Quadratic algebras, quadratic operads, quadratic categories and quantum cohomology

Question

Motivated by the quantisation of the symmetric laws in physics, the category of quadratic algebras has been endowed with two tensor products by Manin in his Montreal lectures notes. These products have been extended to the category of quadratic operads by Ginzburg and Kapranov. A long time ago in my master thesis, (I have started to work on this topic in June 1995 and I have defended my master's thesis in September 1995), I have defined the notion of quadratic category which is a category endowed with two tensor products, quadratic algebras and quadratic operads are examples of quadratic categories. More precisely:

A quadratic category $(C,\bullet, \circ)$ is a category $C$ endowed with two tensor products $\bullet$ and $\circ$ such that:

$I_{\circ}$ and $I_{\bullet}$ are the respective neutral elements of $(C,\circ)$ and $(C,\bullet)$.

We denote by $c^{\bullet}:(A\bullet B)\bullet C\rightarrow A\bullet (B\bullet C)$ the associative constraint of $\bullet$.

For every $A\in C$, there exists $A^!$ in $C$,

morphisms:

$b_A:I_{\bullet}\rightarrow A\circ A^!$

$d_A:A^!\bullet A\rightarrow I_{\circ}$

two natural morphisms:

$f^1_{A,B,C}:(A\circ B)\bullet C\rightarrow A\circ (B\bullet C)$

$f^2_{A,B,C}: A\bullet (B\circ C)\rightarrow (A\bullet B)\circ C$

which verifies:

$f^1_{A\bullet B,C,D}(f^2_{A,B,C}\bullet id_D)=f^2_{A,B,C\bullet D}(Id_A\bullet f^1_{B,C,D})c^{\bullet}_{A,B\circ C,D}$

$(Id_A\circ f^2_{B,C,D})f^1_{A,B,C\circ D}=c^{\circ}_{A,B\bullet C,D}(f^1_{A,B,C}\circ Id_D)f^2_{A\circ B,C,D}$

and

$(Id_A\circ d_A)f^1_{A,A^!,A}(b_A\bullet Id_A)=Id_A$

$(Id_{A^!}\circ d_A)f^2_{A^!,A,A^!}(Id_{A^!}\bullet b_A)=Id_{A^!}$.

We have the following result:

Theorem.

Let $C$ be a quadratic category and $B,D$ two objects of $C$, the functor $A\rightarrow Hom_C(A\bullet B,D)$ is representable by $D\circ B^!$.

Questions.

I would like to know if there exist other examples of such quadratic categories related or not related to the theory of quantum groups ?

In a recent note, Manin studies the interaction between quadratic algebras, quadratic operad, a notion of enriched category due to Kelly and quantum cohomology? Can these relations be interpreted with this framework of quadratic category ?

Reference.

V. Ginzburg, M. Kapranov. Koszul duality for operads. Duke Math 1994.

Yu. Manin Higher structures, quantum group and genus zero operad

https://arxiv.org/pdf/1802.04072.pdf

Yu. Manin. Quantum groups and non–commutative geometry. Publ. de CRM, Universit´e de Montr´eal (1988),

Tsemo Aristide M\'emoire de D.E.A 1995.

Answer 1

A similar place where two interacting monoidal structures come up is in duoidal categories.

I believe that any star-autonomous category, i.e. a symmetric monoidal closed category with a dualizing object, is an example of a quadratic category. I have only checked a fraction of the coherence diagrams, though.

Examples:

• There are examples coming from linear logic; in deference to that subject where "$!$" means something different, I will use write $A^\ast$ instead of $A^!$.

• There are also examples from algebraic geometry: if $X$ is a smooth scheme of pure dimension $d$ over a field $k$, then the derived category of $X$ is an example (because of Serre duality). There are many more examples; see here for some (in that paper, the theory of $\ast$-autonomous categories is essentially rediscovered).

• I believe there are many more examples.

Details:

Let $(\mathcal C,\otimes, I)$ be a symmetric monoidal closed category with dualizing object $D$, and write $[-,-]$ for the internal hom. We may define $I_\bullet = I$, $\bullet = \otimes$, $A^\ast = [A,D]$, $I_\circ= D$, and $A \circ B = [[A,D]\otimes[B,D],D] = (A^\ast \bullet A^\ast)^\ast$. Note that

1. There is a canonical isomorphism $A^{\ast\ast} = A$.
2. There is a canonical isomorphism $Hom(A \otimes B, C^\ast) = Hom(A,(B\otimes C)^\ast)$

I believe that $\mathcal C$ is a quadratic category:

• The map $d_A: A^\ast \bullet A \to I_\circ$ is given by the evaluation $[A,D]\otimes A \to D$.

• The map $b_A: I_\bullet \to A \circ A^\ast$ is given by the map $I \to [[A,D],[A,D]]$ representing the identity, since $A \circ A^\ast = (A^\ast\otimes A^{\ast\ast})^\ast = (A^\ast \otimes A)^\ast = [[A,D]\otimes A,D] = [[A,D],[A,D]]$.

• The map $f^1_{A,B,C}: (A\circ B) \bullet C \to A\circ (B\bullet C)$, i.e. $(A^\ast \otimes B^\ast)^\ast \otimes C \to (A^\ast \otimes (B \otimes C)^\ast)^\ast$ corresponds via (2) to a map $(A^\ast \otimes B^\ast)^\ast \to (C \otimes A^\ast \otimes (B \otimes C)^\ast)^\ast = [C \otimes (B\otimes C)^\ast,A] = [(B\otimes C)^\ast,[C,A]] = [[C,B^\ast],[C,A]]$; the domain can be rewritten as $[B^\ast,A]$. The morphism $[B^\ast,A] \to [[C,B^\ast],[C,A]]$ to which $f^1$ corresponds is the morphism representing postcomposition.

• The map $f^2_{A,B,C}$ is defined similarly.

Proviso: I have only checked the unit equations.