decide cartesian filtered colimits for Haus, Meas, Top, Top*

Author: ScriptRaccoon

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

@@ -67,7 +67,7 @@ VALUES
 ),
 (
 	'Haus',
-	'cartesian closed',
+	'cartesian filtered colimits',
 	FALSE,
 	'It is shown in <a href="https://math.stackexchange.com/questions/1255678">MSE/1255678</a> that $\mathbb{Q} \times - : \mathbf{Top} \to \mathbf{Top}$ does not preserve sequential colimits (so that it cannot be a left adjoint). The same example also works in $\mathbf{Haus}$: Surely $\mathbb{Q}$ is Hausdorff, $X_n$ is Hausdorff, as is their colimit $X$, and the colimit (taken in $\mathbf{Top}$) of the $X_n \times \mathbb{Q}$ admits a bijective continuous map to a Hausdorff space, therefore is also Hausdorff, meaning it is also the colimit taken in $\mathbf{Haus}$.'
 ),

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

@@ -67,13 +67,7 @@ VALUES
 ),
 (
 	'Meas',
-	'cartesian closed',
-	FALSE,
-	'The functors $X \times -$ do not preserve filtered colimits by <a href="https://math.stackexchange.com/questions/5027218" target="_blank">MSE/5027218</a>, hence cannot be left adjoints.'
-),
-(
-	'Meas',
-	'exact filtered colimits',
+	'cartesian filtered colimits',
 	FALSE,
 	'See <a href="https://math.stackexchange.com/questions/5027218" target="_blank">MSE/5027218</a>.'
 ),

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

@@ -73,9 +73,9 @@ VALUES
 ),
 (
 	'Top',
-	'cartesian closed',
+	'cartesian filtered colimits',
 	FALSE,
-	'The functor $\mathbb{Q} \times - : \mathbf{Top} \to \mathbf{Top}$ does not preserve colimits, hence has no right adjoint. See <a href="https://math.stackexchange.com/questions/2969372" target="_blank">MSE/2969372</a> for a proof.'
+	'The functor $\mathbb{Q} \times - : \mathbf{Top} \to \mathbf{Top}$ does not preserve colimits, see <a href="https://math.stackexchange.com/questions/2969372" target="_blank">MSE/2969372</a>.'
 ),
 (
 	'Top',
@@ -95,12 +95,6 @@ VALUES
 	FALSE,
 	'If $X$ is a set, consider the discrete space $X_d$ on $X$ and the indiscrete space $X_i$ on $X$. The identity map $X \to X$ lifts to a continuous map $X_d \to X_i$, which is bijective and therefore both a mono- and an epimorphism, but it is not an isomorphism unless $X$ has at most one element.'
 ),
-(
-	'Top',
-	'exact filtered colimits',
-	FALSE,
-	'See <a href="https://math.stackexchange.com/questions/1255678" target="_blank">MSE/1255678</a>.'
-),
 (
 	'Top',
 	'skeletal',

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

@@ -103,9 +103,9 @@ VALUES
 ),
 (
 	'Top*',
-	'exact filtered colimits',
+	'cartesian filtered colimits',
 	FALSE,
-	'See <a href="https://math.stackexchange.com/questions/1255678" target="_blank">MSE/1255678</a> (the counterexample also works for pointed spaces).'
+	'The functor $\mathbb{Q} \times - : \mathbf{Top}_* \to \mathbf{Top}_*$ does not preserve colimits, see <a href="https://math.stackexchange.com/questions/2969372" target="_blank">MSE/2969372</a>. The counterexample also works for pointed spaces.'
 ),
 (
 	'Top*',

scripts/expected-data/Ab.json

@@ -23,6 +23,7 @@
 	"epi-regular": true,
 	"equalizers": true,
 	"exact filtered colimits": true,
+	"cartesian filtered colimits": true,
 	"filtered colimits": true,
 	"cofiltered limits": true,
 	"directed limits": true,

scripts/expected-data/Set.json

@@ -25,6 +25,7 @@
 	"epi-regular": true,
 	"equalizers": true,
 	"exact filtered colimits": true,
+	"cartesian filtered colimits": true,
 	"filtered colimits": true,
 	"cofiltered limits": true,
 	"directed limits": true,

scripts/expected-data/Top.json

@@ -78,6 +78,7 @@
 	"essentially finite": false,
 	"essentially small": false,
 	"exact filtered colimits": false,
+	"cartesian filtered colimits": false,
 	"finitary algebraic": false,
 	"finite": false,
 	"Grothendieck abelian": false,