathbf{Ab}$ is abelian, it has effective cocongruences. Now, suppose a cocongruence $X \rightrightarrows E$ is the cokernel pair of $h : Z \to X$, where $X$ and $E$ are free abelian groups. If we find a surjective map $g : F \to Z$, then $h \circ g$ has the same cokernel pair as $h$, and $h \circ g$ is a morphism in $\mathbf{FreeAb}$.' +), ( 'FreeAb', 'countable powers', @@ -84,4 +90,4 @@ VALUES 'sequential colimits', FALSE, 'See MO/509715.' -); \ No newline at end of file +); diff --git a/databases/catdat/data/004_property-assignments/Met.sql b/databases/catdat/data/004_property-assignments/Met.sql index de319d56..0434c286 100644 --- a/databases/catdat/data/004_property-assignments/Met.sql +++ b/databases/catdat/data/004_property-assignments/Met.sql @@ -144,4 +144,10 @@ VALUES 'If $(N,z,s)$ is a natural numbers object in $\mathbf{Met}$, then

$1 \xrightarrow{z} N \xleftarrow{s} N$

is a coproduct cocone by Johnstone, Part A, Lemma 2.5.5. Since there is a map $1 \to N$, we have $N \neq \varnothing$. However, the coproduct of two non-empty metric spaces does not exist, see MSE/1778408.' -); \ No newline at end of file +), +( + 'Met', + 'effective congruences', + FALSE, + 'Any kernel pair of $h : X \to Z$ in $\mathbf{Met}$ corresponds to a closed subset of $X\times X$. However, there are plenty of non-closed congruences, such as $\Delta \cup (\mathbb{Q} \times \mathbb{Q}) \subseteq \mathbb{R} \times \mathbb{R}$ with the subspace metric.' +); diff --git a/databases/catdat/data/004_property-assignments/Rel.sql b/databases/catdat/data/004_property-assignments/Rel.sql index cf6e361d..1210ab25 100644 --- a/databases/catdat/data/004_property-assignments/Rel.sql +++ b/databases/catdat/data/004_property-assignments/Rel.sql @@ -71,6 +71,12 @@ VALUES TRUE, 'A proof can be found here.' ), +( + 'Rel', + 'effective congruences', + TRUE, + 'A proof can be found here.' +), ( 'Rel', 'preadditive', diff --git a/databases/catdat/data/004_property-assignments/Set.sql b/databases/catdat/data/004_property-assignments/Set.sql index b587ee04..f6d0ea82 100644 --- a/databases/catdat/data/004_property-assignments/Set.sql +++ b/databases/catdat/data/004_property-assignments/Set.sql @@ -29,6 +29,12 @@ VALUES TRUE, 'Use the empty algebraic theory.' ), +( + 'Set', + 'effective cocongruences', + TRUE, + 'Since the contravariant powerset functor $\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$ is monadic, we conclude that $\mathbf{Set}^{\mathrm{op}}$ is Barr-exact and therefore has effective congruences.' +), ( 'Set', 'skeletal', diff --git a/databases/catdat/data/004_property-assignments/Set_c.sql b/databases/catdat/data/004_property-assignments/Set_c.sql index bfe7298d..f2ac9f3f 100644 --- a/databases/catdat/data/004_property-assignments/Set_c.sql +++ b/databases/catdat/data/004_property-assignments/Set_c.sql @@ -77,6 +77,18 @@ VALUES TRUE, 'This is because $\{0,1\}$ is a subobject classifier in $\mathbf{Set}$, which is countable, and the monomorphisms coincide.' ), +( + 'Set_c', + 'effective congruences', + TRUE, + 'For a countable kernel pair $E$ of $h : X \to Z$ where $X$ is countable, replacing $Z$ with the image of $h$ gives the same kernel pair.' +), +( + 'Set_c', + 'effective cocongruences', + TRUE, + 'For a countable cokernel pair $E$ of $h : Z \to X$ where $X$ is countable, replacing $h$ with the inclusion map of its image gives the same cokernel pair.' +), ( 'Set_c', 'small', diff --git a/databases/catdat/data/004_property-assignments/Set_f.sql b/databases/catdat/data/004_property-assignments/Set_f.sql index e884cd5c..134a0b9d 100644 --- a/databases/catdat/data/004_property-assignments/Set_f.sql +++ b/databases/catdat/data/004_property-assignments/Set_f.sql @@ -41,6 +41,12 @@ VALUES TRUE, 'A congruence on a set $X$ in $\mathbf{Set}_\mathrm{f}$ is the same as an equivalence relation $R$ on $X$ whose equivalence classes are finite. In that case, the usual quotient map $p : X \to X/R$ is finite-to-one. Moreover, if $h : X/R \to Y$ is a map such that $h \circ p : X \to Y$ is finite-to-one, then $h$ is finite-to-one as well because $h^*(\{y\}) \subseteq p^*((h \circ p)^*(\{y\}))$ for all $y \in Y$. Therefore, $p$ is also the quotient in $\mathbf{Set}_\mathrm{f}$.' ), +( + 'Set_f', + 'effective congruences', + TRUE, + 'If $E \rightrightarrows X$ is the kernel pair of $h : X \to Z$ and both maps $E \to X$ are finite-to-one, then that means the equivalence classes of $E$ are finite. Thus, necessarily $h$ was finite-to-one also.' +), ( 'Set_f', 'locally cartesian closed', @@ -132,4 +138,4 @@ VALUES FALSE, 'We will prove that the family of singleton sets $(1)_{n \in \mathbb{N}}$ has no multi-coproduct, generalizing the proof that the family does not have a coproduct given above. A cocone is just a map of sets $f : \mathbb{N} \to X$. A morphism from $f : \mathbb{N} \to X$ to $g : \mathbb{N} \to Y$ is a finite-to-one map $h : X \to Y$ with $g = h \circ f$. This describes the category of cocones, and we need to show that it has no multi-initial object. To this end, we claim that the connected component of the unique map $! : \mathbb{N} \to 1$ consists precisely of the maps $f : \mathbb{N} \to X$ with finite image. Once that is established, we can recycle the proof for missing coproducts since there we have only used finite codomains.
Let $g = h \circ f$ be as above. If $\mathrm{im}(f)$ is finite, then $\mathrm{im}(g) = h_*(\mathrm{im}(f))$ is finite as well. Conversely, if $\mathrm{im}(g)$ is finite, then $\mathrm{im}(f) \subseteq \bigcup_{y \in \mathrm{im}(g)} h^*(\{y\})$ is finite as well. This shows that the connected component of $!$ is contained in the collection of maps with finite image. Conversely, if $f$ has finite image, then there is a morphism from the corestriction $f'' : \mathbb{N} \to \mathrm{im}(f)$ to $f$, and also a morphism from $f''$ to $!$. This proves the remaining inclusion.' -); \ No newline at end of file +); diff --git a/databases/catdat/data/004_property-assignments/Set_pointed.sql b/databases/catdat/data/004_property-assignments/Set_pointed.sql index 15bec3c1..d6e32b8d 100644 --- a/databases/catdat/data/004_property-assignments/Set_pointed.sql +++ b/databases/catdat/data/004_property-assignments/Set_pointed.sql @@ -67,6 +67,12 @@ VALUES TRUE, 'The coproduct (wedge sum) of a family of pointed sets $(X_i)_{i \in I}$ can be realized as the subset of $\prod_{i \in I} X_i$ consisting of those tuples $x$ such that $x_i = 0$ for all but (at most) one index.' ), +( + 'Set*', + 'effective cocongruences', + TRUE, + 'We have that $\mathbf{Set}_*^{\mathrm{op}}$ is a slice category of $\mathbf{Set}^{\mathrm{op}}$, which in turn is monadic over $\mathbf{Set}$. Therefore, $\mathbf{Set}_*^{\mathrm{op}}$ is Barr-exact, and in particular it has effective congruences.' +), ( 'Set*', 'skeletal', diff --git a/databases/catdat/data/004_property-assignments/Setne.sql b/databases/catdat/data/004_property-assignments/Setne.sql index 6a4954f2..e348cb10 100644 --- a/databases/catdat/data/004_property-assignments/Setne.sql +++ b/databases/catdat/data/004_property-assignments/Setne.sql @@ -114,6 +114,12 @@ VALUES TRUE, 'This follows from this lemma applied to the forgetful functor to $\mathbf{Set}$.' ), +( + 'Setne', + 'effective congruences', + TRUE, + 'If a congruence $E \rightrightarrows X$ is the kernel pair of $h : X \to Z$, with both $E$ and $X$ non-empty, then certainly $Z$ must also be non-empty.' +), ( 'Setne', 'sequential limits', @@ -137,4 +143,11 @@ VALUES 'coquotients of cocongruences', FALSE, 'The two maps $\{0\} \rightrightarrows \{0,1\}$ form a cocongruence on $\{0\}$ — namely the cofull cocongruence on $\{0\}$ — but they do not have an equalizer.' -) +), +( + 'Setne', + 'effective cocongruences', + FALSE, + 'The two maps $\{0\} \rightrightarrows \{0,1\}$ form a cocongruence on $\{0\}$ — namely the cofull cocongruence on $\{0\}$ — but there is no map $Z \to \{0\}$ making the required commutative diagram, much less a cocartesian square.' +); + diff --git a/databases/catdat/data/004_property-assignments/SetxSet.sql b/databases/catdat/data/004_property-assignments/SetxSet.sql index 78eb6657..56871c7c 100644 --- a/databases/catdat/data/004_property-assignments/SetxSet.sql +++ b/databases/catdat/data/004_property-assignments/SetxSet.sql @@ -23,6 +23,12 @@ VALUES TRUE, 'Take the two-sorted algebraic theory with no operations and no equations.' ), +( + 'SetxSet', + 'effective cocongruences', + TRUE, + 'Suppose we have a cocongruence $X \rightrightarrows E$ in $\mathbf{Set} \times \mathbf{Set}$. Then each component is a cocongruence, so they are the kernel pairs of some maps $Z_1\to X_1$, $Z_2 \to X_2$. Then $E$ is the cokernel pair of $(Z_1, Z_2) \to (X_1, X_2)$.' +), ( 'SetxSet', 'skeletal', diff --git a/databases/catdat/data/004_property-assignments/Z.sql b/databases/catdat/data/004_property-assignments/Z.sql index dedd72fb..2d3ef0ba 100644 --- a/databases/catdat/data/004_property-assignments/Z.sql +++ b/databases/catdat/data/004_property-assignments/Z.sql @@ -53,6 +53,18 @@ VALUES TRUE, 'This follows immediately from the fact for $\mathbf{Set}$.' ), +( + 'Z', + 'effective congruences', + TRUE, + 'If we have a congruence $E \rightrightarrows X$ in $[\mathbf{CRing}, \mathbf{Set}]$, then each image at a commutative ring gives a congruence in $\mathbf{Set}$. Defining $Y$ pointwise to be the quotient of this congruence, we get a morphism of functors $h : X \to Y$ whose kernel pair is $E$.' +), +( + 'Z', + 'effective cocongruences', + TRUE, + 'If we have a congruence $X\rightrightarrows E$ in $[\mathbf{CRing}, \mathbf{Set}]$, then each image at a commutative ring gives a cocongruence in $\mathbf{Set}$. Defining $Y$ pointwise to be the equalizer of the pair, we get a morphism of functors $h : Y \to X$ whose cokernel pair is $E$.' +), ( 'Z', 'locally essentially small', @@ -100,4 +112,4 @@ VALUES 'cofiltered-limit-stable epimorphisms', FALSE, 'We already know that $\mathbf{Set}$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\mathbf{Set} \to [\mathbf{CRing}, \mathbf{Set}]$ that maps a set to its constant functor.' -); \ No newline at end of file +); diff --git a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql index 592ddc66..86cbba3e 100644 --- a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql +++ b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql @@ -76,6 +76,13 @@ VALUES 'In a thin category, any congruence pair is a reflexive fork, and thus consists of equal isomorphisms. Therefore, the congruence is the kernel pair of the identity morphism on the target.', FALSE ), +( + 'left_cancellative_effective_congruences', + '["left cancellative"]', + '["effective congruences"]', + 'For any congruence $f, g : E \rightrightarrows X$, if $r$ is the reflexivity morphism we have $r\circ f = r\circ g = \mathrm{id}_X$, so $f = g$. As here, that implies that $E$ is the kernel pair of $\mathrm{id}_X$.', + FALSE +), ( 'products_consequences', '["products"]', From 885bbc30b98b18c766718a99d9c50638dc08937f Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 13:22:04 -0400 Subject: [PATCH 05/18] Fix and expand proofs of relations with cancellative properties --- .../001_limits-colimits-existence-implications.sql | 9 ++++++++- 1 file changed, 8 insertions(+), 1 deletion(-) diff --git a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql index 86cbba3e..35411925 100644 --- a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql +++ b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql @@ -80,7 +80,14 @@ VALUES 'left_cancellative_effective_congruences', '["left cancellative"]', '["effective congruences"]', - 'For any congruence $f, g : E \rightrightarrows X$, if $r$ is the reflexivity morphism we have $r\circ f = r\circ g = \mathrm{id}_X$, so $f = g$. As here, that implies that $E$ is the kernel pair of $\mathrm{id}_X$.', + 'For any congruence $f, g : E \rightrightarrows X$, if $r$ is the reflexivity morphism we have $r$ is both an epimorphism and a split monomorphism, so it is an isomorphism. Therefore, $f\circ r = g\circ r = \mathrm{id}_X$ implies $f = g$. As here, that implies that $E$ is the kernel pair of $\mathrm{id}_X$.', + FALSE +), +( + 'right_cancellative_effective_congruences', + '["right cancellative"]', + '["effective congruences"]', + 'For any congruence $f, g : E \rightrightarrows X$, if $r$ is the reflexivity morphism we have $f\circ r = g\circ r = \mathrm{id}_X$, so $f = g$. As here, that implies that $E$ is the kernel pair of $\mathrm{id}_X$.', FALSE ), ( From 3529b1f5e90d83f34be91d11219134550651d8e1 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 13:30:53 -0400 Subject: [PATCH 06/18] Add proof Top_* does not have effective congruences, based on the fact Top does not --- .../catdat/data/004_property-assignments/Top_pointed.sql | 9 ++++++--- 1 file changed, 6 insertions(+), 3 deletions(-) diff --git a/databases/catdat/data/004_property-assignments/Top_pointed.sql b/databases/catdat/data/004_property-assignments/Top_pointed.sql index 644cfa6b..9c89323e 100644 --- a/databases/catdat/data/004_property-assignments/Top_pointed.sql +++ b/databases/catdat/data/004_property-assignments/Top_pointed.sql @@ -167,7 +167,10 @@ VALUES 'cofiltered-limit-stable epimorphisms', FALSE, 'We already know that $\mathbf{Set}_*$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\mathbf{Set}_* \to \mathbf{Top}_*$ that equips a pointed set with the indiscrete topology.' +), +( + 'Top*', + 'effective congruences', + FALSE, + 'Suppose that $\mathbf{Top}_*$ had effective congruences. Then for any congruence $E \rightrightarrows X$ in $\mathbf{Top}$, we can expand it to a congruence $E + \{*\} \rightrightarrows X + \{*\}$ in $\mathbf{Top}_*$. If $E + \{*\}$ is the kernel pair of $h : X + \{*\} \to Z$, then $E$ is the kernel pair of $h$ restricted to $X$. This contradicts the fact that $\mathbf{Top}$ does not have effective congruences.' ); - - - From 172f39dfbce4845610585bb270cabb1854c5e27f Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 14:55:19 -0400 Subject: [PATCH 07/18] Add some assignments based on well-known equivalence relations which happen to be representable but not effective --- databases/catdat/data/004_property-assignments/Man.sql | 6 ++++++ databases/catdat/data/004_property-assignments/Meas.sql | 6 ++++++ databases/catdat/data/004_property-assignments/PMet.sql | 6 ++++++ databases/catdat/data/004_property-assignments/Prost.sql | 6 ++++++ databases/catdat/data/004_property-assignments/Top.sql | 6 ++++++ .../catdat/data/004_property-assignments/Top_pointed.sql | 6 ++++++ 6 files changed, 36 insertions(+) diff --git a/databases/catdat/data/004_property-assignments/Man.sql b/databases/catdat/data/004_property-assignments/Man.sql index 1fda6e42..2c974990 100644 --- a/databases/catdat/data/004_property-assignments/Man.sql +++ b/databases/catdat/data/004_property-assignments/Man.sql @@ -80,6 +80,12 @@ VALUES Because $r \circ (p \circ i_1) : X \to X$ is the identity, the image of $p \circ i_1$ is the equalizer of $\mathrm{id}_E$ and $(p \circ i_1) \circ r$, hence closed. Likewise, the image of $p \circ i_2$ is closed. Thus, the image of $p$, which is the union of these images, is closed.
Now, since the pushforward maps of tangent spaces compose to the identity, we see that $p$ must be a local immersion and $r$ must be a submersion. Also, since the fibers of $r$ have one or two points each, we see that the dimension of $E$ must locally be the same as the dimension of $X$. This implies that in fact $p$ and $r$ are local diffeomorphisms. Therefore, the cardinality of the fiber of $r$ is locally constant. Thus, if $U$ is the subset of $X$ where $r$ has fiber of a single point, with the subspace topology, then $U$ is a clopen submanifold of $X$ which serves as the equalizer of $p \circ i_1$ and $p \circ i_2$.' ), +( + 'Man', + 'effective cocongruences', + TRUE, + 'From the proof that $\mathbf{Man}$ has coquotients of cocongruences, we know that for any cocongruence $X \rightrightarrows E$, there is a clopen submanifold $U$ of $X$ such that the fibers of $r : E \twoheadrightarrow X$ have one point on $U$, and two points on $X \setminus U$. Therefore, $E$ is the cokernel pair of the inclusion map $U \hookrightarrow X$.' +), ( 'Man', 'small', diff --git a/databases/catdat/data/004_property-assignments/Meas.sql b/databases/catdat/data/004_property-assignments/Meas.sql index 37c9e8f9..8331d58a 100644 --- a/databases/catdat/data/004_property-assignments/Meas.sql +++ b/databases/catdat/data/004_property-assignments/Meas.sql @@ -100,4 +100,10 @@ VALUES 'cofiltered-limit-stable epimorphisms', FALSE, 'We already know that $\mathbf{Set}$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\mathbf{Set} \to \mathbf{Meas}$ which equips a set with the trivial $\sigma$-algebra.' +), +( + 'Meas', + 'effective cocongruences', + FALSE, + 'The proof is similar to the one for $\mathbf{Top}$: Use the trivial $\sigma$-algebra on a two-point space.' ); diff --git a/databases/catdat/data/004_property-assignments/PMet.sql b/databases/catdat/data/004_property-assignments/PMet.sql index 3756102e..9a8208b2 100644 --- a/databases/catdat/data/004_property-assignments/PMet.sql +++ b/databases/catdat/data/004_property-assignments/PMet.sql @@ -138,4 +138,10 @@ VALUES 'If $(N,z,s)$ is a natural numbers object in $\mathbf{PMet}$, then

$1 \xrightarrow{z} N \xleftarrow{s} N$

is a coproduct cocone by Johnstone, Part A, Lemma 2.5.5. Since there is a map $1 \to N$, we have $N \neq \varnothing$. However, the coproduct of two non-empty pseudo-metric spaces does not exist, see MSE/1778408.' +), +( + 'PMet', + 'effective cocongruences', + FALSE, + 'The proof is similar to the one for $\mathbf{Top}$: Use the two-point space with the zero metric, which represents the functor taking a pseudo-metric space to the pairs of points with $d(x,y) = 0$. In this case, once you conclude $Z = \varnothing$, the map $h : Z \to 1$ does not have any cokernel pair, since that would have to be a coproduct $1+1$.' ); diff --git a/databases/catdat/data/004_property-assignments/Prost.sql b/databases/catdat/data/004_property-assignments/Prost.sql index 4a2d9a52..9e0ebd2d 100644 --- a/databases/catdat/data/004_property-assignments/Prost.sql +++ b/databases/catdat/data/004_property-assignments/Prost.sql @@ -100,4 +100,10 @@ VALUES 'cofiltered-limit-stable epimorphisms', FALSE, 'We know that $\mathbf{Set}$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the functor $\mathbf{Set} \to \mathbf{Prost}$ that equips a set with the chaotic preorder.' +), +( + 'Prost', + 'effective cocongruences', + FALSE, + 'Consider the proset $E := \{ a, b \}$ with the chaotic preorder. This represents the functor which sends a proset to the pairs of elements $x,y$ with $x \le y$ and $y \le x$. Therefore, it defines a cocongruence $1 \rightrightarrows E$, where the maps are the two possible functions. However, this cannot be effective: for any map $h : Z \to 1$ which equalizes the two functions, $Z$ must be empty. But that means the cokernel pair of $h$ is the two-element proset with the trivial preorder.' ); diff --git a/databases/catdat/data/004_property-assignments/Top.sql b/databases/catdat/data/004_property-assignments/Top.sql index dcd4464d..65beca91 100644 --- a/databases/catdat/data/004_property-assignments/Top.sql +++ b/databases/catdat/data/004_property-assignments/Top.sql @@ -130,4 +130,10 @@ VALUES 'cofiltered-limit-stable epimorphisms', FALSE, 'We already know that $\mathbf{Set}$ does not have this property. Now apply the contrapositive of the dual of this lemma to the functor $\mathbf{Set} \to \mathbf{Top}$ which equips a set with the indiscrete topology.' +), +( + 'Top', + 'effective cocongruences', + FALSE, + 'Consider the indiscrete topological space $I$ on two points. This represents the functor which takes a topological space $X$ to the pairs of indistinguishable points of $X$. Therefore, we get a cocongruence $1 \rightrightarrows I$, where the maps are the two possible functions. However, this cannot be effective: if we have $h : Z\to 1$ which equalizes the two maps, then $Z$ must be empty. But that means the cokernel pair of $h$ is the discrete space on two points.' ); diff --git a/databases/catdat/data/004_property-assignments/Top_pointed.sql b/databases/catdat/data/004_property-assignments/Top_pointed.sql index 9c89323e..33360aad 100644 --- a/databases/catdat/data/004_property-assignments/Top_pointed.sql +++ b/databases/catdat/data/004_property-assignments/Top_pointed.sql @@ -173,4 +173,10 @@ VALUES 'effective congruences', FALSE, 'Suppose that $\mathbf{Top}_*$ had effective congruences. Then for any congruence $E \rightrightarrows X$ in $\mathbf{Top}$, we can expand it to a congruence $E + \{*\} \rightrightarrows X + \{*\}$ in $\mathbf{Top}_*$. If $E + \{*\}$ is the kernel pair of $h : X + \{*\} \to Z$, then $E$ is the kernel pair of $h$ restricted to $X$. This contradicts the fact that $\mathbf{Top}$ does not have effective congruences.' +), +( + 'Top*', + 'effective cocongruences', + FALSE, + 'Consider the pointed topological space $I := \{ *, a, b \}$ with topology $\{ \varnothing, \{ * \}, \{ a, b \}, \{ *, a, b \} \}$. This represents the functor which sends a pointed topological space $X$ to the pairs of indistinguishable points of $X$. Therefore, we get a cocongruence $\{ *, a \} \rightrightarrows I$ on the discrete space $\{ *, a \}$, where the maps are the two possible pointed functions. However, this cannot be effective: if we have $h : Z \to \{ *, a \}$ which equalizes the cocongruence, then $Z$ must be the singleton pointed space. But that means the cokernel pair of $h$ is the discrete space on $\{ *, a, b \}$.' ); From 2b0a2ea5a698f6b2d5a139868e6cab81b6e71111 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 15:25:52 -0400 Subject: [PATCH 08/18] Update expected results --- databases/catdat/scripts/expected-data/Ab.json | 2 ++ databases/catdat/scripts/expected-data/Set.json | 2 ++ databases/catdat/scripts/expected-data/Top.json | 4 +++- 3 files changed, 7 insertions(+), 1 deletion(-) diff --git a/databases/catdat/scripts/expected-data/Ab.json b/databases/catdat/scripts/expected-data/Ab.json index b511fc1b..efca47d0 100644 --- a/databases/catdat/scripts/expected-data/Ab.json +++ b/databases/catdat/scripts/expected-data/Ab.json @@ -104,6 +104,8 @@ "filtered-colimit-stable monomorphisms": true, "quotients of congruences": true, "coquotients of cocongruences": true, + "effective congruences": true, + "effective cocongruences": true, "cartesian closed": false, "locally cartesian closed": false, diff --git a/databases/catdat/scripts/expected-data/Set.json b/databases/catdat/scripts/expected-data/Set.json index 44db92c4..6b976962 100644 --- a/databases/catdat/scripts/expected-data/Set.json +++ b/databases/catdat/scripts/expected-data/Set.json @@ -99,6 +99,8 @@ "filtered-colimit-stable monomorphisms": true, "quotients of congruences": true, "coquotients of cocongruences": true, + "effective congruences": true, + "effective cocongruences": true, "Grothendieck abelian": false, "Malcev": false, diff --git a/databases/catdat/scripts/expected-data/Top.json b/databases/catdat/scripts/expected-data/Top.json index e70a439c..c95e68aa 100644 --- a/databases/catdat/scripts/expected-data/Top.json +++ b/databases/catdat/scripts/expected-data/Top.json @@ -153,5 +153,7 @@ "cofiltered-limit-stable epimorphisms": false, "exact cofiltered limits": false, "gaunt": false, - "core-thin": false + "core-thin": false, + "effective congruences": false, + "effective cocongruences": false } From 19acec93a8e6b4094fa7e2136c7a36ec4e44f9b8 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 18:29:42 -0400 Subject: [PATCH 09/18] Update to use new display mode and amscd rendering --- .../002_limits-colimits-existence.sql | 20 +++++++++++++++++-- 1 file changed, 18 insertions(+), 2 deletions(-) diff --git a/databases/catdat/data/003_properties/002_limits-colimits-existence.sql b/databases/catdat/data/003_properties/002_limits-colimits-existence.sql index 30b9d1bc..2497db35 100644 --- a/databases/catdat/data/003_properties/002_limits-colimits-existence.sql +++ b/databases/catdat/data/003_properties/002_limits-colimits-existence.sql @@ -346,7 +346,15 @@ VALUES ( 'effective congruences', 'has', - 'A congruence $f, g : E \rightrightarrows X$ (see definition here) is effective if it is the kernel pair of some morphism, i.e. if there is a morphism $h : X \to Y$ such that we have a cartesian square [put diagram here]. A category has effective congruences if every congruence in the category is effective.', + 'A congruence $f, g : E \rightrightarrows X$ (see definition here) is effective if it is the kernel pair of some morphism, i.e. if there is a morphism $h : X \to Y$ such that we have a cartesian square + $$ + \begin{CD} + E @> f >> X \\ + @V g VV @V h VV \\ + X @> h >> Y. + \end{CD} + $$ + A category has effective congruences if every congruence in the category is effective.', 'https://ncatlab.org/nlab/show/congruence', 'effective cocongruences', TRUE @@ -354,7 +362,15 @@ VALUES ( 'effective cocongruences', 'has', - 'A cocongruence $f, g : X \rightrightarrows E$ (see definition here) is effective if it is the cokernel pair of some morphism, i.e. if there is a morphism $h : Y \to X$ such that we have a cocartesian square [put diagram here]. A category has effective cocongruences if every cocongruence in the category is effective.', + 'A cocongruence $f, g : X \rightrightarrows E$ (see definition here) is effective if it is the cokernel pair of some morphism, i.e. if there is a morphism $h : Y \to X$ such that we have a cocartesian square + $$ + \begin{CD} + Y @> h >> X \\ + @V h VV @V f VV \\ + X @> g >> E. + \end{CD} + $$ + A category has effective cocongruences if every cocongruence in the category is effective.', NULL, 'effective congruences', TRUE From c170f55729aa717cd72d7be7228b4540a7f1ca08 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 20:04:55 -0400 Subject: [PATCH 10/18] Reorganize proofs of implications from cancellativity --- .../001_limits-colimits-existence-implications.sql | 14 -------------- .../004_morphism-behavior-implications.sql | 4 ++-- 2 files changed, 2 insertions(+), 16 deletions(-) diff --git a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql index fdec8b48..f236f413 100644 --- a/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql +++ b/databases/catdat/data/005_implications/001_limits-colimits-existence-implications.sql @@ -76,20 +76,6 @@ VALUES 'In a thin category, any congruence pair is a reflexive fork, and thus consists of equal isomorphisms. Therefore, the congruence is the kernel pair of the identity morphism on the target.', FALSE ), -( - 'left_cancellative_effective_congruences', - '["left cancellative"]', - '["effective congruences"]', - 'For any congruence $f, g : E \rightrightarrows X$, if $r$ is the reflexivity morphism we have $r$ is both an epimorphism and a split monomorphism, so it is an isomorphism. Therefore, $f\circ r = g\circ r = \mathrm{id}_X$ implies $f = g$. As here, that implies that $E$ is the kernel pair of $\mathrm{id}_X$.', - FALSE -), -( - 'right_cancellative_effective_congruences', - '["right cancellative"]', - '["effective congruences"]', - 'For any congruence $f, g : E \rightrightarrows X$, if $r$ is the reflexivity morphism we have $f\circ r = g\circ r = \mathrm{id}_X$, so $f = g$. As here, that implies that $E$ is the kernel pair of $\mathrm{id}_X$.', - FALSE -), ( 'products_consequences', '["products"]', diff --git a/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql b/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql index 28f49344..4cf98ce6 100644 --- a/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql +++ b/databases/catdat/data/005_implications/004_morphism-behavior-implications.sql @@ -37,8 +37,8 @@ VALUES ( 'reflexive_pair_trivial', '["left cancellative"]', - '["reflexive coequalizers", "coreflexive equalizers"]', - 'Any parallel pair of morphisms with a common section (or retraction) must be a pair of equal isomorphisms.', + '["reflexive coequalizers", "coreflexive equalizers", "effective congruences", "effective cocongruences"]', + 'Any parallel pair of morphisms with a common section (or retraction) must be a pair of equal isomorphisms. In particular, they are the kernel pair of the identity morphism on the target, and the cokernel pair of the identity morphism on the source.', FALSE ), ( From cb80c4f3848aaa16b604c24e198ba8fd68c38476 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Thu, 23 Apr 2026 21:38:42 -0400 Subject: [PATCH 11/18] Elementary topoi have effective cocongruences --- .../005_implications/008_topos-theory-implications.sql | 9 +++++++++ 1 file changed, 9 insertions(+) diff --git a/databases/catdat/data/005_implications/008_topos-theory-implications.sql b/databases/catdat/data/005_implications/008_topos-theory-implications.sql index 2a694e13..1ae0fc24 100644 --- a/databases/catdat/data/005_implications/008_topos-theory-implications.sql +++ b/databases/catdat/data/005_implications/008_topos-theory-implications.sql @@ -174,6 +174,15 @@ VALUES 'We assume that every congruence is effective, and the regularity condition implies that every effective congruence has a quotient.', FALSE ), +( + 'regular_epiregular_extensive_consequences', + '["regular", "epi-regular", "extensive"]', + '["effective cocongruences", "co-Malcev"]', + 'Suppose we have a coreflexive corelation on $X$, $X+X \overset{p}{\twoheadrightarrow} E \overset{r}{\twoheadrightarrow} X$. Let $Y$ be the equalizer of $p\circ i_1, p\circ i_2 : X \to E$. Then by the assumptions $p$ is a regular epimorphism. By regularity, $p$ is the coequalizer of its kernel pair, which can be expressed as the equalizer of $p\circ p_1, p\circ p_2 : (X+X) \times (X+X) \to E$. By distributivity and extensitivity, it is sufficient to calculate the equalizer on each quadrant of $(X+X) \times (X+X)$. On the $(1,1)$ quadrant, this is the equalizer of $p\circ i_1\circ p_1, p\circ i_1\circ p_2$, which is isomorphic to $X$ since $p\circ i_1$ is a split monomorphism. On the $(1,2)$ quadrant, it is the equalizer of $p\circ i_1\circ p_1, p\circ i_2\circ p_2$. Since $r$ is a common section of $p\circ i_1$ and $p\circ i_2$, any generalized element of this equalizer has equal components; thus, the equalizer is isomorphic to the equalizer $Y$ of $p\circ i_1, p\circ i_2$. Similarly, on the $(2,1)$ quadrant, it is isomorphic to $Y$, and on the $(2,2)$ quadrant, it is isomorphic to $X$.
+ + Since the $X$ pieces are already equalized by the kernel pair of $p$, and the second $Y$ piece is redundant, we thus get that $p$ is the coequalizer of $i_1 \circ \mathrm{inc}_Y$ and $i_2 \circ \mathrm{inc}_Y$, which is equivalent to the cokernel pair of $\mathrm{inc}_Y$ and thus an effective cocongruence.', + FALSE +), ( 'pretopos_balanced', '["effective congruences", "extensive"]', From b0e5929c074de69ca23ce07df8f1a83c69e7f594 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Fri, 24 Apr 2026 09:18:29 -0400 Subject: [PATCH 12/18] Refine implication to multi-algebraic -> effective congruences: thanks to Yuto Kawase for the reference --- .../007_locally-presentable-implications.sql | 14 +++++++------- 1 file changed, 7 insertions(+), 7 deletions(-) diff --git a/databases/catdat/data/004_category-implications/007_locally-presentable-implications.sql b/databases/catdat/data/004_category-implications/007_locally-presentable-implications.sql index 5ef549d7..b801b7bc 100644 --- a/databases/catdat/data/004_category-implications/007_locally-presentable-implications.sql +++ b/databases/catdat/data/004_category-implications/007_locally-presentable-implications.sql @@ -153,13 +153,6 @@ VALUES 'This is trivial.', TRUE ), -( - 'locally_strongly_finitely_presentable_exact', - '["locally strongly finitely presentable"]', - '["effective congruences"]', - 'By one of the equivalent conditions, the category is monadic over a small power of $\mathbf{Set}$, and therefore is Barr-exact.', - FALSE -), ( 'generalized_variety_require_sifted_colimit', '["generalized variety"]', @@ -230,6 +223,13 @@ VALUES 'This follows from one of equivalent formulations of multi-algebraic categories.', TRUE ), +( + 'multi-algebraic_implies_effective_congruences', + '["multi-algebraic"]', + '["effective congruences"]', + 'This is Thm. 4.0 in Yves Diers, Catégories Multialgébriques or its English translation.', + FALSE +), ( 'locally-finitely-multi-presentable_stable-monos', '["locally finitely multi-presentable"]', From 1bdcfc9c4d093f5ac0c3a748dfd7fb622de3aae7 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 25 Apr 2026 13:28:12 -0400 Subject: [PATCH 13/18] Add an easy assigment --- .../catdat/data/003_category-property-assignments/Set_f.sql | 6 ++++++ 1 file changed, 6 insertions(+) diff --git a/databases/catdat/data/003_category-property-assignments/Set_f.sql b/databases/catdat/data/003_category-property-assignments/Set_f.sql index 134a0b9d..09c2c221 100644 --- a/databases/catdat/data/003_category-property-assignments/Set_f.sql +++ b/databases/catdat/data/003_category-property-assignments/Set_f.sql @@ -47,6 +47,12 @@ VALUES TRUE, 'If $E \rightrightarrows X$ is the kernel pair of $h : X \to Z$ and both maps $E \to X$ are finite-to-one, then that means the equivalence classes of $E$ are finite. Thus, necessarily $h$ was finite-to-one also.' ), +( + 'Set_f', + 'effective cocongruences', + TRUE, + 'Suppose we have a cocongruence $f, g : X \rightrightarrows E$ in $\mathbf{Set}_f$. Then it is a coreflexive corelation in $\mathbf{Set}$. Since $\mathbf{Set}$ is co-Malcev and has effective cocongruences, that implies $E$ is the cokernel pair of some function $h : Z \to X$. If we replace $h$ with the inclusion map of its image, then this function is in $\mathbf{Set}_\mathrm{f}$ and gives the same cokernel pair.' +), ( 'Set_f', 'locally cartesian closed', From f77a3a9db945b6ec2e13a52e274dca490f93b4ea Mon Sep 17 00:00:00 2001 From: Script Raccoon Date: Sun, 26 Apr 2026 01:17:22 +0200 Subject: [PATCH 14/18] Grp has effective cocongruences --- .../003_category-property-assignments/Grp.sql | 6 + static/pdf/.gitignore | 7 ++ static/pdf/cocongruences_of_groups.pdf | Bin 0 -> 164476 bytes static/pdf/cocongruences_of_groups.tex | 108 ++++++++++++++++++ 4 files changed, 121 insertions(+) create mode 100644 static/pdf/.gitignore create mode 100644 static/pdf/cocongruences_of_groups.pdf create mode 100644 static/pdf/cocongruences_of_groups.tex diff --git a/databases/catdat/data/003_category-property-assignments/Grp.sql b/databases/catdat/data/003_category-property-assignments/Grp.sql index 3f88bec9..1e0a8cbb 100644 --- a/databases/catdat/data/003_category-property-assignments/Grp.sql +++ b/databases/catdat/data/003_category-property-assignments/Grp.sql @@ -47,6 +47,12 @@ VALUES TRUE, 'Since epimorphisms are surjective (see below), this is the first isomorphism theorem for groups.' ), +( + 'Grp', + 'effective cocongruences', + TRUE, + 'A proof can be found here.' +), ( 'Grp', 'normal', diff --git a/static/pdf/.gitignore b/static/pdf/.gitignore new file mode 100644 index 00000000..372e1325 --- /dev/null +++ b/static/pdf/.gitignore @@ -0,0 +1,7 @@ +# Files from Latex Workshop +*.aux +*.fdb_latexmk +*.fls +*.log +*.synctex.gz +*.out \ No newline at end of file diff --git a/static/pdf/cocongruences_of_groups.pdf b/static/pdf/cocongruences_of_groups.pdf new file mode 100644 index 0000000000000000000000000000000000000000..f5bdf0b6b6758fe49d41941094048c6d0ff68ded GIT binary patch literal 164476 zcma&MQ;;r95T@C-PusR_+qP}{YumPQ+I`x#ZQHhu`DbEgFLw80@~Sd2qB3u)BAViz8LThxL{cZIjDDV}M9plid=DUT 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amscd, amsthm, mathtools} +\usepackage{hyperref} + +\theoremstyle{plain} +\newtheorem{prop}{Proposition} +\newtheorem{cor}[prop]{Corollary} + +\theoremstyle{definition} +\newtheorem{defi}[prop]{Definition} + +\DeclareMathOperator{\id}{id} +\DeclareMathOperator{\eq}{eq} +\DeclareMathOperator{\Grp}{\mathbf{Grp}} + +\newcommand{\C}{\mathcal{C}} + +\title{Cocongruences on groups are effective} +\author{Martin Brandenburg} +\date{\today} + +\begin{document} +\maketitle + +Our goal is to prove that every cocongruence in the category $\Grp$ is effective. We will establish a more general result for categories in which pushouts and monomorphisms interact in a suitable way. + +\begin{defi} +We shall say that a category $\C$ has \emph{good pushouts of monomorphisms} if it has pushouts of monomorphisms and if, for every diagram of monomorphisms +\[\begin{tikzcd} +B \ar{r} & B' \\ +A \ar{r} \ar{u} \ar{d} & A' \ar{u} \ar{d} \\ +C \ar{r} & C' +\end{tikzcd}\] +in which each square is a pullback, the induced morphism +\[B \sqcup_A C \to B' \sqcup_{A'} C'\] +is also a monomorphism. +\end{defi} + +\begin{prop} +The category $\Grp$ has good pushouts of monomorphisms. +\end{prop} + +\begin{proof} +Consider a diagram as above. We regard every monomorphism in it as an inclusion. Choose a system of representatives $S \subseteq B$ for the right $A$-cosets in $B$, meaning that the multiplication map $\cdot : A \times S \to B$ is bijective. Likewise, choose $T \subseteq C$ such that the multiplication map $\cdot : A \times T \to C$ is bijective. We may assume that $1 \in S$ and $1 \in T$. + +It is well known (see, for example, Serre's book \emph{Trees}, Ch.\ I, §1, Thm.\ 1) that every element of the amalgamated free product $B \sqcup_A C$ has a unique representation of the form +\[w = a \cdot x_1 \cdots x_n,\] +where $a \in A$, each $x_i$ lies either in $S \setminus \{1\}$ or in $T \setminus \{1\}$, and these choices alternate. + +The map +\[A \backslash B \to A' \backslash B', \, Ab \mapsto A'b\] +is injective. Indeed, if $b_1,b_2 \in B$ satisfy $A' b_1 = A' b_2$, then $b_1 b_2^{-1} \in A'$. Since $B \cap A' = A$, it follows that $b_1 b_2^{-1} \in A$, and hence $A b_1 = A b_2$. + +Therefore, we may extend $S$ to a system of representatives $S' \subseteq B'$ for the right $A'$-cosets in $B'$. Likewise, we may extend $T$ to a system of representatives $T' \subseteq C'$ for the right $A'$-cosets in $C'$. + +With respect to these systems, an element $w \in B \sqcup_A C$ written in normal form as above remains in normal form after being mapped to $B' \sqcup_{A'} C'$. This shows that the induced map is injective. +\end{proof} + +\begin{prop} +Let $\C$ be a balanced category with good pushouts of monomorphisms and equalizers of monomorphisms. Then every cocongruence in $\C$ is effective. +\end{prop} + +\begin{proof} +Let $X \in \C$ be an object, and let $i_1,i_2 : X \rightrightarrows Y$ be a cocongruence. Since it is coreflexive, there exists a morphism $r : Y \to X$ satisfying +\[r \circ i_1 = \id_X, \quad r \circ i_2 = \id_X.\] +In particular, $i_1$ and $i_2$ are monomorphisms. Since the cocongruence is cotransitive, there exists a morphism +\[c : Y \to Y \sqcup_{i_2,X,i_1} Y\] +satisfying +\[c \circ i_1 = u_1 \circ i_1, \quad c \circ i_2 = u_2 \circ i_2,\] +where $u_1,u_2 : Y \rightrightarrows Y \sqcup_{i_2,X,i_1} Y$ are the pushout inclusions satisfying $u_1 i_2 = u_2 i_1$. We will not use the fact that the cocongruence is cosymmetric; this will follow automatically. Define the monomorphism +\[E := \eq(i_1,i_2) \hookrightarrow X.\] + +Since $i_1$ and $i_2$ agree on $E$, there exists a unique morphism +\[\varphi : X \sqcup_E X \to Y\] +defined by $\varphi \circ j_1 = i_1$ and $\varphi \circ j_2 = i_2$, where $j_1,j_2 : X \rightrightarrows X \sqcup_E X$ are the two inclusions. + +We must show that $\varphi$ is an isomorphism. It is clearly an epimorphism, since $i_1$ and $i_2$ are jointly epimorphic by assumption. Since $\C$ is balanced, it therefore suffices to prove that $\varphi$ is a monomorphism. + +We will show that even the morphism +\[\gamma := c \circ \varphi : X \sqcup_E X \to Y \sqcup_{i_2,X,i_1} Y\] +is a monomorphism. It is characterized by +\[\gamma \circ j_1 = c \circ \varphi \circ j_1 = c \circ i_1 = u_1 \circ i_1,\] +\[\gamma \circ j_2 = c \circ \varphi \circ j_2 = c \circ i_2 = u_2 \circ i_2.\] + +In other words, $\gamma$ is induced by the diagram of monomorphisms +\[\begin{tikzcd} +X \ar{r}{i_1} & Y \\ +E \ar{r} \ar{u} \ar{d} & X \ar{u}[swap]{i_2} \ar{d}{i_1} \\ +X \ar{r}[swap]{i_2} & Y +\end{tikzcd}\] + +Since $\C$ has good pushouts of monomorphisms, it suffices to verify that both squares are pullbacks. Observe that the two squares are symmetric, so it is enough to consider one of them. To verify the universal property, let $a : T \to X$ and $b : T \to X$ be morphisms satisfying $i_1 \circ a = i_2 \circ b$. Applying $r : Y \to X$, we obtain +\[a = r \circ i_1 \circ a = r \circ i_2 \circ b = b.\] + +Thus, $a$ is simply a morphism equalizing $i_1$ and $i_2$, so it factors uniquely through $E \hookrightarrow X$. +\end{proof} + +\begin{cor} +Every cocongruence in the category $\Grp$ is effective. +\end{cor} + + +\end{document} From f89d31f050f337314e685df22131600e31046054 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sun, 26 Apr 2026 14:15:39 -0400 Subject: [PATCH 15/18] Make link target an absolute path Co-authored-by: Script Raccoon --- .../catdat/data/003_category-property-assignments/PMet.sql | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/databases/catdat/data/003_category-property-assignments/PMet.sql b/databases/catdat/data/003_category-property-assignments/PMet.sql index 24a079ff..d62762ac 100644 --- a/databases/catdat/data/003_category-property-assignments/PMet.sql +++ b/databases/catdat/data/003_category-property-assignments/PMet.sql @@ -143,5 +143,5 @@ VALUES 'PMet', 'effective cocongruences', FALSE, - 'The proof is similar to the one for $\mathbf{Top}$: Use the two-point space with the zero metric, which represents the functor taking a pseudo-metric space to the pairs of points with $d(x,y) = 0$. In this case, once you conclude $Z = \varnothing$, the map $h : Z \to 1$ does not have any cokernel pair, since that would have to be a coproduct $1+1$.' + 'The proof is similar to the one for $\mathbf{Top}$: Use the two-point space with the zero metric, which represents the functor taking a pseudo-metric space to the pairs of points with $d(x,y) = 0$. In this case, once you conclude $Z = \varnothing$, the map $h : Z \to 1$ does not have any cokernel pair, since that would have to be a coproduct $1+1$.' ); From 4e77752d47a9192a45d7e231d3bb4b8d357309c3 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sun, 26 Apr 2026 14:32:56 -0400 Subject: [PATCH 16/18] Add clarification Co-authored-by: Script Raccoon --- .../catdat/data/003_category-property-assignments/Set_f.sql | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/databases/catdat/data/003_category-property-assignments/Set_f.sql b/databases/catdat/data/003_category-property-assignments/Set_f.sql index 09c2c221..45407866 100644 --- a/databases/catdat/data/003_category-property-assignments/Set_f.sql +++ b/databases/catdat/data/003_category-property-assignments/Set_f.sql @@ -45,7 +45,7 @@ VALUES 'Set_f', 'effective congruences', TRUE, - 'If $E \rightrightarrows X$ is the kernel pair of $h : X \to Z$ and both maps $E \to X$ are finite-to-one, then that means the equivalence classes of $E$ are finite. Thus, necessarily $h$ was finite-to-one also.' + 'If $E \rightrightarrows X$ is the kernel pair of $h : X \to Z$ in $\mathbf{Set}$ and both maps $E \to X$ are finite-to-one, then that means the equivalence classes of $E$ are finite. Thus, necessarily $h$ was finite-to-one also.' ), ( 'Set_f', From 025cdbf0f7ef3c125bcf3074ea0087949fbca173 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sun, 26 Apr 2026 15:53:14 -0400 Subject: [PATCH 17/18] Address review comments --- .../100_related-category-properties.sql | 4 ++++ .../catdat/data/003_category-property-assignments/FS.sql | 6 ------ .../data/003_category-property-assignments/FinSet.sql | 6 ------ .../catdat/data/003_category-property-assignments/Set.sql | 6 ------ .../data/003_category-property-assignments/Set_c.sql | 6 ------ .../data/003_category-property-assignments/Set_pointed.sql | 3 ++- .../001_limits-colimits-existence-implications.sql | 7 ------- .../005_additional-structure-implications.sql | 4 +++- .../008_topos-theory-implications.sql | 2 +- 9 files changed, 10 insertions(+), 34 deletions(-) diff --git a/databases/catdat/data/002_category-properties/100_related-category-properties.sql b/databases/catdat/data/002_category-properties/100_related-category-properties.sql index 956c6b59..4e05b383 100644 --- a/databases/catdat/data/002_category-properties/100_related-category-properties.sql +++ b/databases/catdat/data/002_category-properties/100_related-category-properties.sql @@ -298,8 +298,10 @@ VALUES ('coquotients of cocongruences', 'effective cocongruences'), ('effective congruences', 'normal'), ('effective congruences', 'quotients of congruences'), +('effective congruences', 'mono-regular'), ('effective cocongruences', 'conormal'), ('effective cocongruences', 'coquotients of cocongruences'), +('effective cocongruences', 'epi-regular'), ('direct', 'one-way'), ('direct', 'skeletal'), ('inverse', 'one-way'), @@ -345,7 +347,9 @@ VALUES ('cofiltered', 'finitely complete'), ('cofiltered', 'cofiltered limits'), ('mono-regular', 'normal'), +('mono-regular', 'effective congruences'), ('epi-regular', 'conormal'), +('epi-regular', 'effective cocongruences'), ('normal', 'zero morphisms'), ('normal', 'mono-regular'), ('normal', 'kernels'), diff --git a/databases/catdat/data/003_category-property-assignments/FS.sql b/databases/catdat/data/003_category-property-assignments/FS.sql index f107a8c6..bb030e59 100644 --- a/databases/catdat/data/003_category-property-assignments/FS.sql +++ b/databases/catdat/data/003_category-property-assignments/FS.sql @@ -65,12 +65,6 @@ VALUES TRUE, 'The empty set and a singleton give a multi-terminal object.' ), -( - 'FS', - 'effective congruences', - TRUE, - 'In fact, for any congruence $E \rightrightarrows X$ in $\mathbf{FS}$, we must have the reflexivity morphism $X \to E$ surjective. That implies $E$ is the kernel pair of $\mathrm{id}_X$.' -), ( 'FS', 'small', diff --git a/databases/catdat/data/003_category-property-assignments/FinSet.sql b/databases/catdat/data/003_category-property-assignments/FinSet.sql index eb490cd6..219cd972 100644 --- a/databases/catdat/data/003_category-property-assignments/FinSet.sql +++ b/databases/catdat/data/003_category-property-assignments/FinSet.sql @@ -47,12 +47,6 @@ VALUES 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', - 'effective cocongruences', - TRUE, - 'For a finite cokernel pair $E$ of $h : Z \to X$ where $X$ is a finite set, replacing $h$ with the inclusion map of its image gives the same cokernel pair.' -), ( 'FinSet', 'small', diff --git a/databases/catdat/data/003_category-property-assignments/Set.sql b/databases/catdat/data/003_category-property-assignments/Set.sql index f6d0ea82..b587ee04 100644 --- a/databases/catdat/data/003_category-property-assignments/Set.sql +++ b/databases/catdat/data/003_category-property-assignments/Set.sql @@ -29,12 +29,6 @@ VALUES TRUE, 'Use the empty algebraic theory.' ), -( - 'Set', - 'effective cocongruences', - TRUE, - 'Since the contravariant powerset functor $\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}$ is monadic, we conclude that $\mathbf{Set}^{\mathrm{op}}$ is Barr-exact and therefore has effective congruences.' -), ( 'Set', 'skeletal', diff --git a/databases/catdat/data/003_category-property-assignments/Set_c.sql b/databases/catdat/data/003_category-property-assignments/Set_c.sql index f2ac9f3f..373224d0 100644 --- a/databases/catdat/data/003_category-property-assignments/Set_c.sql +++ b/databases/catdat/data/003_category-property-assignments/Set_c.sql @@ -83,12 +83,6 @@ VALUES TRUE, 'For a countable kernel pair $E$ of $h : X \to Z$ where $X$ is countable, replacing $Z$ with the image of $h$ gives the same kernel pair.' ), -( - 'Set_c', - 'effective cocongruences', - TRUE, - 'For a countable cokernel pair $E$ of $h : Z \to X$ where $X$ is countable, replacing $h$ with the inclusion map of its image gives the same cokernel pair.' -), ( 'Set_c', 'small', diff --git a/databases/catdat/data/003_category-property-assignments/Set_pointed.sql b/databases/catdat/data/003_category-property-assignments/Set_pointed.sql index d6e32b8d..d5dba878 100644 --- a/databases/catdat/data/003_category-property-assignments/Set_pointed.sql +++ b/databases/catdat/data/003_category-property-assignments/Set_pointed.sql @@ -68,10 +68,11 @@ VALUES 'The coproduct (wedge sum) of a family of pointed sets $(X_i)_{i \in I}$ can be realized as the subset of $\prod_{i \in I} X_i$ consisting of those tuples $x$ such that $x_i = 0$ for all but (at most) one index.' ), ( + -- TODO: rework this when Barr-exact is added 'Set*', 'effective cocongruences', TRUE, - 'We have that $\mathbf{Set}_*^{\mathrm{op}}$ is a slice category of $\mathbf{Set}^{\mathrm{op}}$, which in turn is monadic over $\mathbf{Set}$. Therefore, $\mathbf{Set}_*^{\mathrm{op}}$ is Barr-exact, and in particular it has effective congruences.' + 'We have that $\mathbf{Set}_*^{\mathrm{op}}$ is a slice category of $\mathbf{Set}^{\mathrm{op}}$, which in turn is monadic over $\mathbf{Set}$. Therefore, by combining results from Borceux and Bourn Appendix A and nLab, $\mathbf{Set}_*^{\mathrm{op}}$ is Barr-exact, and in particular it has effective congruences.' ), ( 'Set*', diff --git a/databases/catdat/data/004_category-implications/001_limits-colimits-existence-implications.sql b/databases/catdat/data/004_category-implications/001_limits-colimits-existence-implications.sql index be2f4b72..a29adac7 100644 --- a/databases/catdat/data/004_category-implications/001_limits-colimits-existence-implications.sql +++ b/databases/catdat/data/004_category-implications/001_limits-colimits-existence-implications.sql @@ -69,13 +69,6 @@ VALUES 'For any congruence $E$ on an object $X$ of a preadditive category, let $E_0$ be the kernel of $p_2 : E \to X$. The restriction of $p_1$ to $E_0$ is a monomorphism. We can then see that $E$ must be the pullback of $p_1 - p_2 : E \to X$ and $E_0 \hookrightarrow X$. Then the cokernel of $E_0 \hookrightarrow X$ is a quotient of $E$.', FALSE ), -( - 'thin_effective_congruences', - '["thin"]', - '["effective congruences"]', - 'In a thin category, any congruence pair is a reflexive fork, and thus consists of equal isomorphisms. Therefore, the congruence is the kernel pair of the identity morphism on the target.', - FALSE -), ( 'products_consequences', '["products"]', diff --git a/databases/catdat/data/004_category-implications/005_additional-structure-implications.sql b/databases/catdat/data/004_category-implications/005_additional-structure-implications.sql index 6942f746..278436ba 100644 --- a/databases/catdat/data/004_category-implications/005_additional-structure-implications.sql +++ b/databases/catdat/data/004_category-implications/005_additional-structure-implications.sql @@ -31,7 +31,9 @@ VALUES 'preadditive_kernels_normal_imply_effective_congruences', '["preadditive", "kernels", "normal"]', '["effective congruences"]', - 'Let $f, g : E \rightrightarrows X$ be a congruence. Then let $E_0$ be the kernel of $g$. We see that $f$ restricted to $E_0$ is a monomorphism $E_0 \hookrightarrow X$. If this is the kernel of a morphism $h : X \to Y$, then $E$ is the kernel pair of $h$.', + 'Let $f, g : E \rightrightarrows X$ be a congruence. Then let $E_0$ be the kernel of $g$. We see that $f$ restricted to $E_0$ is a monomorphism $E_0 \hookrightarrow X$. Let $f |_{E_0}$ be the kernel of a morphism $h : X \to Y$. We claim that $E$ is also the kernel pair of $h$.
+ To see this, suppose we have a pair of generalized elements $x_1, x_2 : T \rightrightarrows X$. Then the pair $x_1, x_2$ factors through $E$ if and only if $x_1 - x_2, 0$ does. This is equivalent to the condition that $x_1 - x_2$ factors through $E_0$. That in turn is equivalent to $h \circ (x_1 - x_2) = 0$, which is equivalent to $h \circ x_1 = h \circ x_2$.
+ In particular, applying the forward implications in the case $T := E, x_1 := f, x_2 := g$, we conclude that $h \circ f = h \circ g$, so we get the required commutative diagram. From there, the reverse implications show this diagram is a cartesian square.', FALSE ), ( diff --git a/databases/catdat/data/004_category-implications/008_topos-theory-implications.sql b/databases/catdat/data/004_category-implications/008_topos-theory-implications.sql index 7d41ffcf..0159f2ff 100644 --- a/databases/catdat/data/004_category-implications/008_topos-theory-implications.sql +++ b/databases/catdat/data/004_category-implications/008_topos-theory-implications.sql @@ -122,7 +122,7 @@ VALUES 'topos_consequence', '["elementary topos"]', '["finitely cocomplete", "disjoint finite coproducts", "epi-regular", "effective congruences"]', - 'See Mac Lane & Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. 4.7.8.', + 'See Mac Lane & Moerdijk, Cor. IV.5.4, Cor. IV.10.5, Thm. 4.7.8; and Johnstone, Part A, Proposition 2.4.1.', FALSE ), ( From 2c5ff6625d3bae057f30796a40401b3c773458d0 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sun, 26 Apr 2026 17:23:36 -0400 Subject: [PATCH 18/18] Adjust some wording --- .../catdat/data/003_category-property-assignments/FinGrp.sql | 2 +- .../catdat/data/003_category-property-assignments/FreeAb.sql | 2 +- .../catdat/data/003_category-property-assignments/Set_c.sql | 2 +- 3 files changed, 3 insertions(+), 3 deletions(-) diff --git a/databases/catdat/data/003_category-property-assignments/FinGrp.sql b/databases/catdat/data/003_category-property-assignments/FinGrp.sql index a3e49cb4..35c11cd6 100644 --- a/databases/catdat/data/003_category-property-assignments/FinGrp.sql +++ b/databases/catdat/data/003_category-property-assignments/FinGrp.sql @@ -69,7 +69,7 @@ VALUES 'FinGrp', 'effective congruences', TRUE, - 'For a finite kernel pair $E$ of $h : X \to Z$ where $X$ is a finite group, replacing $Z$ with the image of $h$ gives the same kernel pair.' + 'For a kernel pair $E$ of $h : X \to Z$ where $E$ and $X$ are finite groups, replacing $Z$ with the image of $h$ gives the same kernel pair.' ), ( 'FinGrp', diff --git a/databases/catdat/data/003_category-property-assignments/FreeAb.sql b/databases/catdat/data/003_category-property-assignments/FreeAb.sql index b8f8d8cb..d8ba847c 100644 --- a/databases/catdat/data/003_category-property-assignments/FreeAb.sql +++ b/databases/catdat/data/003_category-property-assignments/FreeAb.sql @@ -65,7 +65,7 @@ VALUES 'FreeAb', 'effective cocongruences', TRUE, - 'Since $\mathbf{Ab}$ is abelian, it has effective cocongruences. Now, suppose a cocongruence $X \rightrightarrows E$ is the cokernel pair of $h : Z \to X$, where $X$ and $E$ are free abelian groups. If we find a surjective map $g : F \to Z$, then $h \circ g$ has the same cokernel pair as $h$, and $h \circ g$ is a morphism in $\mathbf{FreeAb}$.' + 'Since $\mathbf{Ab}$ is abelian, it has effective cocongruences. Now, suppose a cocongruence $X \rightrightarrows E$ is the cokernel pair of $h : Z \to X$, where $X$ and $E$ are free abelian groups. If we find a surjective map $g : F \to Z$ where $F$ is a free abelian group, then $h \circ g$ has the same cokernel pair as $h$, and $h \circ g$ is a morphism in $\mathbf{FreeAb}$.' ), ( 'FreeAb', diff --git a/databases/catdat/data/003_category-property-assignments/Set_c.sql b/databases/catdat/data/003_category-property-assignments/Set_c.sql index 373224d0..86b08391 100644 --- a/databases/catdat/data/003_category-property-assignments/Set_c.sql +++ b/databases/catdat/data/003_category-property-assignments/Set_c.sql @@ -81,7 +81,7 @@ VALUES 'Set_c', 'effective congruences', TRUE, - 'For a countable kernel pair $E$ of $h : X \to Z$ where $X$ is countable, replacing $Z$ with the image of $h$ gives the same kernel pair.' + 'For a kernel pair $E$ of $h : X \to Z$ where $E$ and $X$ are countable, replacing $Z$ with the image of $h$ gives the same kernel pair.' ), ( 'Set_c',