A Grothendieck topology $J$ on a category $\mathcal{C}$ is an assignment $J$ sending any object $X \in \mathcal{C}$ to a collection $J(X)$ of sieves on $\mathcal{C}$ such that the following properties are satisfied.

Before we view the properties

we denote $S \in J(X)$ with $S \mathbin{⪧}_J X$ or $X \mathbin{⪦}_J S$ .
We need to define maximal sieve.

Definition Maximal sieve [IGSL]

In a category $\mathcal{C}$ , a maximal sieve of an object $X \in \mathcal{C}$ is defined as

M_X := \{ f \mid \text{cod}(f) = X \}

Property Maximality axiom [BEIE]

The maximal sieve $M_X \mathbin{⪧}_J X$ .

Property Pullback stability [FXZI]

If $S \mathbin{⪧}_J X$ and $f : Y \to X$ , then

f^*(S) \mathbin{⪧}_J Y

$f^*(S)$ defined by

f^*(S) := \{ g : Z \to Y \mid f \circ g \in S \}

We can represent it as

Property Transitivity [MACB]

Let $S \mathbin{⪧} X$ be a sieve on $X$ , and $T \mathbin{⪧}_J X$ be a sieve in $J(X)$ . If for all $f \in T$ we have $f^*(S) \mathbin{⪧}_J \text{dom}(f)$ , then $S \mathbin{⪧}_J X$ .

Definition. Grothendieck topology [ZD97]

Definition Maximal sieve [IGSL]

Property Maximality axiom [BEIE]

Property Pullback stability [FXZI]

Property Transitivity [MACB]