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Consider the following examples:

  • The forgetful functor $U_1: \operatorname{Vect}_\mathbb{C} \to \operatorname{Vect}_\mathbb{R}$
  • The forgetful functor $U_2: \operatorname{Diff}^{\text{or}} \to \operatorname{Diff}$ (oriented manifolds and orientation-preserving smooth maps to unoriented manifolds and smooth maps)
  • The forgetful functor $U_3: \operatorname{Bord}^{\text{or}} \to \operatorname{Bord}$ (oriented manifolds and oriented cobordisms to unoriented manifolds and unoriented cobordisms)

I want to abstract these functors into a common abstract framework. They have these properties in common:

  • They're faithful.
  • The source category is an involutive monoidal category, the target isn't. An involution is a functor $X \mapsto \overline{X}$ satisfying $\overline{X \otimes Y} \cong \overline{Y} \otimes \overline{X}$ and $\overline{\overline{X}} \cong X$ naturally. The involution on $\operatorname{Vect}_\mathbb{C}$ is complex conjugation, on manifolds it's orientation reversal.
  • There is some trivial isomorphism $U_iX \cong U_i\overline{X}$, so $U$ forgets precisely the involutive structure (but notice that in the target category, there are "nonorientable" objects and morphisms which aren't in the image of $U_i$).

But possibly, there are other important similarities I might have missed.

Is this kind of functor known? How would an abstract definitions just in terms of involutive monoidal categories look like? What are the correct coherence axioms to the isomorphisms $u_X: UX \cong U\overline{X}$ such that they can be strictified?

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