I heard two quotes, one from Alain Connes and an other one from Orlov. 
Alain Connes was talking about noncommutative geometry and he said the following:
**" a noncommutative algebra creates its own internal time "** 


In a talk by Orlov about Mirror symmetry, he was asked if he considers the monoidal structure on the derived category of a scheme. Orlov said that "**the monoidal structure is not natural in this context and we should not base our theory on this structure** " he added "**The tensor product is a NATURAL structure to consider in the category of Motives**"


I will be happy if someone can put some enlightenment to what Connes and Orlov meant (if the quotations above make sense ) ?



Mysterious quotes (at least for me)
Here is a guess about the remark of Orlov. Suppose that one wants to define a good notion of *noncommutative scheme*, given that an affine noncommutative scheme is an associative algebra. Trying to define the spectrum of an associative algebra leads to various problems (c.f. [this answer](https://mathoverflow.net/questions/159449/why-is-naive-definition-of-non-commutative-spectrum-bad/159573#159573)), so a different approach is needed. On the other hand there are various invariants of assocative algebras, like algebraic K-theory, which all happen to be *Morita invariant*: they depend only on derived categories. Further, a theorem of Bondal and Van den Bergh says that the derived category of a commutative scheme is equivalent to the derived category of some dg-algebra. This suggests to define a noncommutative scheme as the derived category of a dg-algebra. This will then include both affine noncommutative schemes and commutative schemes. This is the approach practised by the Moscow school (Bondal, Orlov, Kaledin, Lunts, etc.).


This explains the first remark: it is essential to consider noncommutative spaces only up to Morita equivalence. If we would consider the symmetric monoidal structure in the definition of noncommutative scheme, then any two commutative schemes would be isomorphic in the noncommutative world iff they were isomorphic in the usual commutative sense. See work of Balmer for this. In fact most interesting morphisms in the noncommutative world do not respect the symmetric monoidal structures coming from the commutative world.


Regarding the second remark, suppose that we want a theory of motives for noncommutative schemes. In fact, such theories already exist after Cisinski-Tabuada and Robalo. For example, following Robalo, we can echo the Morel-Voevodsky construction in our setting: take the (infinity-)category of noncommutative schemes, pass to the free cocompletion (infinity-presheaves), impose A^1-homotopy invariance and descent with respect to a noncommutative version of the Nisnevich topology, and then formally stabilize with respect to P^1. This gives a noncommutative version of the stable motivic homotopy category. Now suppose that we were to include symmetric monoidal structures in the picture, i.e. start with the infinity-category of *symmetric monoidal* dg-categories, and repeat the same construction. I have not checked this, but I think that one would just recover (an enlargement of) the Morel-Voevodsky stable motivic homotopy category of commutative schemes. My guess is that Orlov may have had something along these lines in mind.


