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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeSep 24th 2013
   • (edited Sep 24th 2013)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexThe filtration in dual category is called the cofiltration. A subobject in the opposite category is a quotient object, so instead of having a sequence of inclusions, we have a sequence of projections. If $V$ is $\mathbb{N}$-filtered vector space, then the dual space is $\mathbb{N}$-cofiltered (or should one say $\mathbb{N}^{op}$-cofiltered). I am these days interested in a special case of, so to speak, pro-finite-dimensional cofiltrations of vector spaces. Let $$ V_0\subset V_1\subset V_2\subset\ldots\subset V $$ be an exhaustive filtration of a vector space $V$ (i.e. $\cup_{i=0}^\infty V_i = V$) by finite dimensional vector subspaces $V_i\subset V$ over a ground field $F$. We also write $V_i = F_i V$. Such vector spaces equipped with exhaustive $\mathbb{N}$-filtrations by finite-dimensional vector spaces form a category which I will denote by $Filt_{fd}Vec_F$ which is symmetric monoidal with respect to the filtered algebraic tensor product: $(V\otimes W)_n = \sum_{k=0}^n V_k\otimes_F W_{n-k}\subset V\otimes_F W$. Consider the full algebraic dual $Vec_F\to Vec_F$, $V\mapsto V^*= Hom_F(V,F)$; it extends to a contravariant functor; the dual of a morphism $f:V\to W$ is also called the transpose $f^T=f^*$, and is defined by $f^T(\omega)(v) = \omega(f(v))$. Now the vector spaces $V^*$ comes with cofiltration by finite-dimensional vector spaces $(V_k)^*$; the connecting projections $\pi_{k+l,k}: (V_{k+l})^*\to (V_k)^*$ and the canonical projections $\pi_k:V^*\to (V_k)^*$ are mutually compatible and simply the restriction maps along the inclusions $V_k\subset V_{k+l}$ and $V_k\subset V$. The inverse limit $lim_k (V_k)^*$ is a vector subspace of the direct product $(V_n)^*$ whose elements are threads $(\omega_n)_n\in \prod_n (V_n)^*$ such that for all $k,n$ $\omega_n = \pi_{n+k,n}(\omega_{n+k})$. Clearly, one can identify $$ V^* \cong lim_k (V_k)^*. $$ Indeed the thread $(\omega_n)_n$ corresponds to the unique linear functional $\omega$ on $V^*$ such that $\pi_n(\omega) = \omega_n$. Now consider the category $cFilt_{fd} Vec_F$ of vector space cofiltered by finite-dimensional vector spaces and complete in the sense that they are equipped with an isomorphism $Z \cong lim Z_n$ with the inverse limit of its own cofiltration; for simplicity the morphisms will strictly respect the cofiltration (with a little more relaxed notion of a morphism we can get a subcategory of the category of pro-finite-dimensional vector spaces, but this would make the theory more cumbersome for my present purposes). I am claiming that $cFilt_{fd} Vec_F$ is provided with a notion of a completed tensor product $\widehat\otimes$ making $cFilt_{fd} Vec_F$ into a symmetric monoidal category in such a way that the duality $V\mapsto V^*$ is equipped with a structure of a strong monoidal functor $(Filt_{fd} Vec_F)^{op}\to cFilt_{fd} Vec_F$. In fact there is also the inverse functor, though $V^*$ is not isomorphic to $V$ for infinite-dimensional $V$, the cofiltration is made out of finite-dimensional pieces which are rigid/dualizable objects with respect to the usual tensor product. This is likely true in much more general context than filtered vector spaces.

   The filtration in dual category is called the cofiltration. A subobject in the opposite category is a quotient object, so instead of having a sequence of inclusions, we have a sequence of projections. If $V$ is $\mathbb{N}$-filtered vector space, then the dual space is $\mathbb{N}$-cofiltered (or should one say $\mathbb{N}^{op}$-cofiltered).

   I am these days interested in a special case of, so to speak, pro-finite-dimensional cofiltrations of vector spaces.

   Let

   $$V_0\subset V_1\subset V_2\subset\ldots\subset V$$

   be an exhaustive filtration of a vector space $V$ (i.e. $\cup_{i=0}^\infty V_i = V$) by finite dimensional vector subspaces $V_i\subset V$ over a ground field $F$. We also write $V_i = F_i V$. Such vector spaces equipped with exhaustive $\mathbb{N}$-filtrations by finite-dimensional vector spaces form a category which I will denote by $Filt_{fd}Vec_F$ which is symmetric monoidal with respect to the filtered algebraic tensor product: $(V\otimes W)_n = \sum_{k=0}^n V_k\otimes_F W_{n-k}\subset V\otimes_F W$. Consider the full algebraic dual $Vec_F\to Vec_F$, $V\mapsto V^*= Hom_F(V,F)$; it extends to a contravariant functor; the dual of a morphism $f:V\to W$ is also called the transpose $f^T=f^*$, and is defined by $f^T(\omega)(v) = \omega(f(v))$. Now the vector spaces $V^*$ comes with cofiltration by finite-dimensional vector spaces $(V_k)^*$; the connecting projections $\pi_{k+l,k}: (V_{k+l})^*\to (V_k)^*$ and the canonical projections $\pi_k:V^*\to (V_k)^*$ are mutually compatible and simply the restriction maps along the inclusions $V_k\subset V_{k+l}$ and $V_k\subset V$. The inverse limit $lim_k (V_k)^*$ is a vector subspace of the direct product $(V_n)^*$ whose elements are threads $(\omega_n)_n\in \prod_n (V_n)^*$ such that for all $k,n$ $\omega_n = \pi_{n+k,n}(\omega_{n+k})$. Clearly, one can identify

   $$V^* \cong lim_k (V_k)^*.$$

   Indeed the thread $(\omega_n)_n$ corresponds to the unique linear functional $\omega$ on $V^*$ such that $\pi_n(\omega) = \omega_n$.

   Now consider the category $cFilt_{fd} Vec_F$ of vector space cofiltered by finite-dimensional vector spaces and complete in the sense that they are equipped with an isomorphism $Z \cong lim Z_n$ with the inverse limit of its own cofiltration; for simplicity the morphisms will strictly respect the cofiltration (with a little more relaxed notion of a morphism we can get a subcategory of the category of pro-finite-dimensional vector spaces, but this would make the theory more cumbersome for my present purposes). I am claiming that $cFilt_{fd} Vec_F$ is provided with a notion of a completed tensor product $\widehat\otimes$ making $cFilt_{fd} Vec_F$ into a symmetric monoidal category in such a way that the duality $V\mapsto V^*$ is equipped with a structure of a strong monoidal functor $(Filt_{fd} Vec_F)^{op}\to cFilt_{fd} Vec_F$. In fact there is also the inverse functor, though $V^*$ is not isomorphic to $V$ for infinite-dimensional $V$, the cofiltration is made out of finite-dimensional pieces which are rigid/dualizable objects with respect to the usual tensor product. This is likely true in much more general context than filtered vector spaces.

2. • CommentRowNumber2.
   • CommentAuthorzskoda
   • CommentTimeSep 24th 2013
   • (edited Sep 24th 2013)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexWe proceed toward the study of the completed tensor product in $cFilt_{fd}Vec_F$ We first see that there is an "ordinary" tensor product of cofiltered spaces $Z\otimes Y$ (which is still cofiltered, but does not belong to $cFilt_{fd}Vec_F$ in general as it is not complete) whose $n$-th cofiltered piece is just $$ (Z\otimes Y)_n = \frac{Z\otimes W}{\cap_l Ker(\pi_l^Z\otimes\pi_{n-l}^Y)}. $$ and the connecting projections $\pi_{n+k,n}$ are the unique epimorphisms between $(Z\otimes Y)_n$ for varrying $n$ commuting with the projections $\pi_n$ from $Z\otimes W$. The completed tensor product $Z\widehat{\otimes} Y$ in $cFilt_{fd}Vec_F$ is given by the inverse limit $$ Z\widehat\otimes W = lim_n (Z\otimes W)_n $$ (Is there a universal property of this tensor product not referencing to the construction of $(Z\otimes W)_n$ and corresponding projections ?) From now on we skip the field $F$ from the notation. We now recall that for finite-dimensional vector spaces $M,N$, $(M\otimes N)^*\cong N^*\otimes M^*$. We now claim that for $V,W\in Filt_{fd}Vec$, the following diagram commutes $$\array{ (\sum_l V_{l}\otimes W_{n+k-l})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_{n+k} \\ i_{n,n+k}^T\downarrow &&\downarrow \pi_{n+k,n} \\ (\sum_s V_{s}\otimes W_{n-s})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_n }$$ where $i_{n,n+k} : \sum_s V_{s}\otimes W_{n-s}\longrightarrow \sum_l V_{l}\otimes W_{n+k-l}$ are the canonical embeddings. The right hand side of the diagram defines the cofiltration which together with its limit, by definition, forms the completed tensor product $$ W^*\widehat{\otimes} V^*. $$ The left hand side, is on the other hand, constructed by duality from the filtration on $V\otimes W$ (indeed the connecting projections are dual to connecting injections) and it defines an isomorphic object in $cFilt_{fd}Vec$. Heuristically we behave as if $$\array{ (\sum_l V_{l}\otimes W_{n+k-l})^* &\stackrel\cong \longrightarrow &\sum_l (W_l)^*\otimes (V_{n+k-l})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_{n+k} \\ i_{n,n+k}^T\downarrow &&\downarrow&&\downarrow \pi_{n+k,n} \\ (\sum_s V_{s}\otimes W_{n-s})^* &\stackrel\cong \longrightarrow &\sum_s (W_{n-s})^*\otimes (V_{s})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_n }$$ The middle column is at this point just a heuristics (the sum is not direct has to be internal, but in which vector space...) and it should be in fact skipped. In the same vain, the linear maps $\sum_s (W_{n-s})^*\otimes (V_{s})^* \stackrel\cong \longrightarrow (W^*\otimes V^*)_n$ need more explanation. First of all, $W^*_s=(W^*)_s$ by the definition and $(i_s)^* = \pi^{W^*}_s : W^*\to (W^*)_s$ are the canonical projections.

   We proceed toward the study of the completed tensor product in $cFilt_{fd}Vec_F$ We first see that there is an “ordinary” tensor product of cofiltered spaces $Z\otimes Y$ (which is still cofiltered, but does not belong to $cFilt_{fd}Vec_F$ in general as it is not complete) whose $n$-th cofiltered piece is just

   $$(Z\otimes Y)_n = \frac{Z\otimes W}{\cap_l Ker(\pi_l^Z\otimes\pi_{n-l}^Y)}.$$

   and the connecting projections $\pi_{n+k,n}$ are the unique epimorphisms between $(Z\otimes Y)_n$ for varrying $n$ commuting with the projections $\pi_n$ from $Z\otimes W$.

   The completed tensor product $Z\widehat{\otimes} Y$ in $cFilt_{fd}Vec_F$ is given by the inverse limit

   $$Z\widehat\otimes W = lim_n (Z\otimes W)_n$$

   (Is there a universal property of this tensor product not referencing to the construction of $(Z\otimes W)_n$ and corresponding projections ?)

   From now on we skip the field $F$ from the notation.

   We now recall that for finite-dimensional vector spaces $M,N$, $(M\otimes N)^*\cong N^*\otimes M^*$.

   We now claim that for $V,W\in Filt_{fd}Vec$, the following diagram commutes

   $$\array{ (\sum_l V_{l}\otimes W_{n+k-l})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_{n+k} \\ i_{n,n+k}^T\downarrow &&\downarrow \pi_{n+k,n} \\ (\sum_s V_{s}\otimes W_{n-s})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_n }$$

   where $i_{n,n+k} : \sum_s V_{s}\otimes W_{n-s}\longrightarrow \sum_l V_{l}\otimes W_{n+k-l}$ are the canonical embeddings. The right hand side of the diagram defines the cofiltration which together with its limit, by definition, forms the completed tensor product

   $$W^*\widehat{\otimes} V^*.$$

   The left hand side, is on the other hand, constructed by duality from the filtration on $V\otimes W$ (indeed the connecting projections are dual to connecting injections) and it defines an isomorphic object in $cFilt_{fd}Vec$.

   Heuristically we behave as if

   $$\array{ (\sum_l V_{l}\otimes W_{n+k-l})^* &\stackrel\cong \longrightarrow &\sum_l (W_l)^*\otimes (V_{n+k-l})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_{n+k} \\ i_{n,n+k}^T\downarrow &&\downarrow&&\downarrow \pi_{n+k,n} \\ (\sum_s V_{s}\otimes W_{n-s})^* &\stackrel\cong \longrightarrow &\sum_s (W_{n-s})^*\otimes (V_{s})^* &\stackrel\cong \longrightarrow & (W^*\otimes V^*)_n }$$

   The middle column is at this point just a heuristics (the sum is not direct has to be internal, but in which vector space…) and it should be in fact skipped. In the same vain, the linear maps $\sum_s (W_{n-s})^*\otimes (V_{s})^* \stackrel\cong \longrightarrow (W^*\otimes V^*)_n$ need more explanation. First of all, $W^*_s=(W^*)_s$ by the definition and $(i_s)^* = \pi^{W^*}_s : W^*\to (W^*)_s$ are the canonical projections.

3. • CommentRowNumber3.
   • CommentAuthorzskoda
   • CommentTimeSep 24th 2013
   • (edited Sep 24th 2013)
   • PermaLink
   Author: zskoda
   Format: MarkdownItexLet us construct the isomorphism $$ (W^*\otimes V^*)/\cap_l Ker(\pi_l^{W^*}\otimes\pi_{n-l}^{V^*})}\longrightarrow (\sum_s V_s\otimes W_{n-s})^* $$ So let $\sum_\alpha \omega_\alpha\otimes \omega'_\alpha$ be a representative on the lhs, $\omega_\alpha\in W^*$, $\omega'_\alpha\in V^*$; to start, we need to evaluate it on elements $v\otimes w\in V_{n-s}\otimes W_s$ for some $s$ (by linearity on linear combination of such; if something belongs to tensor products for different $s$ we need to get the same). The formula is $$ \sum_\alpha (\omega_\alpha\otimes \omega'_\alpha) (v\otimes w) = \sum_\alpha \omega_\alpha(i_s^W(w))\omega'_\alpha(i_{n-s}^V(v)) $$ Now this does not depend on a representative, as if we add to $\omega_\alpha$ something in $$\left(\cap_l Ker(\pi_l^{W^*}\otimes\pi_{n-l}^{V^*})\right)\subset Ker((i_s^W)^T\otimes (i_{n-s}^V)^T),$$ hence the added term gives zero when evaluated on $i_s^W(w)\otimes i_{n-s}^V(v)$. Suppose now $w\otimes v\in W_s\otimes V_{n-s}\cap W_t\otimes V_{n-t}$ for $s\neq t$. Then $(i_s\otimes i_{n-s})(w\otimes v) = (i_t\otimes i_{n-t})(w\otimes v)$ hence again no difference. Thus the map above is well defined. Now we have to show that it is an isomorphism.

   Let us construct the isomorphism

   So let $\sum_\alpha \omega_\alpha\otimes \omega'_\alpha$ be a representative on the lhs, $\omega_\alpha\in W^*$, $\omega'_\alpha\in V^*$; to start, we need to evaluate it on elements $v\otimes w\in V_{n-s}\otimes W_s$ for some $s$ (by linearity on linear combination of such; if something belongs to tensor products for different $s$ we need to get the same). The formula is

   $$(W^*\otimes V^*)/\cap_l Ker(\pi_l^{W^*}\otimes\pi_{n-l}^{V^*})} \sum_\alpha (\omega_\alpha\otimes \omega'_\alpha) (v\otimes w) = \sum_\alpha \omega_\alpha(i_s^W(w))\omega'_\alpha(i_{n-s}^V(v))$$

   Now this does not depend on a representative, as if we add to $\omega_\alpha$ something in

   $$\left(\cap_l Ker(\pi_l^{W^*}\otimes\pi_{n-l}^{V^*})\right)\subset Ker((i_s^W)^T\otimes (i_{n-s}^V)^T),$$

   hence the added term gives zero when evaluated on $i_s^W(w)\otimes i_{n-s}^V(v)$.

   Suppose now $w\otimes v\in W_s\otimes V_{n-s}\cap W_t\otimes V_{n-t}$ for $s\neq t$. Then $(i_s\otimes i_{n-s})(w\otimes v) = (i_t\otimes i_{n-t})(w\otimes v)$ hence again no difference. Thus the map above is well defined.

   Now we have to show that it is an isomorphism.