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Questions that are about research in mathematics, or about the job of a research mathematician, without being mathematical problems or statements in the strictest sense. Do not use this tag for easy or supposedly easy mathematical questions.

35 votes
12 answers
3k views

No canonical isomorphism [duplicate]

I thought that it would be interesting to collect into a big list various instances of isomorphic structures with no preferred isomorphism between them. I expect the examples to be interesting since i …
35 votes
1 answer
2k views

Are there topological versions of the idea of divisor?

I am trying to extract a particular, more lightweight and more focussed at the same time, case of my recent question Which of the physics dualities are closest in essence to the Spanier-Whitehead dual …
მამუკა ჯიბლაძე's user avatar
18 votes
3 answers
7k views

What is so special about set theory anyway? [closed]

(Later edit - tried to clarify a couple of vague places concerning interpretations of theories that became evident in comments (thanks to Andrej Bauer, Mauro ALLEGRANZA and Emil Jeřábek). (To closers …
მამუკა ჯიბლაძე's user avatar
13 votes
1 answer
577 views

Which nice/deep elaborations on the (operators <-> sheaves) / (endomorphisms <-> objects) th...

A linear operator $T:V\to V$ on a (say) vector space over a field $k$ is just a $k[T]$-module, and may be viewed as the sheaf $\mathscr F_T$ over $\mathbb A^1_k$, with fibre over $\lambda\in k$ equal …
მამუკა ჯიბლაძე's user avatar
12 votes
1 answer
351 views

Multiplicative infinitesimals in q-analogs?

Risking to be downvoted, here is a very lightweight question. In various fields - say, algebraic geometry, nonstandard analysis, synthetic differential geometry - infinitely small quantities, i. e. th …
მამუკა ჯიბლაძე's user avatar
7 votes
0 answers
234 views

$q$-crystals - is there such a thing?

There are several important facts that I first heard about here on MO. One of the most enlightening of these is that $\mathscr D$-modules on a scheme $X$ may be viewed as sheaves on the groupoid of in …
მამუკა ჯიბლაძე's user avatar
6 votes
2 answers
468 views

Is the logic of the ($\infty$-?)topos of simplicial sets "contradictory up to homotopy"?

In a sense this is a followup to my earlier question Does the (1-)topos structure on simplicial sets have any homotopy-theoretic significance?. In the topos of simplicial sets, the subobject classifie …
მამუკა ჯიბლაძე's user avatar
4 votes
0 answers
111 views

Is there a simple algebraic setup to accomodate fibres and cofibres at the same time?

If I understand it correctly, there are two mutually dual "leading principles" in homotopy theory: never perform quotients, add structure instead; never require subobjects, take fibres instead. Al …
მამუკა ჯიბლაძე's user avatar
0 votes
0 answers
415 views

Solving the equation $\operatorname{Powerset}(X)=\varnothing$

There are (at least) two variants of this question. Is it possible to modify the axioms of set theory, without arriving at obvious contradiction, in such a way that in a model of the theory there is …
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