All Questions

I recently noticed the following categorical/universal way to describe the passage from $\mathbb{Z}$ to $\mathbb{Q}$: We start with the categroy $\mathsf{Sets}^{\mathrm{actv}}_*$ of pointed sets and —...

Define a homomorphism $\varphi : A \to B$ of commutative discrete rings or commutative ring spectra to be a (effective) descent morphism if the comparison functor from $\mathsf{Mod}_A$ to the category ...

There is this (folklore?) fact: for a commutative ring $R$, the category of $R$-modules is equivalent to the category of internal abelian groups in the slice category $\operatorname{Commutative rings}/...

Let $S$ be the sphere spectrum, and for $X$ a topological space, let $S(X)$ be the mapping spectrum from the free loop spectrum on $X$ to the sphere spectrum. This is an $E_\infty$ ring spectrum (also ...

A 1991 paper of Lewis, titled “Is there a convenient category of spectra?” proves that there is no category $\mathrm{Sp}$ satisfying the following desiderata$^1$: There is a symmetric monoidal smash ...

I am currently trying to wrap my head around all kinds of different definitions of the notion of (generalized) Thom spectrum. My setup is as follows: Suppose I have a commutative (symmetric) ring ...