Showing that a certain number is equal to some infinite sum of numbers using power series. For example, showing that $\ln(2)=1-\frac{1}{2}+\frac{1}{3}-\cdots$ or $e=2+\frac{1}{2}+\frac{1}{3!}+\cdots$.

An obvious problem in algebraic topology would be the computation of the homotopy groups of spheres.

So this entire discussion is in Goerss and Jardine's "Simplicial homotopy theory" and also in May's "Simplicial objects in algebraic topology". Also Curtis' papers and monographs are very nice and ...

From the website Chris Dionne mentioned in the comments: MODULE. A JSTOR search found the English term in E. T. Bell’s “Successive Generalizations in the Theory of Numbers,” American Mathematical ...

One "duality principle" that occurs in category theory is that of Isbel Duality. My feeling is (feel free to correct me if I am wrong) is that this encapsulates stone duality, Gelfand duality, and the ...

I am not sure if this is the type of direct proof that you are looking for, but here it goes. I will start with a more general theorem: Let $M$ be a CANCELABLE monoid, and $K$ be the left adjoint to ...

I do not know what kind of effect this has on your question, but it might be a good thing to look at in this context. In 1995, a formula for the $n$-th digit of $\pi$ written in hexadecimal was found. ...

While I agree with the above answers, I know of some situations where it might be good to take a skeleton (or something like it). This has to do with large categories which are essentially small. You ...

a mobius strip roller coaster. http://en.wikipedia.org/wiki/M%C3%B6bius_Loop_roller_coaster

Stoney Brook math videos: http://www.math.sunysb.edu/Videos/dfest/ http://www.math.sunysb.edu/html/videos.shtml

Here are some video lectures that John Morgan gave at Stony Brook http://www.math.sunysb.edu/Videos/dfest/ This also has many other nice mathematical videos.

The answer to this question depends on the type of set theory that you are working in, and the way you decide to code the integers and real numbers inside set theory. For instance in material set ...

I'm not precicely sure what you are looking for, but the following references I think are relevant. The Grothendieck Theory of Dessins d'Enfants (London Mathematical Society Lecture Note Series) LMs ...

Here I will sketch a proof, leaving many of the details out. The proof will be by induction on the number of edges of the rooted tree, which is the same as the dimension of the PS-complex (prod-...

Their is a notion of a category that is homotopicaly small. These categories have a nerve that is well defined up to homotopy. The notion is as follows: Let $\mathcal{C}$ be a large category. It is ...

Any time you have a topological groupoid, you have a simplicial groupoid since geometric realization preserves finite limits. Although this seems like a rather trivial remark this allows for one to ...

This answer is sort of an analogy, I am not quite sure how to make it precise. Further, It addresses that part of the question about a fiber product being anything from an intersection to a product (...

Another class of metric spaces that are of interest are length spaces. Roughly speaking, these are spaces in which you can measure the length of paths. The distance is then the inf of all lengths ...

This is that a small category may be regarded as a simplicial set with unique inner horn fillers.