Timeline for A non-conventional definition of topoi
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
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| Nov 1, 2021 at 9:15 | comment | added | D.-C. Cisinski | @AndreasBlass Given any small site, one can define $W$ as the smallest class of maps between presheaves closed under finite limits as well as under colimits and containing inclusions of covering sieves. The localization of the category presheaves by $W$ is then (equivalent to) the category of sheaves. | |
| Nov 1, 2021 at 2:19 | history | edited | David Roberts♦ | CC BY-SA 4.0 |
added 125 characters in body
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| Oct 31, 2021 at 23:43 | comment | added | Andreas Blass | @მამუკაჯიბლაძე Yes, but the point of the quoted material seems be mainly that, in the case of a sheaf subtopos $S$ of a presheaf topos $P$ on a space, "the class of those morphisms in $P$ which are carried to an isomorphism under the left adjoint of the inclusion from $S$ into $P$" can be defined without knowing anything about $S$, namely as the morphisms that induce isomorphisms on all stalks. | |
| Oct 31, 2021 at 22:27 | comment | added | მამუკა ჯიბლაძე | Well, appropriately formulated, it can be generalized. A presheaf morphism induces isomorphism on stalks if and only if it is carried to an isomorphism by the left adjoint of the inclusion of sheaves into presheaves. And any subtopos $S$ of any topos $P$ is equivalent to $W^{-1}P$, where $W$ is the class of those morphisms in $P$ which are carried to an isomorphism under the left adjoint of the inclusion from $S$ into $P$. | |
| Oct 31, 2021 at 19:01 | vote | accept | user234212323 | ||
| Oct 31, 2021 at 18:49 | history | answered | Dan Petersen | CC BY-SA 4.0 |