***Forwarding - Sorry for multiple messages *** Dear all, I am writing to let you know that, in the scope of the /DuaLL/ project, *Friedrich Wehrung* is coming to Nice to give a mini-course on "Constructing (counter)examples via condensates", in the week *May 20 - 24* (you can find the abstract below). [I apologize to those who are receiving this e-mail twice] Here is the planned schedule: Mon 20: 15h -> 16h Tue 21: 15h -> 16h Wed 22: 15h -> 16h Thu 23: 11h -> 12h Fri 24: 11h -> 12h For logistic reasons (and a request from the speaker), please let me know by the ***May 10*** whether you'll be attending this course. In case you wish to attend but the schedule doesn't suit you, please let me know as soon as possible. You can also see the following website for next planned meetings of /DuaLL/: https://math.unice.fr/~cborlido/GdT.html Please don't hesitate to let me know if you wish to be added to our mailing list. Best regards, C??lia -------------------------------------------------------------------------------------------------------------------------------------- *Abstract:* For categories A and B, the determination of the range of a given functor F: A\to B gives often rise to seemingly intractable problems, even at the most basic level ??? that is, does every object of B belong to the range of F (up to isomorphism of course)? Similar, apparently stronger questions can be stated for arrows in B, or, more generally, for commutative diagrams in B. It turns out that due to a 2011 construction of the author with Pierre Gillibert, that we called the condensate construction, all those questions are, under fairly general conditions, equivalent. For instance, representing an arrow is ?? not really ?? harder than representing an object. However, this equivalence comes with a cost: going from a diagram counterexample to an object counterexample (outside the range of F) requires a cardinality jump, the amplitude of which depends of a natural number called the Kuratowski index (often equal to the order-dimension) of the shape of the diagram in question. Recent improvements of the condensate construction made it possible to prove stronger negative results, stating that the range of the functor F is not even closed under elementary equivalence with respect to infinitary languages of the form L_{\infty,\lambda}. Such negative results are inferred from the existence of a (necessarily non-commutative) diagram D in A such that F(D) is not F(X) for any commutative diagram X in A. For example, the long-standing problem of the characterization of the spectra of all Abelian lattice-ordered groups finds there a negative solution: namely, the class of all Stone duals of such spectra (which are special kinds of distributive lattices with zero) is not closed under L_{\infty,\lambda}-elementary equivalence for any infinite cardinal \lambda; in particular, it is not the class of all models of any class of L_{\infty,\lambda} sentences. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]