Переписка с E.S.
Продолжение этого -- http://mathoverflow.net/questions/86821/homotopic-morphisms-between-curved-a-infinity-algebras
E.S.: <...> I think I understand why for dg-modules if we look at A_\infty-morphisms then we correctly compute the morphisms in the derived category. Is there a simple reason why it goes wrong for curved dg-modules? <...>
me: I don't know whether it is simple enough or not, but the reason is explained in terms of the philosophy of "two kinds of derived categories". You may want to look up the introduction to my paper, which is now available from the arXiv at http://arxiv.org/abs/1202.2697 .
Basically, there are a number of explanations as to why the derived category of the first kind (what we conventionally call the derived category, say, of DG-modules over a DG-algebra) has no meaning for CDG-modules over a CDG-algebra over a field.
Speaking of A_\infty-algebras, an A_\infty-algebra or module whose differential is so nondegenerate that its cohomology is zero is viewed as a zero object in the conventional theory. So if m_1 is nondegenerate enough, all the operations starting from m_2 and higher no longer matter. When you attempt to extend this theory to curved algebras, it is m_0 that becomes the dominating term. If it is nondegenerate enough, all the operations starting from m_1 no longer matter. And it is all too easy for an element of a vector space to be "nondegenerate enough" -- being a nonzero vector is largely sufficient.
One alternative is to reverse the order of domination, having m_2 dominate over m_1 and m_1 over m_0. This leads to the constructions of the derived categories of the second kind. These are the exotic derived categories of curved modules discussed in various recent papers of mine (on Koszul duality, matrix factorizations, etc).
In the A_\infty setting, this would mean prohibiting your curved A_\infty-algebras, modules, morphisms of algebras and modules etc. from having an infinite number of nonzero higher operations. So only a finite number of higher multiplication or action maps would be allowed in each particular algebra, module, or morphism.
You may be familiar with these phenomena from your work on the Hochschild (co)homology of matrix factorizations. The choice between taking infinite direct sums or infinite products along the diagonals of various complexes is exactly one of choosing the order of domination of various terms ( = in the direction of which differential to take the cohomology first when constructing the spectral sequence, etc).
Recall that when Caldararu-Tu define their Borel-Moore Hochschild homology or compactly supported Hochschild cohomology of curved A_\infty-algebras, they have to make this assumption of a finite number of nonzero higher operations only, or otherwise their formulas for the total differentials in the Hochschild complexes would be divergent.
Actually, it is this divergence that gives their definition a chance to be nontrivial. The conventional definition of Hochschild (co)homology does not have this convergence issue and can be extended to arbitrary curved A_\infty-algebras. This fact alone sort of guarantees triviality of this definition in the case of algebras with nonzero curvature over a field.
So you can do the theory of curved A_\infty-algebras with a finite number of higher operations only if you wish to, and it is supposed to be nontrivial, but my feeling is that it would be quite unpleasant from the technical point of view. Analogues of the conventional results like "derived category of DG-modules = homotopy category of A_\infty-modules" would be hard to obtain, if they are true at all (though they are certainly expected to be closer to the truth in this bounded higher operations setting then in the unbounded one).
The preferrable setting for a curved A_\infty theory of the second kind is that of curved A_\infty-coalgebras. In this situation, one would impose the local "convergence condition" (the requirement that the sequence of images of any fixed element under the higher comultiplication maps contains only a finite number of nonzero terms) instead of the stronger global finiteness condition. This theory is discussed in "Two kinds of derived categories...", Sections 7.4-7.6. It is reasonably well-behaved.
The other alternative is to work over a ring rather than over a field, and prohibit your curvature elements from being nondegenerate by requiring them to be divisible by a fixed ideal. This allows to have a nontrivial curved A_\infty theory of the first kind, or something close to that. This is the subject of my new preprint.
<...> I would say that the natural formalism for discussing homotopies is that of model categories, but in this particular case I cannot immediately think of an appropriate model category structure. E.g., I don't know what the natural model category structure on CDG-algebras might be.
If the idea is to work with curved A_\infty things, my best suggestion would be to find a way to replace curved algebras with curved coalgebras; then conilpotent curved A_\infty-coalgebras are roughly the same thing as cofibrant DG-algebras. The model structure on DG-algebras might be used here then.
E.S.: Ah, I thought (well, hoped) that might be the answer! Actually the restriction to only finitely-many terms is quite nice from the geometric point of view, it means that your A_\infty structure comes from an actual (polynomial) vector field, not just the formal germ of a vector field. Also this allows you to have morphisms that don't preserve the origin - I used this to repair my paper on Hochschild homology. <...>
me: My feeling is that it looks nice from the geometric point of view because you are visualizing a finite dimensional manifold. Of course, in the finite-dimensional case there is no essential difference between algebras and coalgebras, or local and global finiteness; so with finite-dimensional curved A_\infty algebras and modules of this kind one should not encounter any problems. But once you need to define, e.g., the infinite direct sum of a collection of such curved A_\infty-modules, you may start getting into some trouble.
E.S.: <...> I think I understand why for dg-modules if we look at A_\infty-morphisms then we correctly compute the morphisms in the derived category. Is there a simple reason why it goes wrong for curved dg-modules? <...>
me: I don't know whether it is simple enough or not, but the reason is explained in terms of the philosophy of "two kinds of derived categories". You may want to look up the introduction to my paper, which is now available from the arXiv at http://arxiv.org/abs/1202.2697 .
Basically, there are a number of explanations as to why the derived category of the first kind (what we conventionally call the derived category, say, of DG-modules over a DG-algebra) has no meaning for CDG-modules over a CDG-algebra over a field.
Speaking of A_\infty-algebras, an A_\infty-algebra or module whose differential is so nondegenerate that its cohomology is zero is viewed as a zero object in the conventional theory. So if m_1 is nondegenerate enough, all the operations starting from m_2 and higher no longer matter. When you attempt to extend this theory to curved algebras, it is m_0 that becomes the dominating term. If it is nondegenerate enough, all the operations starting from m_1 no longer matter. And it is all too easy for an element of a vector space to be "nondegenerate enough" -- being a nonzero vector is largely sufficient.
One alternative is to reverse the order of domination, having m_2 dominate over m_1 and m_1 over m_0. This leads to the constructions of the derived categories of the second kind. These are the exotic derived categories of curved modules discussed in various recent papers of mine (on Koszul duality, matrix factorizations, etc).
In the A_\infty setting, this would mean prohibiting your curved A_\infty-algebras, modules, morphisms of algebras and modules etc. from having an infinite number of nonzero higher operations. So only a finite number of higher multiplication or action maps would be allowed in each particular algebra, module, or morphism.
You may be familiar with these phenomena from your work on the Hochschild (co)homology of matrix factorizations. The choice between taking infinite direct sums or infinite products along the diagonals of various complexes is exactly one of choosing the order of domination of various terms ( = in the direction of which differential to take the cohomology first when constructing the spectral sequence, etc).
Recall that when Caldararu-Tu define their Borel-Moore Hochschild homology or compactly supported Hochschild cohomology of curved A_\infty-algebras, they have to make this assumption of a finite number of nonzero higher operations only, or otherwise their formulas for the total differentials in the Hochschild complexes would be divergent.
Actually, it is this divergence that gives their definition a chance to be nontrivial. The conventional definition of Hochschild (co)homology does not have this convergence issue and can be extended to arbitrary curved A_\infty-algebras. This fact alone sort of guarantees triviality of this definition in the case of algebras with nonzero curvature over a field.
So you can do the theory of curved A_\infty-algebras with a finite number of higher operations only if you wish to, and it is supposed to be nontrivial, but my feeling is that it would be quite unpleasant from the technical point of view. Analogues of the conventional results like "derived category of DG-modules = homotopy category of A_\infty-modules" would be hard to obtain, if they are true at all (though they are certainly expected to be closer to the truth in this bounded higher operations setting then in the unbounded one).
The preferrable setting for a curved A_\infty theory of the second kind is that of curved A_\infty-coalgebras. In this situation, one would impose the local "convergence condition" (the requirement that the sequence of images of any fixed element under the higher comultiplication maps contains only a finite number of nonzero terms) instead of the stronger global finiteness condition. This theory is discussed in "Two kinds of derived categories...", Sections 7.4-7.6. It is reasonably well-behaved.
The other alternative is to work over a ring rather than over a field, and prohibit your curvature elements from being nondegenerate by requiring them to be divisible by a fixed ideal. This allows to have a nontrivial curved A_\infty theory of the first kind, or something close to that. This is the subject of my new preprint.
<...> I would say that the natural formalism for discussing homotopies is that of model categories, but in this particular case I cannot immediately think of an appropriate model category structure. E.g., I don't know what the natural model category structure on CDG-algebras might be.
If the idea is to work with curved A_\infty things, my best suggestion would be to find a way to replace curved algebras with curved coalgebras; then conilpotent curved A_\infty-coalgebras are roughly the same thing as cofibrant DG-algebras. The model structure on DG-algebras might be used here then.
E.S.: Ah, I thought (well, hoped) that might be the answer! Actually the restriction to only finitely-many terms is quite nice from the geometric point of view, it means that your A_\infty structure comes from an actual (polynomial) vector field, not just the formal germ of a vector field. Also this allows you to have morphisms that don't preserve the origin - I used this to repair my paper on Hochschild homology. <...>
me: My feeling is that it looks nice from the geometric point of view because you are visualizing a finite dimensional manifold. Of course, in the finite-dimensional case there is no essential difference between algebras and coalgebras, or local and global finiteness; so with finite-dimensional curved A_\infty algebras and modules of this kind one should not encounter any problems. But once you need to define, e.g., the infinite direct sum of a collection of such curved A_\infty-modules, you may start getting into some trouble.