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. Although having encountered instances of both many times, I must confess that I understand the first one much better than the second one. I understand conceptual reasons behind … Read more

What do orbital integrals have to do with reciprocity?

Hi, this is my first question (of many). I am blogging for the Fields Medal Symposium and would like to get into the mathematics involved with our program. In an attempt to sort through the articles and through all the conversations that I’ve been having in and around the Fields Institute, I’m having trouble seeing … Read more

Toric ideal of slice of a polytope?

Given a collection A:={a1,…,an} of different integer points in Nd, which span an affine hyperplane when viewed in Rd, one can define a toric ideal IA from a monomial homomophism: ϕA:k[x1,…,xn]→k[y1,…,yd],xi↦yai:=d∏j=1yai,jj according to IA:=kerϕA (suppose k is algebraically closed of characteristic zero). Let H be a hyperplane in kd such that conv(A) (when viewed as … Read more

Examples of rich theories that started out as an infinite-dimensional inquiry

It seems that when a mathematical theory was newly invented, or a particular phenomenon was discovered, it is often while tackling a specific hard problem, but as more of the theory was developed it found use in a variety of problems, many of which are “easier”, as in not assuming as much prerequisite. Thus, when … Read more

What kinds of limits does localization of commutative rings reflect?

Localization of commutative rings is a left exact left adjoint, so it behaves nicely with plenty of things. Local-to-global principles are also abundant in commutative algebra, and I thought some of them might actually be reflecting limits in disguise – that some types of limits in localizations must arise from limits in the original ring. … Read more

Are the paradoxes of material or strict implication used anywhere to prove theorems in mathematics

In the Stanford Encyclopedia of Philosophy entry “Relevance Logic“, the following inference is listed as classically valid: The moon is made of green cheese. Therefore, it is raining in Ecuador now or it is not. This inference can be tweaked slightly to make it more mathematical: The moon is made of green cheese. Therefore, CH … Read more

About the cone being unique up to non-unique isomorphism

In an answer to this MO question [link] Fernando Muro sais: the mapping cone of a morphism in a triangulated category is unique up to non-unique isomorphism. This fact has originated a lot of research in this topic, and it still does. I would be curious to know (not necessarily from the author of that … Read more

“$\kappa$ strongly inaccessible” = “every function $f:V_\kappa\to V_\kappa$ can be self-applied”?

Strongly inaccessible cardinals are usually introduced either as (a) cardinalities of models of ZFC or (b) cardinals which are not the power set of a smaller cardinal nor the supremum of a set with hereditarily lesser cardinality. These seem to represent the model-theoretic and set-theoretic perspectives on strong inaccessibility. Recently I learned that if $\kappa$ … Read more

Intrinsic vs. Extrinsic [closed]

Closed. This question is off-topic. It is not currently accepting answers. Want to improve this question? Update the question so it’s on-topic for MathOverflow. Closed 10 years ago. Improve this question Undoubtedly, these terms play essential roles in (pure) mathematics. My problem is that I have feelings what they mean in different fields, such as, … Read more