Let $Vect_f$ be the category of finite-dimensional vector spaces. This category comes with a very well-behaved duality functor. Now the ind-completion of this category (if I understand correctly) is the category $Vect$ of all vector spaces, and the pro-completion will then be the dual categor, $ProVect_f$ (which includes things like formal power series).

Suppose I take an object $V$ in $ProVect_f$. Then $V\otimes V^*$ can be viewed as either an object of the ind-completion of $ProVect_f$ or an object of the pro-completion of $Vect = IndVect_f$. My question is whether it is meaningful to view $V\otimes V^*$ as an algebra object in an appropriate monoidal category.

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    $\begingroup$ The title and body ask different questions! If $V$ is finite-dimensional then $\text{End}(V)$ isn't $V \otimes V^{\ast}$; the latter is endomorphisms $V \to V$ of finite rank, which is not an algebra object due to failing to have an identity. $\endgroup$ – Qiaochu Yuan Sep 26 '16 at 19:19
  • $\begingroup$ Right. What I'm looking for should be something like an idempotented algebra (it is supposed to admit a map from the Hecke algebra of a p-adic group) $\endgroup$ – Dmitry Vaintrob Sep 26 '16 at 20:28
  • $\begingroup$ Who said algebras must have identities :) $\endgroup$ – Benjamin Steinberg Sep 26 '16 at 23:59

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