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.

• ## Weighted limits, ends, and Day convolution (Part 2)

[See part 1 here]

Using the idea of weighted limits, defined in the last post, we can now talk about ends. The idea of an end is that, given some functor $F\colon \mathcal{C}^\mathrm{op}\times\mathcal{C}\to\mathcal{D}$, which we can think of as defining both a left and a right action on $\prod_{c\in\mathcal{C}}F(c,c)$, we wish to construct some sort of universal subobject1 where the two actions coincide. Dually, a motivation behind the coend is in asking for some universal quotient of $\coprod_{c\in\mathcal{C}}F(c,c)$ that forces the two actions to agree.

1. A subobject of an object $y$ is a (class of isomorphisms of) monomorphism(s) into $y$.

• ## Loop spaces, spectra, and operads (Part 2)

[See part 1 here and part 3 here]

In the previous post of this series I talked a bit about basic loop space stuff and how this gave birth to the idea of ‘homotopically-associative algebras’. I’m going to detour slightly from what I was going to delve into next and speak about delooping for a bit first. Then I’ll introduce spectra as sort of a generalisation of infinite deloopings. I’ll probably leave the stuff about $E_\infty$-algebras for another post, but will definitely at least mention about how it ties in to all this stuff.

• ## 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.

• ## 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.