# Twisted duality in a symmetric monoidal category

I would like to know if the following definition is already known or even well-known. Is there a reference in the literature? Do you know prominent examples?

Definition. Let $\mathcal{C}$ be a symmetric monoidal category and $X,Y,L \in \mathcal{C}$. Then $X$ is dual to $Y$ twisted by $L$ if there are morphisms $X \otimes Y \to L$ and $L \to Y \otimes X$ which satisfy the triangular identities:

• $X \otimes L \to X \otimes Y \otimes X \to L \otimes X$ equals the symmetry
• $L \otimes Y \to Y \otimes X \otimes Y \to Y \otimes L$ equals the symmetry

One may also say that $X \otimes Y \to L$ is a perfect pairing.

Observe that $X$ is dual to $Y$ twisted by $1$ iff $X$ is dual to $Y$ in the usual sense. More generally, when $L$ is invertible, then $X$ is dual to $Y$ twisted by $L$ iff $X$ is dual to $Y \otimes L^{-1}$.

It seems to me that this twisted notion of duality is more natural than the usual one where only $L=1$ is allowed. Namely, the common examples follow this pattern: a) If $V$ is a vector space or even vector bundle of dimension $n$ and $0 \leq k \leq n$, then the exterior power $\wedge^k(V) \otimes \wedge^{n-k}(V) \to \wedge^n(V)$ is a perfect pairing. b) If $X$ is a smooth projective variety of dimension $n$ with dualizing sheaf $\omega^{\circ}$, $F$ is a coherent sheaf on $X$ and $0 \leq k \leq n$, then $H^k(F) \otimes H^{n-k}(F^* \otimes \omega^{\circ}) \to H^n(\omega)$ is a perfect pairing (Serre duality). c) Likewise Poincaré duality for oriented manifolds.

• Do you have any examples where the twisting object is not invertible? The notions of dualizing object and Serre functor are well-established, but as far as I know, they are limited to the invertible case. – S. Carnahan Feb 4 '13 at 13:55
• Up to now I only have boring examples where $L$ is not invertible. 1) If $\mathcal{C}$ is discrete, then $L=X \otimes Y$ always works. 2) More generally $L=X \otimes Y$ (with the obvious morphisms) works if $X$ and $Y$ are symtrivial (mathoverflow.net/questions/119689). For example $\mathbb{Q} \otimes_{\mathbb{Z}} \mathbb{Q} \to \mathbb{Q}$ is a perfect pairing of $\mathbb{Z}$-modules. 3) If $\mathcal{C}$ is additive and $L= 1^{\oplus I}$ for a finite set $I$, then $X$ is dual to $Y$ twisted by $L$ iff for every $i \in I$ we have a duality between $X$ and $Y$ and these are "orthogonal". – Martin Brandenburg Feb 4 '13 at 14:50