TITLE: Do smooth ind schemes have Dualizing sheafs?
QUESTION [11 upvotes]: Say I have an ind scheme $X = \cup_i X_i$ over a field $k$. I have its tangent bundle $\hom_k(k[\epsilon], X)$ which I can think of as ind scheme via $\cup_i \hom_k(k[\epsilon],X_i)$.  The problem is even if $X$ is smooth it might be the case that most of the $X_i$ are not smooth.  I believe this happens at least for polynomial loop groups $G[z^\pm]$.  In this case the sheaf of differentials is not locally free.  This seems to be an obstruction to constructing the canonical sheaf inductively.  
Additionally if each $X_i$ is infinite dimensional, which happens for the formal loop group, then it seems like top exterior power doesn't make much sense.  And finally if you looked at say $\mathbb{P}^\infty := \cup_n \mathbb{P}^n$ then it also seems unclear what a canonical sheaf should be.  If it were a line bundle it could be described as a line bundle $L_n$ on each $\mathbb{P}^n$ which are compatible under pull backs.  But then each $L_n$ would have the same degree $d$.  But the canonical line bundles $O(-n-1)$ have a different degree on each $\mathbb{P}^n$!
So is there any sense in asking for something like a canonical sheaf or dualizing sheaf for smooth ind schemes?
UPDATE: Brian Conrad shared the following with me:
If $f:X \to Y$ is a map between finite type schemes over a field (or one can be much more general...) then for a relative dualizing complex $\omega_Y$ on $Y$ we have that $f^!(\omega_Y)$ is a relative dualizing complex on $X$ (for suitable functor $f^!$ at derived category level).  In other words, one does have "compatibility" for relative dualizing complexes, but with respect to the appropriate "derived pullback" operation $f^!$.   (One has to think about duality and "canonical sheaf" in a much broader sense than Serre duality over a field in order to define "relative dualizing object" in a derived category.)
The upshot is that one has to work in derived categories (and so develop a suitable formalism of direct/inverse limits in derived categories) to make a good theory of duality on ind-schemes.

REPLY [3 votes]: By judicious use of Kan extension, one can define a functorial !-pullback, and hence a dualizing complex, for a broad class of geometric objects, including arbitrary DG prestacks (Gaitsgory section 10.1), and in particular smooth ind-schemes as you ask.
This is used in Theorem 10.1.1 of Gaitsgory-Rozenblyum, where tensoring with the dualizing complex yields an equivalence between quasicoherent sheaves and ind-coherent sheaves in the case of formally smooth DG ind-schemes that are weakly $\aleph_0$ and locally almost of finite type.  In this case, the dualizing complex is an ind-coherent sheaf that presumably looks something like a shifted line bundle, but I couldn't extract that information directly from the paper.  Serre duality for DG ind-schemes locally almost of finite type also appears as Corollary 2.6.2 of the same paper, but its statement doesn't seem to use the dualizing complex directly.