nLab Galois category

Introduction

A Galois category is a category equipped with structure allowing for a construction of a ‘fundamental group’ from it, akin to the construction of the fundamental group of a topological space from the category of covering spaces of it. See Grothendieck's Galois theory for more on the latter.

Details

There are two ways to define a Galois category. We give them both below, following SGA1.

Definition

A Galois category is a category equivalent to the classifying topos of a profinite group?.

Definition

A Galois category is a category $G$ for which there exists a functor $F: G \rightarrow \mathsf{FinSet}$, where $\mathsf{FinSet}$ is the category FinSet of finite sets, such that the following hold.

1. $G$ has finite limits.
2. $G$ has finite colimits.
3. Every arrow $f: X \rightarrow Y$ of $G$ factorizes as a composition of a strict epimorphism $u: X \rightarrow Z$ and the coprojection $v: Z \rightarrow Y$ for a coproduct decomposition $Y \cong Z \sqcup Z'$, where $v$ is a monomorphism.
4. $F$ is exact, that is to say, preserves finite limits and finite colimits.
5. $F$ is conservative.

Remark

The original form of 2. in §4 of SGA1 is slightly weaker, and the axiom 4. was originally two axioms which together were slightly weaker than our axiom, namely that $F$ preserved finite limits, and that $F$ preserved the colimits required to exist in the weaker form of 2. However, as discussed in Remark 4.2 of SGA1, if the weaker axioms hold and the other axioms hold, then 2. and 4. as we have given them hold, so we prefer to use them for simplicity and brevity.

References

• Revêtements étales et groupe fondamental (SGA 1), Alexander Grothendieck, 1971, Springer-Verlag, vol. 224 of Lecture notes in mathematics.