diff --git a/database/data/categories/FinVect.yaml b/database/data/categories/FinVect.yaml index dcf12144e..7a6fed25e 100644 --- a/database/data/categories/FinVect.yaml +++ b/database/data/categories/FinVect.yaml @@ -12,6 +12,7 @@ tags: related: - Vect - Ab_fg + - Vect_c satisfied_properties: - property: locally small diff --git a/database/data/categories/Grp_c.yaml b/database/data/categories/Grp_c.yaml index 19222b462..2c1671c91 100644 --- a/database/data/categories/Grp_c.yaml +++ b/database/data/categories/Grp_c.yaml @@ -13,6 +13,7 @@ related: - Grp - FinGrp - Set_c + - Vect_c satisfied_properties: - property: locally small diff --git a/database/data/categories/Set_c.yaml b/database/data/categories/Set_c.yaml index ba19a412b..e3f21981b 100644 --- a/database/data/categories/Set_c.yaml +++ b/database/data/categories/Set_c.yaml @@ -13,6 +13,7 @@ related: - FinSet - Set - Grp_c + - Vect_c satisfied_properties: - property: locally small diff --git a/database/data/categories/Vect.yaml b/database/data/categories/Vect.yaml index d7045dcad..437fae58e 100644 --- a/database/data/categories/Vect.yaml +++ b/database/data/categories/Vect.yaml @@ -14,6 +14,7 @@ related: - FinVect - FiltVect - FreeAb + - Vect_c satisfied_properties: - property: split abelian diff --git a/database/data/categories/Vect_c.yaml b/database/data/categories/Vect_c.yaml new file mode 100644 index 000000000..74f255bff --- /dev/null +++ b/database/data/categories/Vect_c.yaml @@ -0,0 +1,82 @@ +id: Vect_c +name: category of countable-dimensional vector spaces +notation: $\Vect^\c_K$ +objects: vector spaces over a field $K$ of countable dimension +morphisms: linear maps +description: This is a full subcategory of $\Vect_K$ containing $\FinVect_K$. Up to isomorphism, the only object in $\Vect^\c_K \setminus \FinVect_K$ is $K^{\oplus \IN}$. +nlab_link: null + +tags: + - algebra + +related: + - FinVect + - Vect + - Grp_c + - Set_c + +satisfied_properties: + - property: locally small + proof: It is a full subcategory of $\Vect_K$, which is locally small. + + - property: essentially small + proof: 'Every object is isomorphic to one of the vector spaces in the set $\{K^n : n \in \IN\} \cup \{K^{\oplus \IN}\}$.' + + - property: countable coproducts + proof: This follows from the fact that a countable union of countable sets is countable. + + - property: split abelian + proof: This follows easily from the corresponding property of $\Vect_K$. + + - property: generator + proof: We know that $K$ is a generator even in $\Vect_K$. + + - property: cogenerator + proof: We know that $K$ is a cogenerator even in $\Vect_K$. + +unsatisfied_properties: + - property: small + proof: Even the collection of trivial vector spaces is not a set. + + - property: skeletal + proof: This is trivial. + + - property: ℵ₂-small copowers + proof: >- + Let $I$ be a set of cardinality $\aleph_1$. Assume that the coproduct $C = \coprod_{i \in I} K$ exists; we need to be careful here, since we cannot assume that its underlying vector space is the direct sum. The coproduct inclusions correspond to elements $c_i \in C$ for $i \in I$, and the universal property says that for every countable-dimensional vector space $V$ and every family of vectors $v_i \in V$, there is a unique linear map $\varphi : C \to V$ such that $\varphi(c_i)=v_i$. We claim that the $c_i$ are linearly independent. + + Let $\{i_1,\dotsc,i_n\}$ be a finite subset of $I$ and suppose that $\sum_{k=1}^n \lambda_k c_{i_k} = 0$. For $i \in I$, define the vector $v_i \in K^n$ by $v_i = e_k$ if $i = i_k$ and $v_i = 0$ otherwise. By the universal property, there is a linear map $\varphi : C \to K^n$ such that $\varphi(c_i) = v_i$. Thus, $0 = \sum_{k=1}^n \lambda_k \varphi(c_{i_k}) = \sum_{k=1}^n \lambda_k e_k$. This shows that $\lambda_k=0$ for all $k$, as claimed. + + Since the set $\{c_i : i \in I\}$ is linearly independent and has cardinality $\aleph_1$, $C$ cannot have countable dimension. + + - property: countable powers + proof: First, the inclusion functor $\Vect^\c_K \hookrightarrow \Vect_K$ is continuous since the forgetful functor $\Vect_K \to \Set$ is continuous and conservative, while the forgetful functor $\Vect^\c_K \to \Set$ is representable and hence continuous. Thus, it suffices to show that $\Vect^\c_K$ is not closed under countable powers in $\Vect_K$. Specifically, we will show that $K^{\IN}$ does not have countable dimension. If it did, it would be isomorphic to $V \coloneqq K^{\oplus \IN}$. But then $V^* \cong V$, which contradicts MO/13322. + label: Vect_c_continuous_inclusion + + - property: ℵ₁-cofiltered limits + proof: >- + We have already seen that the inclusion functor $\Vect^\c_K \hookrightarrow \Vect_K$ is continuous, so it suffices to prove that $\Vect^\c_K$ is not closed under $\aleph_1$-cofiltered limits. Pick any uncountable set $I$ and consider the poset $P_{<\aleph_1}(I)$ of countable subsets of $I$. This is $\aleph_1$-filtered, so its dual is $\aleph_1$-cofiltered. Define the diagram $P_{<\aleph_1}(I)^{\op} \to \Vect^\c_K$ by sending a countable subset $C \subseteq I$ to $K^{\oplus C}$ and an inclusion $C \subseteq D$ to the projection $K^{\oplus D} \to K^{\oplus C}$ that discards the coordinates in $D \setminus C$. + The limit of this diagram is a subspace of $K^I$ consisting of those functions $f : I \to K$ such that, for every countable subset $C \subseteq I$, the restriction $f|_C : C \to K$ has finite support. It follows that $f$ has finite support, because otherwise there would be an infinite countable subset $C \subseteq I$ such that $f(c)$ is non-zero for all $c \in C$, and then $f|_C$ would not have finite support. Therefore, the limit is just $K^{\oplus I}$, which does not have countable dimension. + references: + - Vect_c_continuous_inclusion + +special_objects: + initial object: + description: trivial vector space + terminal object: + description: trivial vector space + coproducts: + description: '[countable case] direct sums' + products: + description: '[finite case] direct products' + +special_morphisms: + isomorphisms: + description: bijective linear maps + proof: It is a full subcategory of $\Vect_K$, in which the isomorphisms are known to be the bijective linear maps. + monomorphisms: + description: injective linear maps + proof: The non-trivial direction follows since the forgetful functor to $\Set$ is representable (by $K$). + epimorphisms: + description: surjective linear maps + proof: By construction of the finite colimits, the inclusion functor to $\Vect_K$ preserves finite colimits. Hence, it preserves epimorphisms. Since it is faithful, it also reflects epimorphisms. Therefore, the description of epimorphisms in $\Vect_K$ carries over to this subcategory.