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• CommentRowNumber1.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 20th 2021

This entry defines Grothendieck topologies using sieves.

However, in the original definition (Michael Artin’s seminar notes “Grothendieck topologies”), a Grothendieck topology on a category $C$ is defined as a set $T$ of coverings.

More precisely (to cite from Artin’s notes), a Grothendieck topology is defined as families of maps $\{\phi_i\colon U_i\to U\}_{i\in I}$ such that

• for any isomorphism $\phi$ we have $\{\phi\}\in T$;

• if $\{U_i\to U\}\in T$ and $\{V_{i,j}\to U_i\}\in T$ for each $i$, then $\{V_{i,j}\to U\}\in T$;

• if $\{U_i\to U\}\in T$ and $V\to U$ is a morphism, then $U_i\times_U V$ exist and $\{U_i\times_U V\to V\}\in T$.

This is almost identical to the current definition of Grothendieck pretopology, except that in Artin’s definition only the relevant pullbacks are required to exist.

It seems to me that the original definition by Artin is the one used most often in algebraic geometry.

• CommentRowNumber2.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 20th 2021

Moved the content from historical note on Grothendieck topology here.

• CommentRowNumber3.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 20th 2021

Added the original definition.

• CommentRowNumber4.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 20th 2021

Added the original reference.

• CommentRowNumber5.
• CommentAuthorDmitri Pavlov
• CommentTimeFeb 21st 2021

Added names for the axioms. Added a reference to SGA 4.

I removed the following axiom from the definition of a Grothendieck topology, since it is redundant (i.e., implied by the other axioms).

1. Two sieves $F, G$ of $c$ cover $c$ if and only if their intersection $F \cap G$ covers $c$. (Here the saturation condition is important.)