Is I consider an ind scheme such as $G(k((t)))$ for a reductive connected group over $k=\bar{k}$


I have the conjugacy action of $G(k[[t]])$.


In what category can I make the quotient $[G(k((t))/ad(G(k[[t]])]$?


In the category of presheaves? The category of fppf sheaves?


And if I want to make the fiber product of two such object over $k$, same question.


Moreover, if $f:X\rightarrow Y$ is a morphism of presheaves, i.e. contravariant functors from the category of schemes to sets, is the notion of formall smoothness (by using the infinitesimal lifting property as the definition of formal smoothness) stable under base change?



quotient of ind scheme
Not sure what you are really asking. You can always do quotients in fppf sheaves, as explained by ancient greeks. Just quotient presheaves and sheafify around.


The point about the affine grassmanian is that it is turning out to be an ind-scheme, which is not true for quotients of general ind-schemes.


I am not sure about the last question. You can say that your thingy is formally smooth if you have the lifting property along the usual schemes. But I do not know what it means, what the examples and utility are, and whether anyone looked into them. In general, you should probably avoid even thinking about it: consider spending your life in fppf sheaves!


