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Handy link: microlocalization.
Microlocal analysis is a wider idea than just the procedure microlocalization. It deals with certain properties, which are microlocal. Algebraic microlocalization is discovered by Van den Bergh and van Oystaeyen much later than microlocalization of Kyoto school and which applies to broader algebraic context, which can not have the same interpretation as microlocality in analysis does and applies to certain noncommutative filtered rings.
I do not understand what ind-sheaves have to do with microlocalization. They are useful indeed for the distribution spaces and so on, but this is not specific to microlocalization. The idea that ind-objects are close in spirit to bringing topologies into algebraic world and to extending spaces of distributions has many incarnations. For example, the de Rham cohomology for singular spaces can be modelled by taking the usual de Rham currents instead of forms, as shown by MacPherson and collaborators (for noncommutative exxtension of these ideas look at Cuntz-Quillen work from 1990s). Another case of this idea is the Morris-Pareigis formal scheme theory.
Thanks, put some of these remarks in $n$Lab…
The second paper says:
Our aim in this paper is to present the noncommutative analogue of the approach of Deligne [D] and Hartshorne [H] to de Rham cohomology in algebraic geometry. In this approach de Rham cohomology is first obtained for a nonsingular algebraic variety by means of the de Rham complex of differential forms. An arbitrary variety is then treated by embedding it in a nonsingular variety and completing the de Rham complex of the latter along the subvariety.
On the other hand, another model, due MacPherson is explained in Chriss-Ginzburg book. One take just the de Rham complex of differential forms on a (complex) singular variety whose coefficients are generalized functions, so one works with the de Rham complex of currents. The first approach is having a spirit of ind-objects (and probably could be restated in their terms) and the second in the sense of distributions. Cuntz and Quillen notice that the noncommutative version of de Rham is the cyclic homology and try to do a similar program there.
The Morris-Pareigis work generalized the Yoneda lemma to have some topology, so that they have a representing object a topological ring, not just a ring. It is like a formal power series ring representing some formal scheme. Further they represent formal schemes using formal colimit of certain presheaves, satisfying some axioms. It is well known that formal schemes can be considered as forming a subcategory of ind-schemes. The usual picture with formal spectra has some topologically ringed spaces, the topology is kind of replacing the flexibility which can be also achieved via formal systems. The connection is more explicit in a flexible approach of Morris-Pareigis.
Related notions: algebraic microlocalization
Page name changed from capitalized to ordinary as the usual convention in $n$Lab.
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