Grothendieck coverage

Last updated January 27, 2022

Idea

An abstraction of the manner in which open sets cover a topological space.

A category with a Grothendieck coverage is a site.

Definition

Let $C$ be a category.

1. A Grothendieck coverage of $C$ assigns to each $c\in\text{Obj}(C)$ a collection of “covering” sieves $J(c)$ such that
   1. (stability under base change) if $F\in J(c)$ and $g:d\to c$, then $$g^\ast F\in J(d).$$
   2. (local character condition) let $S\in J(c)$ and $T$ any sieve on $c$. If for every $d\in\text{Obj}(C)$ and $g:d\to c\in S$ we have that $g^\ast T\in J(d)$, then $T\in J(c)$.
   3. (maximial sieve is a covering sieve) $\text{Obj}(C/c)\in J(c)$

Remarks on Definition 1

• Stability under base change
  • Let $C=\text{Op}(X)$, the category of open sets of a topological space $X$ under inclusion. This condition tells us that if ${U_i}$ covers $U$ and $g:V\hookrightarrow U$, it must be the case that $g^\ast {U_i}={U_i\cap V}$ covers $V$.
• Local character condition
  • This is a sort of converse for the first statement, saying that if the pullback along every arrow in a covering sieve is also a covering sieve, then it itself is a covering sieve.
  • Going with our example above, let ${U_i}$ be a covering sieve for $U$. Every arrow in this sieve is of the form $g:U_i\to U$. Now suppose we had a set ${V_{ij}}$ such that $g^\ast {V_{ij}}$ is a covering sieve, i.e. ${V_{ij}}$ covers each $U_i$.Then it should (and is) the case that the ${V_{ij}}$ cover $U$.
  • So another way of seeing this is that if a sieve covers each local component of a covering sieve, then we ought to be able to piece it all together to conclude that it is globally a covering sieve.
• Maximal sieve is a covering sieve
  • Going with our example above, this tells us that a space ought to be covered by the collection of all its (open?) subsets