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 3)

[See part 1 here and part 2 here]

This post is a weird one: it’s not really aimed at any one audience, but is more of a dump of a bunch of information that I’m trying to process.

• ## 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 $y$.

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.