nLab nInsights

This page aims to gather priceless snippets of mathematical wisdom about specific subjects, garnered from around the n-categorical blogosphere!

Weighted colimits, explained nicely by John Baez as the categorification of weighted sums.

Bernoulli numbers, link to a problem sheet by John Baez with solution by Jeff Morton on everything you need to know.

Framed cobordisms and TQFT’s, link to a classic comment by John Baez explaining how stable homotopy theory and extended TQFT become one and the same thing in the limit $n \rightarrow \infty$.

The link between duals for objects and adjoint functors. Here Bruce described what extra condition you need, besides $V \otimes -$ being left adjoint to $V^* \otimes -$, to make sure that $V^*$ is a right dual of $V$ in the normal sense, and gave an example showing why this condition was necessary.

Urs: To my mind, the purpose of this page here should be to remind us to eventually create separate $n$Lab entries that actually describe these “$n$Insights”, if only by collecting the material that is currently just being linked to.

Everybody with a minute of free time should feel encouraged to create an entry for one of the above links and start putting in material there, so that we can begin hyperreferencing it, etc. and it becomes part of the big web.

For instance the material on weighted colimits should go to weighted colimit parallel to weighted limit. Notice that some of the aspects that John mentions behind the above link do appear currently at Day convolution.

Similarly the stuff about framed cobordisms should go to either the existing generalized tangle hypothesis, or, better yet, should motivate us to finally create framed bordism? and cobordism hypothesis.

The stuff about duals and adjoints might get its own page or should be added to adjoint functor or else we possibly want a separete page dual object anyway, where this should go.