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Before stating my question I would like to provide afew motivating examples:

Examples:

  1. In the category of Finitely-generated-projective $R$-modules, we have that: $M^{\vee}:=Hom_R(M,R)$ satisfies: $Hom_R(M^{\vee},R)\cong M$.
  2. If $G$ is a locally compact abelian topological group over the circle group $T$, then the set: $G^{\wedge}:=Hom_{CGrp}(G,T)$ of continuous group homomorphisms into the circle group can be given the structure of a locally compact abelian topological group. Moreover the Pontryagin duality theorem states that: $Hom_{CGrp}(G^{\wedge},T)\cong G$ as topological groups (note: here $T$ is the terminal object since all groups are considered over $E$).
  3. In the category of Banach spaces over ${\mathbb{R}}$the above construction fails, however fails for $Hom_{\mathbb{R}}(B,{\mathbb{R}})$ however when restricting our attention to the subcategory of continous linear functionals (CLFs) the $B^{\prime}:=Hom_{CLFs}(B,{\mathbb{R}})$ satisfies the duality relationship: $Hom_{CLFs}(B^{\prime},\mathbb{R})\cong B$ for many Banach spaces $B$.

My goal is to understand when this idea can be extended: Question: When does there exist a faithful subcategory $\mathfrak{C}$ of a category $\mathfrak{D}$ with terminal object $0$ such that, the Hom functor of $\mathfrak{C}$ is internal and for every object $D$ in $\mathfrak{D}$, when considered in $\mathfrak{C}$ there exists some object $D'\in \mathfrak{D}$ satisfying: $Hom_{\mathfrak{C}}(D',0)\cong D$ such that the association $D \mapsto D'$ is functorial?

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Here is a construction that covers the first example but not, I think, the other two. Suppose $C$ is a closed symmetric monoidal category with unit object $1$ and that $c$ is a dualizable object in $C$. Then the dual $c^{\ast}$ can functorially identified with the internal hom $[c, 1]$, where $1$ is the monoidal unit, and in particular we always have the reflexivity condition

$$c \cong [[c, 1], 1]$$

because this always holds for the monoidal dual in a symmetric monoidal category. Examples of such $C$ include the symmetric monoidal categories $\text{Mod}(k)$ of modules over a commutative ring $k$, where the dualizable objects are precisely the finitely presented projective $k$-modules.

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  • $\begingroup$ I think there is a notion of topological tensor making the third one work, not sure though $\endgroup$ – AIM_BLB Oct 16 '15 at 4:04
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    $\begingroup$ It's true that the category of Banach spaces and maps of norm at most $1$ is closed monoidal, with monoidal product given by the projective tensor product. Unfortunately I think the dualizable Banach spaces are the finite-dimensional ones, so they don't exhaust the reflexive Banach spaces. $\endgroup$ – Qiaochu Yuan Oct 16 '15 at 4:38

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