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.

Continue reading →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.

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.