This blog is a collection of reading notes of things that I try to learn. Often they are nothing more than summaries of various pages on the nLab, but collated, (over)simplified, and reordered to be more easily understood by, e.g., me. There will almost certainly be mistakes and bad viewpoints, so caveat lector.

  • Loop spaces, spectra, and operads (Part 1)

    [See part 2 here and part 3 here]

    I have been reading recently about spectra and their use in defining cohomology theories. Something that came up quite a lot was the idea of $E_\infty$-algebras, which I knew roughly corresponded to some commutative version of $A_\infty$-algebras, but beyond that I knew nothing. After some enlightening discussions with one of my supervisors, I feel like I’m starting to see how the ideas of spectra, $E_\infty$-algebras, and operads all fit together. In an attempt to solidify this understanding and pinpoint any difficulties, I’m going to try to write up what I ‘understand’ so far.

    Continue reading ⇾

  • Weighted limits, ends, and Day convolution (Part 1)

    A motto of category theory is that ‘Kan extensions are everywhere’. As a simplification of this, ‘(co)limits are in a lot of places’. By rephrasing the definition of a limit we end up with something that looks invitingly generalisable. This is how we can stumble across the idea of a weighted limit. In this post I’m going to assume that you are already convinced of the usefulness and omnipresence of limits and not talk too much (if at all) about why they are interesting in their own right.

    Continue reading ⇾

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