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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorzskoda
• CommentTimeSep 24th 2013
• (edited Sep 24th 2013)

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.

• CommentRowNumber2.
• CommentAuthorzskoda
• CommentTimeSep 24th 2013
• (edited Sep 24th 2013)

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.

• CommentRowNumber3.
• CommentAuthorzskoda
• CommentTimeSep 24th 2013
• (edited Sep 24th 2013)

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.