Inter-universal geometry と ABC予想 (応援スレ) 44

Inter-universal geometry と ABC予想 (応援スレ) 44at MATH

Inter-universal geometry と ABC予想 (応援スレ) 44 - 暇つぶし2ch30:uarantee13 that the arrows of D are sent to identity maps by X’; in fact if the arrows in the image of X are not invertible, then neither will the arrows in the image of X’. What is going on is that even though one might assume for simplicity that all the objects of the diagram are sent to the same object, assuming that all the arrows in the diagram between them are identity arrows may be an obstruction to the existence of the natural isomorphism a, and hence to the existence of an isomorphism between the (formal) colimits. Another tactic that Scholze?Stix use is looking at diagrams transferred through some equivalence E: C → C’ of categories14. This is particularly useful if the objects and arrows of C’ are a lot simpler to describe, and it may even be the case that C’ has all objects isomorphic, even if there are many non-invertible maps. Note that equivalences of categories commute with colimits, and the free cocompletions of equivalent categories are equivalent, so one is free to consider diagrams in a one-object category C’ as giving elements of the free cocompletion of C. Again, I emphasise that diagrams D → C’, where C’ is a one-object category, can give rise to nontrivial results in the free cocompletion of C’. There is no mathematical reason why calculations cannot proceed in this manner wherever possible. (引用終り) 以上

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