From d2d7585d00be0d264bbab800b146645c377406e8 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Sat, 6 Jun 2026 16:49:21 -0400 Subject: [PATCH 1/3] Add several negative properties of LRS_R deduced from those of Top --- .cspell.json | 1 + content/LRS-not-cartesian-closed.md | 49 +++++++++++++++++++ content/cartesian-closed-results.md | 27 ++++++++++ databases/catdat/data/categories/LRS_R.yaml | 16 ++++++ .../cartesian closed.yaml | 9 ++++ 5 files changed, 102 insertions(+) create mode 100644 content/LRS-not-cartesian-closed.md create mode 100644 content/cartesian-closed-results.md diff --git a/.cspell.json b/.cspell.json index d060b9b6..2e78761b 100644 --- a/.cspell.json +++ b/.cspell.json @@ -46,6 +46,7 @@ "Čech", "characterisation", "clopen", + "closedness", "Clowder", "coaccessible", "cocartesian", diff --git a/content/LRS-not-cartesian-closed.md b/content/LRS-not-cartesian-closed.md new file mode 100644 index 00000000..96af5c59 --- /dev/null +++ b/content/LRS-not-cartesian-closed.md @@ -0,0 +1,49 @@ +--- +title: The category of locally ringed spaces is not cartesian closed +description: A proof that the category of locally ringed spaces over a non-trivial ring R is not cartesian closed +author: Daniel Schepler +--- + +## The category of locally ringed spaces is not cartesian closed + +For most of this development, we will be dealing with the case of $\LRS_k$ where $k$ is a field. We begin by describing $\Top$ as a reflective subcategory of $\LRS_k$. + +::: Lemma 1 +The forgetful functor $U : \LRS_k \to \Top$ has a right adjoint $K : \Top \to \LRS_k$ of equipping a topological space $X$ with the constant sheaf $\underline{k}$. Furthermore, the functor $K$ is fully faithful, thus making $\Top$ into a reflective subcategory of $\LRS_k$. +::: + +_Proof._ In this adjunction, the counit $UK \to \id$ is just the identity. To describe the unit $\id \to KU$, we need to define a morphism $(X, \O_X) \to (X, \underline{k})$ for any locally ringed space $(X, \O_X)$ over $k$. This morphism will be the identity on topological spaces, and the pullback operation $\underline{k} \to \O_X$ will be the unique morphism of sheaves induced by the given structure of $\O_X$ as a sheaf of $k$-algebras. It is now straightforward to check this indeed defines an adjunction; and since the counit is an isomorphism, that implies that $K$ is fully faithful. $\square$ + +We now show that this reflective subcategory is in fact also a coreflective subcategory. Recall that for $f \in \O_X(U)$ we have its vanishing set $V(f) \coloneqq \{x \in U : f(x) = 0\}$, where $f(x) \in \kappa(x)$ is the image of $f_x \in \O_{X,x}$ in the residue field. + +::: Lemma 2 +For each object $X$ of $\LRS_k$, let $X_0$ be the set of points $x \in X$ such that the induced morphism from $k$ to the residue field $\kappa(x)$ is an isomorphism. We give $X_0$ the following strengthening of the subspace topology: it will be the topology where a neighborhood subbasis at $x \in X_0$ is the collection of sets of the form $X_0 \cap V(f)$ where $f \in \O_X(U)$ for some neighborhood $U$ of $x$ in $X$, and $x \in V(f)$. Then $X \mapsto X_0$ defines a functor $S : \LRS_k \to \Top$ that is right adjoint to $K$. +::: + +_Proof._ First, to see that $S$ is a functor, suppose we have a morphism $f : (X, \O_X) \to (Y, \O_Y)$. Then for $x \in X_0$, we have a sequence $k \to \kappa(f(x)) \to \kappa(x)$ where the composition is an isomorphism. Thus, $\kappa(f(x)) \to \kappa(x)$ is a surjective morphism of fields, and therefore an isomorphism. It follows that $k \to \kappa(f(x))$ is also an isomorphism of fields, so $f(x) \in Y_0$. To see that the restriction map $X_0 \to Y_0$ is continuous, suppose $g \in \O_Y(V)$ is such that $f(x) \in V(g)$. Then $x \in V(f^\sharp g)$, and +$$X_0 \cap f^{-1}(Y_0 \cap V(g)) = X_0 \cap V(f^\sharp g)$$ +where $f^\sharp g \in \O_X(f^{-1}(V))$. In other words, we have shown that the inverse image in $X_0$ of any subbasic neighborhood of $f(x)$ is a neighborhood of $x$. + +Now if we apply the functor $S$ to a space of the form $(X, \underline{k})$, then since by definition any section of the constant sheaf $\underline{k}$ is locally constant, we see that we recover exactly $X$ with its original topology. Thus, we can define the unit $\id \to SK$ of the adjunction to be the identity. + +As for the counit $KS \to \id$, for any locally ringed space $(X, \O_X)$ over $k$ we need to define a morphism $(X_0, \underline{k}) \to (X, \O_X)$. The map of topological spaces will be the inclusion map $i : X_0 \hookrightarrow X$, which is continuous since in particular for $U$ an open neighborhood of $x \in X_0$ we have $X_0 \cap U = X_0 \cap V(0_U)$, where $0_U \in \O_X(U)$ is the zero element. The pullback map $\O_X \to i_* \underline{k}$ takes $f \in \O_X(U)$ to the function $X_0 \cap U \to k$ where $x \in X_0 \cap U$ maps to the inverse image of $f(x) \in \kappa(x)$ under the isomorphism $k \to \kappa(x)$. An alternative description of this pullback is that $x \in X_0 \cap U$ maps to the unique $a\in k$ such that $x \in V(f-a)$. Since $X_0 \cap V(f-a)$ is a neighborhood of $x$ in $X_0$ by definition, this shows that we get a locally constant function to $k$ as required. + +From here, it is straightforward to show that this does in fact define an adjunction. $\square$ + +_Remark._ In the special case where $k$ is a finite field, we have +$$\textstyle X_0 \cap V(f) = \bigcap_{a \in k^\times} (X_0 \cap D(f-a)),$$ +which is already open in the subspace topology. Therefore, in this case, $X_0$ is given exactly the subspace topology. + +::: Corollary 3 +If $k$ is a field, then $\LRS_k$ is not cartesian closed. +::: + +_Proof._ We know that $\Top$ is not cartesian closed. By Lemma 2, $\Top$ is a coreflective subcategory of $\LRS_k$. Moreover, the inclusion preserves binary products (in fact, all limits) by Lemma 1. Therefore, the claim follows from Lemma 1 [here](/content/cartesian-closed-results). $\square$ + +Finally, we generalize this result to arbitrary base rings. + +::: Corollary 4 +For any non-trivial commutative ring $R$, the category $\LRS_R$ is not cartesian closed. +::: + +_Proof._ Let $k$ be a residue field of $R$. Then $\LRS_k$ is equivalent to the slice category $\LRS_R / \Spec k$ where $\Spec k$ is subterminal. Therefore, the claim follows from Corollary 3 above and Corollary 2 [here](/content/cartesian-closed-results). $\square$ diff --git a/content/cartesian-closed-results.md b/content/cartesian-closed-results.md new file mode 100644 index 00000000..0c9f8593 --- /dev/null +++ b/content/cartesian-closed-results.md @@ -0,0 +1,27 @@ +--- +title: Results about cartesian closed categories +description: We prove in particular when a coreflective subcategory of a cartesian closed category is again cartesian closed. +author: Daniel Schepler +--- + +## Results about cartesian closed categories + +::: Lemma 1 +Suppose $\D$ is a coreflective subcategory of $\C$ such that $\D$ has binary products and the inclusion functor preserves these binary products. If $\C$ is cartesian closed, then so is $\D$. +::: + +_Proof._ Let $U : \D \to \C$ be the inclusion functor with right adjoint $R : \C \to \D$. Then for any objects $X, Y, Z$ of $\D$ we have natural isomorphisms + +$$ +\begin{align*} +\Hom_\D(Z\times X, Y) & \cong \Hom_\C(UZ \times UX, UY) \\ +& \cong \Hom_\C(UZ, [UX,UY]) \\ +& \cong \Hom_\D\bigl(Z, R([UX,UY])\bigr). +\end{align*} +$$ + +::: Corollary 2 +If $\C$ is a cartesian closed category and $P$ is a [subterminal object](https://ncatlab.org/nlab/show/subterminal+object) of $\C$, then the slice category $\C / P$ is also cartesian closed. +::: + +_Proof._ The forgetful functor $\C / P \to \C$ is fully faithful; it has right adjoint ${-} \times P$; and it preserves binary products (in fact all inhabited limits). Hence, Lemma 1 applies. $\square$ diff --git a/databases/catdat/data/categories/LRS_R.yaml b/databases/catdat/data/categories/LRS_R.yaml index 112b01fe..f5417843 100644 --- a/databases/catdat/data/categories/LRS_R.yaml +++ b/databases/catdat/data/categories/LRS_R.yaml @@ -53,6 +53,22 @@ unsatisfied_properties: Alternatively, using the usual adjunction between affine schemes and locally ringed spaces (EGA I (1971), Ch. 1, Prop. 1.6.3), a generating set in $\LRS_R$ would induce a generating set in the category of affine $R$-schemes, which contradicts the fact that $\CAlg(R)$ does not have a cogenerating set. + - property: cartesian closed + proof: This is proved here. + check_redundancy: false + + - property: cartesian filtered colimits + proof: As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. Therefore, if $\LRS_R$ had cartesian filtered colimits, then $\Top$ would also, giving a contradiction. + + - property: regular + proof: 'As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. Therefore, if $\LRS_R$ were regular, then for every regular epimorphism $X \to Y$ and morphism $Z \to Y$, the pullback $f : Z \times_Y X \to X$ would satisfy that the canonical morphism from the quotient of the kernel pair of $f$ to $X$ is an isomorphism. This would be inherited by the subcategory $\Top$, and since $\Top$ is finitely complete and has coequalizers, that would imply $\Top$ is also regular, giving a contradiction.' + + - property: cofiltered-limit-stable epimorphisms + proof: 'As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. From here, the proof is similar to the one from $\Top$: we apply the contrapositive of the dual of this lemma to the functor $\Set \to \LRS_k$ which equips a set with the indiscrete topology and the constant sheaf of $k$.' + + - property: effective cocongruences + proof: As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. Therefore, if $\LRS_R$ had effective cocongruences, then for every cocongruence $E$ on $X$, the canonical morphism from the cokernel pair of the equalizer of $E$ to $E$ would be an isomorphism. This would be inherited by the subcategory $\Top$, giving a contradiction. + special_objects: initial object: description: empty space diff --git a/databases/catdat/data/category-implications/cartesian closed.yaml b/databases/catdat/data/category-implications/cartesian closed.yaml index a952b113..88d76953 100644 --- a/databases/catdat/data/category-implications/cartesian closed.yaml +++ b/databases/catdat/data/category-implications/cartesian closed.yaml @@ -107,3 +107,12 @@ - locally cartesian closed proof: Each slice is thin, semi-strongly connected, and has a terminal object. Thus, it corresponds to a linear order with a largest element $1$. Every such category is cartesian closed, where the exponential $a \Rightarrow b$ (Heyting implication) is $1$ when $a \leq b$ and otherwise $b$. is_equivalence: false + +- id: cartesian_closed_thin_implies_lcc + assumptions: + - cartesian closed + - thin + conclusions: + - locally cartesian closed + proof: In a thin category, every object is subterminal. Thus, the result follows from Corollary 2 here. + is_equivalence: false From 69ed7468b78623c85894941e3890e8cedd4c50cc Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Mon, 8 Jun 2026 22:23:37 -0400 Subject: [PATCH 2/3] Reorganize arguments as suggested in review --- content/LRS-not-cartesian-closed.md | 17 +++--- content/cartesian-closed-results.md | 27 ---------- content/subcategories.md | 53 ++++++++++++++++++- databases/catdat/data/categories/LRS_R.yaml | 10 ++-- .../cartesian closed.yaml | 2 +- 5 files changed, 68 insertions(+), 41 deletions(-) delete mode 100644 content/cartesian-closed-results.md diff --git a/content/LRS-not-cartesian-closed.md b/content/LRS-not-cartesian-closed.md index 96af5c59..5495deaf 100644 --- a/content/LRS-not-cartesian-closed.md +++ b/content/LRS-not-cartesian-closed.md @@ -14,7 +14,7 @@ The forgetful functor $U : \LRS_k \to \Top$ has a right adjoint $K : \Top \to \L _Proof._ In this adjunction, the counit $UK \to \id$ is just the identity. To describe the unit $\id \to KU$, we need to define a morphism $(X, \O_X) \to (X, \underline{k})$ for any locally ringed space $(X, \O_X)$ over $k$. This morphism will be the identity on topological spaces, and the pullback operation $\underline{k} \to \O_X$ will be the unique morphism of sheaves induced by the given structure of $\O_X$ as a sheaf of $k$-algebras. It is now straightforward to check this indeed defines an adjunction; and since the counit is an isomorphism, that implies that $K$ is fully faithful. $\square$ -We now show that this reflective subcategory is in fact also a coreflective subcategory. Recall that for $f \in \O_X(U)$ we have its vanishing set $V(f) \coloneqq \{x \in U : f(x) = 0\}$, where $f(x) \in \kappa(x)$ is the image of $f_x \in \O_{X,x}$ in the residue field. +We now show that this reflective subcategory is in fact also a coreflective subcategory. Recall that for $f \in \O_X(U)$ we have its vanishing set $V(f) \coloneqq \{x\in U : f \in \m_{X,x}\} = \{x \in U : f(x) = 0\}$, where $f(x) \in \kappa(x)$ is the image of $f_x \in \O_{X,x}$ in the residue field. ::: Lemma 2 For each object $X$ of $\LRS_k$, let $X_0$ be the set of points $x \in X$ such that the induced morphism from $k$ to the residue field $\kappa(x)$ is an isomorphism. We give $X_0$ the following strengthening of the subspace topology: it will be the topology where a neighborhood subbasis at $x \in X_0$ is the collection of sets of the form $X_0 \cap V(f)$ where $f \in \O_X(U)$ for some neighborhood $U$ of $x$ in $X$, and $x \in V(f)$. Then $X \mapsto X_0$ defines a functor $S : \LRS_k \to \Top$ that is right adjoint to $K$. @@ -35,15 +35,18 @@ $$\textstyle X_0 \cap V(f) = \bigcap_{a \in k^\times} (X_0 \cap D(f-a)),$$ which is already open in the subspace topology. Therefore, in this case, $X_0$ is given exactly the subspace topology. ::: Corollary 3 -If $k$ is a field, then $\LRS_k$ is not cartesian closed. +For any non-trivial commutative ring $R$, fix a quotient field $k$. Then the functor $K_R : \Top \to \LRS_R$ of equipping a topological space with the constant sheaf $\underline{k}$ is fully faithful; has a right adjoint; and preserves all inhabited limits. ::: -_Proof._ We know that $\Top$ is not cartesian closed. By Lemma 2, $\Top$ is a coreflective subcategory of $\LRS_k$. Moreover, the inclusion preserves binary products (in fact, all limits) by Lemma 1. Therefore, the claim follows from Lemma 1 [here](/content/cartesian-closed-results). $\square$ - -Finally, we generalize this result to arbitrary base rings. +_Proof._ The functor $K_R$ is the composition of $K : \Top \to \LRS_k$ and the forgetful functor $\LRS_k \to \LRS_R$. Since $\LRS_k$ is equivalent to the slice category of $\LRS_R$ over the subterminal object $\Spec k$, the forgetful functor is fully faithful; has right adjoint ${-} \times \Spec k$; and preserves all inhabited limits. Therefore, from the previously established results on $K$, the result follows. $\square$ ::: Corollary 4 -For any non-trivial commutative ring $R$, the category $\LRS_R$ is not cartesian closed. +Let $R$ be any non-trivial commutative ring. Then:
+(a) $\LRS_R$ is not cartesian closed.
+(b) $\LRS_R$ does not have cartesian filtered colimits.
+(c) $\LRS_R$ is not regular.
+(d) $\LRS_R$ does not have filtered-colimit-stable epimorphisms.
+(e) $\LRS_R$ does not have effective cocongruences. ::: -_Proof._ Let $k$ be a residue field of $R$. Then $\LRS_k$ is equivalent to the slice category $\LRS_R / \Spec k$ where $\Spec k$ is subterminal. Therefore, the claim follows from Corollary 3 above and Corollary 2 [here](/content/cartesian-closed-results). $\square$ +_Proof._ As before, we fix a quotient field of $R$. In each case, this is an easy application of a contrapositive of a result from [here](/content/subcategories) to the functor $K_R$. Namely, (a) follows from Lemma 5; (b) from the dual of Lemma 4; (c) from Lemma 7; (d) from the dual of Lemma 2 with the observation that $K_R$ preserves epimorphisms since it has a right adjoint; and (e) from the dual of Lemma 8. $\square$ diff --git a/content/cartesian-closed-results.md b/content/cartesian-closed-results.md deleted file mode 100644 index 0c9f8593..00000000 --- a/content/cartesian-closed-results.md +++ /dev/null @@ -1,27 +0,0 @@ ---- -title: Results about cartesian closed categories -description: We prove in particular when a coreflective subcategory of a cartesian closed category is again cartesian closed. -author: Daniel Schepler ---- - -## Results about cartesian closed categories - -::: Lemma 1 -Suppose $\D$ is a coreflective subcategory of $\C$ such that $\D$ has binary products and the inclusion functor preserves these binary products. If $\C$ is cartesian closed, then so is $\D$. -::: - -_Proof._ Let $U : \D \to \C$ be the inclusion functor with right adjoint $R : \C \to \D$. Then for any objects $X, Y, Z$ of $\D$ we have natural isomorphisms - -$$ -\begin{align*} -\Hom_\D(Z\times X, Y) & \cong \Hom_\C(UZ \times UX, UY) \\ -& \cong \Hom_\C(UZ, [UX,UY]) \\ -& \cong \Hom_\D\bigl(Z, R([UX,UY])\bigr). -\end{align*} -$$ - -::: Corollary 2 -If $\C$ is a cartesian closed category and $P$ is a [subterminal object](https://ncatlab.org/nlab/show/subterminal+object) of $\C$, then the slice category $\C / P$ is also cartesian closed. -::: - -_Proof._ The forgetful functor $\C / P \to \C$ is fully faithful; it has right adjoint ${-} \times P$; and it preserves binary products (in fact all inhabited limits). Hence, Lemma 1 applies. $\square$ diff --git a/content/subcategories.md b/content/subcategories.md index afb257c0..10078aa4 100644 --- a/content/subcategories.md +++ b/content/subcategories.md @@ -1,7 +1,7 @@ --- title: Results on subcategories description: We prove that several properties of categories descend to suitable subcategories. -author: Martin Brandenburg +author: Martin Brandenburg and Daniel Schepler --- ## Results on subcategories @@ -48,3 +48,54 @@ $$ $$ $\square$ + +::: Lemma 4 +Let $U : \C \to \D$ be a fully faithful functor with a left adjoint $L : \D \to \C$ (i.e. $\C$ is equivalent to a reflective subcategory of $\D$). Assume that $\D$ has cartesian filtered colimits, that $\C$ has finite products, and that $L$ preserves binary products. Then $\C$ also has cartesian filtered colimits. +::: + +_Proof._ +The proof is almost the same as the proof of Lemma 3, restricting to the case where $\I$ is finite and discrete. We just need to treat the case of empty $\I$ as a special case by observing that a filtered colimit of terminal objects is terminal. $\square$ + +::: Lemma 5 +Let $U : \C \to \D$ be a fully faithful functor with a right adjoint $R : \D \to \C$ (i.e. $\D$ is equivalent to a coreflective subcategory of $\D$). Assume that $\C$ has binary products, and that $U$ preserves these binary products. If $\D$ is cartesian closed, then so is $\C$, with exponentials in $\C$ given by +$$[X, Y]_{\C} \cong R([UX, UY]_{\D}).$$ +::: + +_Proof._ For any objects $X, Y, Z$ of $\C$ we have natural isomorphisms + +$$ +\begin{align*} +\Hom_\C(Z\times X, Y) & \cong \Hom_\D(UZ \times UX, UY) \\ +& \cong \Hom_\D(UZ, [UX,UY]) \\ +& \cong \Hom_\C\bigl(Z, R([UX,UY])\bigr). +\end{align*} +$$ + +$\square$ + +::: Corollary 6 +If $\C$ is a cartesian closed category and $P$ is a [subterminal object](https://ncatlab.org/nlab/show/subterminal+object) of $\C$, then the slice category $\C / P$ is also cartesian closed, with exponentials in $\C / P$ given by +$$[X, Y]_{\C / P} \cong [X, Y]_{\C} \times P.$$ +::: + +_Proof._ The forgetful functor $\C / P \to \C$ is fully faithful; it has right adjoint ${-} \times P$; and it preserves binary products (in fact all inhabited limits). Hence, Lemma 4 applies. $\square$ + +::: Lemma 7 +Let $U : \C \to \D$ be a fully faithful functor. Assume that $\C$ has finite limits and coequalizers, and that $U$ preserves pullbacks and coequalizers. If $\D$ is regular, then so is $\C$. +::: + +_Proof._ Since $\C$ has finite limits and coequalizers, the only nontrivial part of proving $\C$ is regular is to check that regular epimorphisms are stable under pullback in $\C$. Since $U$ preserves pullbacks and regular epimorphisms, it suffices to show that $U$ reflects regular epimorphisms. Thus, suppose $f : X \to Y$ is a morphism in $\C$ with $Uf$ a regular epimorphism. Then in $\C$ we have the diagram +$$X \times_Y X \rightrightarrows X \to \im(f) \xrightarrow{i} Y$$ +where $X \times_Y X$ is the kernel pair of $f$, and $\im(f)$ is the coequalizer. By the assumptions, the image under $U$ is equivalent to the diagram in $\D$: +$$UX \times_{UY} UX \rightrightarrows UX \to \im(Uf) \xrightarrow{Ui} UY$$ +where $UX \times_{UY} UX$ is the kernel pair of $Uf$, and $\im(Uf)$ is the coequalizer. Since $Uf$ is a regular epimorphism, we must have $Ui$ is an isomorphism. Since $U$ is fully faithful and therefore conservative, we get $i$ is an isomorphism as well, so $f$ is a regular epimorphism. $\square$ + +::: Lemma 8 +Let $U : \C \to \D$ be a fully faithful functor. Assume that $\C$ has finite limits and coequalizers, and that $U$ preserves inhabited finite limits and coequalizers. If $\D$ has effective congruences, then so does $\C$. +::: + +_Proof._ Suppose we have a congruence $E \hookrightarrow X\times X$ in $\C$. We can then form the quotient $X \to X/E$ as a coequalizer, along with the kernel pair $X \times_{X/E} X$ and the comparison map $i$ in the diagram below: +$$E \xrightarrow{i} X \times_{X/E} X \rightrightarrows X \to X/E.$$ +By the assumptions, the image under $U$ is equivalent to the diagram in $\D$: +$$UE \xrightarrow{Ui} UX \times_{U(X/E)} UX \rightrightarrows UX \to U(X/E).$$ +Here, $UE \rightrightarrows UX$ is a congruence, since the reflexivity and symmetry morphisms for $E$ are easily seen to transform under $U$ to reflexivity and symmetry morphisms for $UE$, and similarly since $U$ preserves pullbacks, the transitivity morphism for $E$ transforms under $U$ to a transitivity morphism for $UE$. This congruence $UE$ of $\D$ is effective, so we must have $Ui$ is an isomorphism. Since $U$ is fully faithful and therefore conservative, we get $i$ is an isomorphism as well, so $E$ is effective. $\square$ diff --git a/databases/catdat/data/categories/LRS_R.yaml b/databases/catdat/data/categories/LRS_R.yaml index f5417843..aac3a84a 100644 --- a/databases/catdat/data/categories/LRS_R.yaml +++ b/databases/catdat/data/categories/LRS_R.yaml @@ -54,20 +54,20 @@ unsatisfied_properties: Alternatively, using the usual adjunction between affine schemes and locally ringed spaces (EGA I (1971), Ch. 1, Prop. 1.6.3), a generating set in $\LRS_R$ would induce a generating set in the category of affine $R$-schemes, which contradicts the fact that $\CAlg(R)$ does not have a cogenerating set. - property: cartesian closed - proof: This is proved here. + proof: This is Corollary 4(a) here. check_redundancy: false - property: cartesian filtered colimits - proof: As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. Therefore, if $\LRS_R$ had cartesian filtered colimits, then $\Top$ would also, giving a contradiction. + proof: This is Corollary 4(b) here. - property: regular - proof: 'As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. Therefore, if $\LRS_R$ were regular, then for every regular epimorphism $X \to Y$ and morphism $Z \to Y$, the pullback $f : Z \times_Y X \to X$ would satisfy that the canonical morphism from the quotient of the kernel pair of $f$ to $X$ is an isomorphism. This would be inherited by the subcategory $\Top$, and since $\Top$ is finitely complete and has coequalizers, that would imply $\Top$ is also regular, giving a contradiction.' + proof: This is Corollary 4(c) here. - property: cofiltered-limit-stable epimorphisms - proof: 'As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. From here, the proof is similar to the one from $\Top$: we apply the contrapositive of the dual of this lemma to the functor $\Set \to \LRS_k$ which equips a set with the indiscrete topology and the constant sheaf of $k$.' + proof: This is Corollary 4(d) here. - property: effective cocongruences - proof: As a corollary of the results here, if we choose a quotient field $k$ of $R$, then the functor $\Top \to \LRS_R$ of equipping a topological space with the constant sheaf of $k$ is fully faithful, and preserves all colimits and all inhabited limits. Therefore, if $\LRS_R$ had effective cocongruences, then for every cocongruence $E$ on $X$, the canonical morphism from the cokernel pair of the equalizer of $E$ to $E$ would be an isomorphism. This would be inherited by the subcategory $\Top$, giving a contradiction. + proof: This is Corollary 4(e) here. special_objects: initial object: diff --git a/databases/catdat/data/category-implications/cartesian closed.yaml b/databases/catdat/data/category-implications/cartesian closed.yaml index 88d76953..a8e6c17e 100644 --- a/databases/catdat/data/category-implications/cartesian closed.yaml +++ b/databases/catdat/data/category-implications/cartesian closed.yaml @@ -114,5 +114,5 @@ - thin conclusions: - locally cartesian closed - proof: In a thin category, every object is subterminal. Thus, the result follows from Corollary 2 here. + proof: In a thin category, every object is subterminal. Thus, the result follows from Corollary 6 here. is_equivalence: false From 9e109eb3aa334e1c793dd06511786a3b2c1f4ea1 Mon Sep 17 00:00:00 2001 From: Daniel Schepler Date: Mon, 8 Jun 2026 22:36:25 -0400 Subject: [PATCH 3/3] Rename LRS content page to reflect its expanded scope --- ...RS-not-cartesian-closed.md => Top-embeds-in-LRS.md} | 8 ++++---- content/subcategories.md | 4 ++-- databases/catdat/data/categories/LRS_R.yaml | 10 +++++----- 3 files changed, 11 insertions(+), 11 deletions(-) rename content/{LRS-not-cartesian-closed.md => Top-embeds-in-LRS.md} (90%) diff --git a/content/LRS-not-cartesian-closed.md b/content/Top-embeds-in-LRS.md similarity index 90% rename from content/LRS-not-cartesian-closed.md rename to content/Top-embeds-in-LRS.md index 5495deaf..c34ee7af 100644 --- a/content/LRS-not-cartesian-closed.md +++ b/content/Top-embeds-in-LRS.md @@ -1,12 +1,12 @@ --- -title: The category of locally ringed spaces is not cartesian closed -description: A proof that the category of locally ringed spaces over a non-trivial ring R is not cartesian closed +title: An embedding of the category of topological spaces in the category of locally ringed spaces +description: Describes a functor which makes the category of topological spaces a coreflective and "almost reflective" subcategory of the category of locally ringed spaces. From the properties of this embedding, we can rule out several properties for the category of locally ringed spaces, using the corresponding failures of these properties for the category of topological spaces. author: Daniel Schepler --- -## The category of locally ringed spaces is not cartesian closed +## An embedding of the category of topological spaces in the category of locally ringed spaces -For most of this development, we will be dealing with the case of $\LRS_k$ where $k$ is a field. We begin by describing $\Top$ as a reflective subcategory of $\LRS_k$. +For much of this development, we will be dealing with the case of $\LRS_k$ where $k$ is a field. We begin by describing $\Top$ as a reflective subcategory of $\LRS_k$. ::: Lemma 1 The forgetful functor $U : \LRS_k \to \Top$ has a right adjoint $K : \Top \to \LRS_k$ of equipping a topological space $X$ with the constant sheaf $\underline{k}$. Furthermore, the functor $K$ is fully faithful, thus making $\Top$ into a reflective subcategory of $\LRS_k$. diff --git a/content/subcategories.md b/content/subcategories.md index 10078aa4..1ef5278b 100644 --- a/content/subcategories.md +++ b/content/subcategories.md @@ -57,7 +57,7 @@ _Proof._ The proof is almost the same as the proof of Lemma 3, restricting to the case where $\I$ is finite and discrete. We just need to treat the case of empty $\I$ as a special case by observing that a filtered colimit of terminal objects is terminal. $\square$ ::: Lemma 5 -Let $U : \C \to \D$ be a fully faithful functor with a right adjoint $R : \D \to \C$ (i.e. $\D$ is equivalent to a coreflective subcategory of $\D$). Assume that $\C$ has binary products, and that $U$ preserves these binary products. If $\D$ is cartesian closed, then so is $\C$, with exponentials in $\C$ given by +Let $U : \C \to \D$ be a fully faithful functor with a right adjoint $R : \D \to \C$ (i.e. $\C$ is equivalent to a coreflective subcategory of $\D$). Assume that $\C$ has binary products, and that $U$ preserves these binary products. If $\D$ is cartesian closed, then so is $\C$, with exponentials in $\C$ given by $$[X, Y]_{\C} \cong R([UX, UY]_{\D}).$$ ::: @@ -78,7 +78,7 @@ If $\C$ is a cartesian closed category and $P$ is a [subterminal object](https:/ $$[X, Y]_{\C / P} \cong [X, Y]_{\C} \times P.$$ ::: -_Proof._ The forgetful functor $\C / P \to \C$ is fully faithful; it has right adjoint ${-} \times P$; and it preserves binary products (in fact all inhabited limits). Hence, Lemma 4 applies. $\square$ +_Proof._ The forgetful functor $\C / P \to \C$ is fully faithful; it has right adjoint ${-} \times P$; and it preserves binary products (in fact all inhabited limits). Hence, Lemma 5 applies. $\square$ ::: Lemma 7 Let $U : \C \to \D$ be a fully faithful functor. Assume that $\C$ has finite limits and coequalizers, and that $U$ preserves pullbacks and coequalizers. If $\D$ is regular, then so is $\C$. diff --git a/databases/catdat/data/categories/LRS_R.yaml b/databases/catdat/data/categories/LRS_R.yaml index aac3a84a..d50bea32 100644 --- a/databases/catdat/data/categories/LRS_R.yaml +++ b/databases/catdat/data/categories/LRS_R.yaml @@ -54,20 +54,20 @@ unsatisfied_properties: Alternatively, using the usual adjunction between affine schemes and locally ringed spaces (EGA I (1971), Ch. 1, Prop. 1.6.3), a generating set in $\LRS_R$ would induce a generating set in the category of affine $R$-schemes, which contradicts the fact that $\CAlg(R)$ does not have a cogenerating set. - property: cartesian closed - proof: This is Corollary 4(a) here. + proof: This is Corollary 4(a) here. check_redundancy: false - property: cartesian filtered colimits - proof: This is Corollary 4(b) here. + proof: This is Corollary 4(b) here. - property: regular - proof: This is Corollary 4(c) here. + proof: This is Corollary 4(c) here. - property: cofiltered-limit-stable epimorphisms - proof: This is Corollary 4(d) here. + proof: This is Corollary 4(d) here. - property: effective cocongruences - proof: This is Corollary 4(e) here. + proof: This is Corollary 4(e) here. special_objects: initial object: