diff --git a/.vscode/settings.json b/.vscode/settings.json index 627e8a24..d4f742a0 100644 --- a/.vscode/settings.json +++ b/.vscode/settings.json @@ -25,6 +25,7 @@ "abelianization", "abelianize", "Adamek", + "adic", "algébriques", "anneaux", "Artin", @@ -203,6 +204,7 @@ "surjective", "tensoring", "Turso", + "uncountably", "unital", "vercel", "Vite", diff --git a/database/data/003_properties/003_limits-colimits-behavior.sql b/database/data/003_properties/003_limits-colimits-behavior.sql index 208234c7..0069c40b 100644 --- a/database/data/003_properties/003_limits-colimits-behavior.sql +++ b/database/data/003_properties/003_limits-colimits-behavior.sql @@ -95,7 +95,15 @@ VALUES 'has', 'In a category $\mathcal{C}$, which we assume to have filtered colimits and finite limits, we say that filtered colimits are exact if for every finite category $\mathcal{I}$ the functor $\lim : [\mathcal{I}, \mathcal{C}] \to \mathcal{C}$ preserves filtered colimits. Equivalently, for every diagram $X : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$, where $\mathcal{I}$ is finite and $\mathcal{J}$ is filtered, the canonical morphism $\mathrm{colim}_{j} \lim_{i} X(i,j) \to \lim_{i} \mathrm{colim}_j X(i,j)$ is an isomorphism.', 'https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits', - NULL, + 'exact cofiltered limits', + TRUE +), +( + 'exact cofiltered limits', + 'has', + 'In a category $\mathcal{C}$, which we assume to have cofiltered limits and finite colimits, we say that cofiltered limits are exact if for every finite category $\mathcal{I}$ the functor $\mathrm{colim} : [\mathcal{I}, \mathcal{C}] \to \mathcal{C}$ preserves cofiltered limits. Equivalently, for every diagram $X : \mathcal{I} \times \mathcal{J} \to \mathcal{C}$, where $\mathcal{I}$ is finite and $\mathcal{J}$ is cofiltered, the canonical morphism $\lim_{j} \mathrm{colim}_{i} X(i,j) \to \mathrm{colim}_{i} \lim_j X(i,j)$ is an isomorphism.', + 'https://ncatlab.org/nlab/show/commutativity+of+limits+and+colimits', + 'exact filtered colimits', TRUE ), ( diff --git a/database/data/003_properties/005_morphism-behavior.sql b/database/data/003_properties/005_morphism-behavior.sql index 86b416e1..e520dd01 100644 --- a/database/data/003_properties/005_morphism-behavior.sql +++ b/database/data/003_properties/005_morphism-behavior.sql @@ -105,4 +105,20 @@ VALUES 'https://ncatlab.org/nlab/show/one-way+category', 'one-way', TRUE +), +( + 'filtered-colimit-stable monomorphisms', + 'has', + 'A category has filtered-colimit-stable monomorphisms if it has filtered colimits and for every filtered diagram of monomorphisms $(X_i \to Y_i)$ also their colimit $\mathrm{colim}_i X_i \to \mathrm{colim}_i Y_i$ is a monomorphism.', + NULL, + 'cofiltered-limit-stable epimorphisms', + TRUE +), +( + 'cofiltered-limit-stable epimorphisms', + 'has', + 'A category has cofiltered-limit-stable epimorphisms if it has cofiltered limits and for every cofiltered diagram of epimorphisms $(X_i \to Y_i)$ also their limit $\lim_i X_i \to \lim_i Y_i$ is an epimorphism.', + NULL, + 'filtered-colimit-stable monomorphisms', + TRUE ); \ No newline at end of file diff --git a/database/data/003_properties/100_related-properties.sql b/database/data/003_properties/100_related-properties.sql index dfd1b997..35ec8b83 100644 --- a/database/data/003_properties/100_related-properties.sql +++ b/database/data/003_properties/100_related-properties.sql @@ -185,11 +185,20 @@ VALUES ('exact filtered colimits', 'filtered colimits'), ('exact filtered colimits', 'finitely complete'), ('exact filtered colimits', 'cartesian filtered colimits'), +('exact cofiltered limits', 'cofiltered limits'), +('exact cofiltered limits', 'finitely cocomplete'), +('exact cofiltered limits', 'cocartesian cofiltered limits'), +('exact cofiltered limits', 'cofiltered-limit-stable epimorphisms'), ('cartesian filtered colimits', 'filtered colimits'), ('cartesian filtered colimits', 'finite products'), ('cartesian filtered colimits', 'exact filtered colimits'), ('cocartesian cofiltered limits', 'cofiltered limits'), ('cocartesian cofiltered limits', 'finite coproducts'), +('cocartesian cofiltered limits', 'exact cofiltered limits'), +('filtered-colimit-stable monomorphisms', 'exact filtered colimits'), +('filtered-colimit-stable monomorphisms', 'filtered colimits'), +('cofiltered-limit-stable epimorphisms', 'exact cofiltered limits'), +('cofiltered-limit-stable epimorphisms', 'cofiltered limits'), ('generator', 'generating set'), ('generating set', 'generator'), ('Grothendieck abelian', 'abelian'), diff --git a/database/data/004_property-assignments/Ab.sql b/database/data/004_property-assignments/Ab.sql index 5785e039..d0c4da72 100644 --- a/database/data/004_property-assignments/Ab.sql +++ b/database/data/004_property-assignments/Ab.sql @@ -34,4 +34,10 @@ VALUES 'skeletal', FALSE, 'This is trivial.' +), +( + 'Ab', + 'cofiltered-limit-stable epimorphisms', + FALSE, + 'We know that $\mathbf{CRing}$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the forgetful functor $\mathbf{CRing} \to \mathbf{Ab}$. Even though it does not preserve epimorphisms, our counterexample for $\mathbf{CRing}$ has used the surjective ring homomorphisms $\mathbb{Z} \to \mathbb{Z}/p^n$, which remain epimorphisms in $\mathbf{Ab}$.' ); diff --git a/database/data/004_property-assignments/CRing.sql b/database/data/004_property-assignments/CRing.sql index 8511ccee..c7957c66 100644 --- a/database/data/004_property-assignments/CRing.sql +++ b/database/data/004_property-assignments/CRing.sql @@ -82,4 +82,10 @@ VALUES 'regular quotient object classifier', FALSE, 'Assume that $P \to \mathbb{Z}$ is a regular quotient object classifier. If $J$ denotes its kernel, this means that every ideal $I \subseteq A$ of any commutative ring has the form $I = \langle \varphi(J) \rangle$ for a unique homomorphism $\varphi : P \to A$. If $\sigma : A \to A$ is an automorphism with $\sigma(I)=I$, then uniqueness gives us $\sigma \circ \varphi = \varphi$, which means that $\varphi(J)$ lies in $A^{\sigma}$, the fixed ring of $\sigma$. But then $I$ is generated by elements in the fixed ring. This fails for $A = \mathbb{Z}[X]$, $I = \langle X \rangle$, $\sigma(X)=-X$. The fixed ring is $\mathbb{Z}[X^2]$, and if $I$ was generated by elements $f \in \mathbb{Z}[X^2] \cap I$, they would be multiples of $X^2$, but $X$ is not a multiple of $X^2$.' +), +( + 'CRing', + 'cofiltered-limit-stable epimorphisms', + FALSE, + 'For a prime $p$ consider the sequence of projections $\cdots \to \mathbb{Z}/p^2 \to \mathbb{Z}/p$ and the constant sequence $\cdots \to \mathbb{Z} \to \mathbb{Z}$. The surjective homomorphisms $\mathbb{Z} \to \mathbb{Z}/p^n$ induce the homomorphism $\mathbb{Z} \to \mathbb{Z}_p$ in the limit, where $\mathbb{Z}_p$ is the ring of $p$-adic integers. It is not surjective since $\mathbb{Z}_p$ is uncountable, but this is not sufficient (at least, for this category): We need to use SP/04W0 to conclude that it is no epimorphism in $\mathbf{CRing}$.' ); diff --git a/database/data/004_property-assignments/FS.sql b/database/data/004_property-assignments/FS.sql index b8b8d8b2..4d56a486 100644 --- a/database/data/004_property-assignments/FS.sql +++ b/database/data/004_property-assignments/FS.sql @@ -53,6 +53,12 @@ VALUES TRUE, 'If $f : X \to Y$ is a surjective map of finite sets, it is the coequalizer of the two projections $p_1, p_2 : X \times_Y X \rightrightarrows X$ in $\mathbf{FinSet}$, but also in $\mathbf{FS}$. Notice that $p_1,p_2$ are surjective. Even though $X \times_Y X$ is not a pullback in $\mathbf{FS}$, we can use this finite set here.' ), +( + 'FS', + 'filtered-colimit-stable monomorphisms', + TRUE, + 'This is because every monomorphism is an isomorphism in this category (see below), and isomorphisms are always stable under any type of colimit.' +), ( 'FS', 'multi-terminal object', diff --git a/database/data/004_property-assignments/Grp.sql b/database/data/004_property-assignments/Grp.sql index 5d71a2e6..47a54348 100644 --- a/database/data/004_property-assignments/Grp.sql +++ b/database/data/004_property-assignments/Grp.sql @@ -92,4 +92,10 @@ VALUES is injective, but often fails to be surjective because the components of an element in the image have bounded free product length (the number of factors appearing in the reduced form). Specifically, consider the free groups $G = \langle y \rangle$ and $H_n = \langle x_1,\dotsc,x_n \rangle$ for $n \in \mathbb{N}$ with the truncation maps $H_{n+1} \to H_n$, $x_{n+1} \mapsto 1$. Define
$p_n := x_1 \, y \, x_2 \, y \, \cdots \, x_{n-1} \, y \, x_n \, y^{-(n-1)} \in G \sqcup H_n$.
If we substitute $x_{n+1}=1$ in $p_{n+1}$, we get $p_n$. Thus, we have $p = (p_n) \in \lim_n (G \sqcup H_n)$. This element does not lie in the image of $\alpha$ since the free product length of $p_n$ (which is well-defined) is $2n$, which is unbounded.' +), +( + 'Grp', + 'cofiltered-limit-stable epimorphisms', + FALSE, + 'We know that $\mathbf{Ab}$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the forgetful functor $\mathbf{Ab} \to \mathbf{Grp}$ (which does preserve epimorphisms).' ); \ No newline at end of file diff --git a/database/data/004_property-assignments/M-Set.sql b/database/data/004_property-assignments/M-Set.sql index 685f8ca3..812c3567 100644 --- a/database/data/004_property-assignments/M-Set.sql +++ b/database/data/004_property-assignments/M-Set.sql @@ -34,4 +34,10 @@ VALUES 'Malcev', FALSE, 'Endow the set $\mathbb{N}$ with the trivial $M$-action, and consider the subset $\{(a,b) : a \leq b \}$ of $\mathbb{N}^2$.' +), +( + 'M-Set', + '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 $\mathbf{Set} \to M{-}\mathbf{Set}$ that equips a set with the trivial $M$-action.' ); diff --git a/database/data/004_property-assignments/Meas.sql b/database/data/004_property-assignments/Meas.sql index a1ef314e..106152a4 100644 --- a/database/data/004_property-assignments/Meas.sql +++ b/database/data/004_property-assignments/Meas.sql @@ -53,6 +53,12 @@ VALUES TRUE, 'Take the two-element set $2$ endowed with the trivial $\sigma$-algebra (where only $\varnothing$ and $2$ are measurable), and use that $2$ is a cogenerator for $\mathbf{Set}$.' ), +( + 'Meas', + 'filtered-colimit-stable monomorphisms', + TRUE, + 'This follows from this lemma applied to the forgetful functor to $\mathbf{Set}$.' +), ( 'Meas', 'semi-strongly connected', diff --git a/database/data/004_property-assignments/Prost.sql b/database/data/004_property-assignments/Prost.sql index 3b0a67b5..4a2d9a52 100644 --- a/database/data/004_property-assignments/Prost.sql +++ b/database/data/004_property-assignments/Prost.sql @@ -94,4 +94,10 @@ VALUES 'co-Malcev', FALSE, 'See MO/509552: Consider the forgetful functor $U : \mathbf{Prost} \to \mathbf{Set}$ and the relation $R \subseteq U^2$ defined by $R(A) := \{(a,b) \in U(A)^2 : a \leq b\}$. Both are representable: $U$ by the singleton preordered set and $R$ by $\{0 \leq 1 \}$. It is clear that $R$ is reflexive, but not symmetric.' +), +( + 'Prost', + '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.' ); diff --git a/database/data/004_property-assignments/Ring.sql b/database/data/004_property-assignments/Ring.sql index e17d1ee5..aab7245a 100644 --- a/database/data/004_property-assignments/Ring.sql +++ b/database/data/004_property-assignments/Ring.sql @@ -90,4 +90,10 @@ VALUES 'Consider the ring $A = \mathbb{Z}[X]$ and the sequence of rings $B_n = \mathbb{Z}[Y]/(Y^{n+1})$ with projections $B_{n+1} \to B_n$, whose limit is $\mathbb{Z}[[Y]]$. Every element in the coproduct of rings $\mathbb{Z}[X] \sqcup \mathbb{Z}[[Y]]$ has a finite "free product" length. Now consider the elements
$w_n = (1 + XY) (1+XY^2) \cdots (1+X Y^n) \in A \sqcup B_n$.
Because of $w_n \equiv w_{n-1} \bmod Y^n$ these form an element $w \in \lim_n (A \sqcup B_n)$. Expanding $w_n$, the longest term is $XY XY^2 \cdots X Y^n$ of "free product" length $2n$, which is unbounded.' +), +( + 'Ring', + 'cofiltered-limit-stable epimorphisms', + FALSE, + 'We know that $\mathbf{CRing}$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the forgetful functor $\mathbf{CRing} \to \mathbf{Ring}$. Even though it is not clear if it preserves epimorphisms, our counterexample for $\mathbf{CRing}$ has used the surjective ring homomorphisms $\mathbb{Z} \to \mathbb{Z}/p^n$, which remain epimorphisms in $\mathbf{Ring}$.' ); \ No newline at end of file diff --git a/database/data/004_property-assignments/Rng.sql b/database/data/004_property-assignments/Rng.sql index 08a6240c..8d5db925 100644 --- a/database/data/004_property-assignments/Rng.sql +++ b/database/data/004_property-assignments/Rng.sql @@ -78,4 +78,10 @@ VALUES 'Consider the ring $A = \mathbb{Z}[X]$ and the sequence of rings $B_n = \mathbb{Z}[Y]/(Y^{n+1})$ with projections $B_{n+1} \to B_n$, whose limit is $\mathbb{Z}[[Y]]$ (both in $\mathbf{Ring}$ and $\mathbf{Rng}$). Every element in the coproduct of rngs $\mathbb{Z}[X] \sqcup \mathbb{Z}[[Y]]$ has a finite "free product" length. Now consider the elements
$w_n = (1 + XY) (1+XY^2) \cdots (1+X Y^n) - 1 \in A \sqcup B_n$.
Because of $w_n \equiv w_{n-1} \bmod Y^n$ these form an element $w \in \lim_n (A \sqcup B_n)$. Expanding $w_n$, the longest term is $XY XY^2 \cdots X Y^n$ of "free product" length $2n$, which is unbounded.' +), +( + 'Rng', + 'cofiltered-limit-stable epimorphisms', + FALSE, + 'We know that $\mathbf{CRing}$ does not have this property. Now use the contrapositive of the dual of this lemma applied to the forgetful functor $\mathbf{CRing} \to \mathbf{Rng}$. Even though it is not clear if it preserves epimorphisms, our counterexample for $\mathbf{CRing}$ has used the surjective ring homomorphisms $\mathbb{Z} \to \mathbb{Z}/p^n$, which remain epimorphisms in $\mathbf{Rng}$.' ); \ No newline at end of file diff --git a/database/data/004_property-assignments/Set.sql b/database/data/004_property-assignments/Set.sql index 8e266208..b587ee04 100644 --- a/database/data/004_property-assignments/Set.sql +++ b/database/data/004_property-assignments/Set.sql @@ -40,4 +40,10 @@ VALUES 'Malcev', FALSE, 'There are lots of non-symmetric reflexive relations, for example $\leq$ on $\mathbb{N}$.' +), +( + 'Set', + 'cofiltered-limit-stable epimorphisms', + FALSE, + 'Pick any decreasing sequence of non-empty sets $X_0 \supseteq X_1 \supseteq \cdots$ with empty intersection, such as $X_n = \mathbb{N}_{\geq n}$. The unique map $X_n \to 1$ is surjective, but their limit $\varnothing \to 1$ is not surjective.' ); diff --git a/database/data/004_property-assignments/Set_pointed.sql b/database/data/004_property-assignments/Set_pointed.sql index a0719751..666cb988 100644 --- a/database/data/004_property-assignments/Set_pointed.sql +++ b/database/data/004_property-assignments/Set_pointed.sql @@ -78,4 +78,10 @@ VALUES 'conormal', FALSE, 'Every cokernel is "injective away from the base point". Formally, if $p : A \to B$ is a cokernel in $\mathbf{Set}_*$, it has the property that $p(x)=p(y) \neq 0$ implies $x=y$ (where $0$ denotes the base point). Clearly this is not satisfied for every surjective pointed map, consider $(\mathbb{N},0) \to (\{0,1\},0)$ defined by $0 \mapsto 0$ and $x \mapsto 1$ for $x > 0$.' +), +( + 'Set*', + '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 $\mathbf{Set} \to \mathbf{Set}_*$ that freely adds a base point.' ); diff --git a/database/data/004_property-assignments/Setne.sql b/database/data/004_property-assignments/Setne.sql index 0f34870f..cb449bff 100644 --- a/database/data/004_property-assignments/Setne.sql +++ b/database/data/004_property-assignments/Setne.sql @@ -108,6 +108,12 @@ VALUES TRUE, 'Any natural numbers object in $\mathbf{Set}$, such as $(\mathbb{N},0,n \mapsto n+1)$, is clearly also one in $\mathbf{Set}_{\neq \varnothing}$.' ), +( + 'Setne', + 'filtered-colimit-stable monomorphisms', + TRUE, + 'This follows from this lemma applied to the forgetful functor to $\mathbf{Set}$.' +), ( 'Setne', 'sequential limits', diff --git a/database/data/004_property-assignments/Top.sql b/database/data/004_property-assignments/Top.sql index 7b8cbdf8..dea6e8f5 100644 --- a/database/data/004_property-assignments/Top.sql +++ b/database/data/004_property-assignments/Top.sql @@ -71,6 +71,12 @@ VALUES TRUE, 'The category has all limits and colimits, and the regular monomorphisms are the subspace inclusions. Thus, it suffices to prove that subspace inclusions are stable under pushouts. For a proof see e.g. Lemma 3.6 at the nLab.' ), +( + 'Top', + 'filtered-colimit-stable monomorphisms', + TRUE, + 'This follows from this lemma applied to the forgetful functor to $\mathbf{Set}$.' +), ( 'Top', 'cartesian filtered colimits', diff --git a/database/data/004_property-assignments/Top_pointed.sql b/database/data/004_property-assignments/Top_pointed.sql index fa4e679d..174f9452 100644 --- a/database/data/004_property-assignments/Top_pointed.sql +++ b/database/data/004_property-assignments/Top_pointed.sql @@ -84,6 +84,12 @@ VALUES 'We continue the proof for $\mathbf{Set}_*$ by showing that the natural bijective map
$\alpha : X \vee \lim_i Y_i \to \lim_i (X \vee Y_i)$
is open. It suffices to consider open sets of two types: (1) If $U \subseteq X$ is open, the $\alpha$-image of $U \vee \lim_i Y_i$ is $p_{i_0}^{-1}(U \vee Y_{i_0})$ for any chosen index $i_0$, hence open. (2) If $i$ is an index and $V_i \subseteq Y_i$ is open, then the $\alpha$-image of $X \vee (p_i^{-1}(V_i) \cap \lim_i Y_i)$ is $p_i^{-1}(X \vee V_i)$, hence open.' ), +( + 'Top*', + 'filtered-colimit-stable monomorphisms', + TRUE, + 'This follows from this lemma applied to the forgetful functor to $\mathbf{Set}$.' +), ( 'Top*', 'coregular', diff --git a/database/data/005_implications/002_limits-colimits-behavior-implications.sql b/database/data/005_implications/002_limits-colimits-behavior-implications.sql index 26cc5805..036a29be 100644 --- a/database/data/005_implications/002_limits-colimits-behavior-implications.sql +++ b/database/data/005_implications/002_limits-colimits-behavior-implications.sql @@ -69,6 +69,15 @@ VALUES 'This holds by definition.', FALSE ), +( + 'exact_filtered_colimits_monos', + '["exact filtered colimits"]', + '["filtered-colimit-stable monomorphisms"]', + 'This is because $X \to Y$ is a monomorphism iff the diagram +

$\begin{array}{ccc} X & \rightarrow & X \\ \downarrow && \downarrow \\ X & \rightarrow & Y \end{array}$

+ is a pullback, and if a functor preserves finite limits, it preserves pullbacks in particular.', + FALSE +), ( 'cartesian_filtered_colimits_condition', '["cartesian filtered colimits"]', diff --git a/database/data/005_implications/004_morphism-behavior-implications.sql b/database/data/005_implications/004_morphism-behavior-implications.sql index 951bc3e4..0deaa2b9 100644 --- a/database/data/005_implications/004_morphism-behavior-implications.sql +++ b/database/data/005_implications/004_morphism-behavior-implications.sql @@ -152,4 +152,18 @@ VALUES '["reflexive coequalizers"]', 'Every reflexive pair is equal: If $f s = g s = \mathrm{id}$, then since $s f = \mathrm{id}$ (one-way), we must have $f = s^{-1}$, and likewise $g = s^{-1}$.', FALSE +), +( + 'filtered_monos_assumption', + '["filtered-colimit-stable monomorphisms"]', + '["filtered colimits"]', + 'This holds by definition.', + FALSE +), +( + 'filtered_monos_trivial', + '["left cancellative", "filtered colimits"]', + '["filtered-colimit-stable monomorphisms"]', + 'This is trivial.', + FALSE ); \ No newline at end of file diff --git a/database/data/010_lemmas/000_lemmas.sql b/database/data/010_lemmas/000_lemmas.sql index f0a7ebc8..5515bc82 100644 --- a/database/data/010_lemmas/000_lemmas.sql +++ b/database/data/010_lemmas/000_lemmas.sql @@ -93,4 +93,13 @@ INSERT INTO lemmas ( 'Coreflection of subobject classifiers', 'Let $\mathcal{D}$ be a category with a (regular) subobject classifier $\Omega$. Assume that $\mathcal{C} \to \mathcal{D}$ is a full subcategory such that (1) any (regular) $\mathcal{D}$-subobject of an object in $\mathcal{C}$ already lies in $\mathcal{C}$, (2) it is coreflective, i.e. there is a functor $R : \mathcal{D} \to \mathcal{C}$ right adjoint to the inclusion. Then $R(\Omega)$ is a (regular) subobject classifier in $\mathcal{C}$.', 'If $X \in \mathcal{C}$, then $\mathrm{Hom}(X,R(\Omega)) \cong \mathrm{Hom}(X,\Omega)$ is isomorphic to the collection of $\mathcal{D}$-subobjects of $X$, which by assumption coincide with the $\mathcal{C}$-subobjects of $X$.' +), +( + 'filtered-monos', + 'Detection of filtered-colimit-stable monomorphisms', + 'Let $\mathcal{C}$ be a category with filtered colimits. Assume that $U : \mathcal{C} \to \mathcal{D}$ is faithful functor which preserves monomorphisms and filtered colimits. If monomorphisms in $\mathcal{D}$ are stable under filtered colimits, then the same is true for $\mathcal{C}$. +

+ For the record, here is the dual statement: Let $\mathcal{C}$ be a category with cofiltered limits. Assume that $U : \mathcal{C} \to \mathcal{D}$ is faithful functor which preserves epimorphisms and cofiltered limits. If epimorphisms in $\mathcal{D}$ are stable under cofiltered limits, then the same is true for $\mathcal{C}$. + ', + 'Since $U$ is faithful, it reflects monomorphisms. From here the proof is straight forward.' ); \ No newline at end of file