infinity-field

**homotopy theory, (∞,1)-category theory, homotopy type theory**

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed…

models: topological, simplicial, localic, …

see also **algebraic topology**

**Introductions**

**Definitions**

**Paths and cylinders**

**Homotopy groups**

**Basic facts**

**Theorems**

The notion of *$A_\infty$-field* is to the notion of A-∞ algebra as that of *field* is to associative algebra/ring.

An A-∞ ring or in fact just an H-space $A$ is a **field** if $\pi_\bullet A$ is a graded field.

For instance (Lurie, lecture 24, def. 3).

For $E$ an $\infty$-field, def. , then it carries the structure of an ∞-module over the $n$th Morava K-theory spectrum $K(n)$, for some $n$.

This follows with the nilpotence theorem.

- For $k$ an ordinary field, the Eilenberg-Mac Lane spectrum of $k$, $H k$, is an $A_\infty$-field.
- The Morava K-theory A-∞ rings $K(n)$ are the basic $A_\infty$-fields. See at
*Morava K-theory – As infinity-Fields*, where $K(0) \simeq H \mathbb{Q}$ and we define $K(\infty)$ as $H \mathbb{F}_p$.

Definition 3 in

- Jacob Lurie,
*Chromatic Homotopy Theory*, Lecture series 2010, Lecture 24*Uniqueness of Morava K-theory*(pdf)

- Jacob Lurie,
*Chromatic Homotopy Theory*, Lecture series 2010,Lecture 25

*The Nilpotence lemma*(pdf)