Timeline for Is the ultraproduct concept fundamentally category-theoretic?
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| Mar 17, 2015 at 21:26 | comment | added | მამუკა ჯიბლაძე | In principle $Sh(\beta X)$ can be also described in more category-theoretic terms in this context. While $Sh(X)$ is sheaves $Sh({\mathscr P}(X),\textrm{can})$ on the powerset of $X$ with the canonical topology, $Sh(\beta X)$ is $Sh({\mathscr P}(X),\textrm{fin})$, sheaves on the same powerset but with the finite cover topology, the geometric inclusion being simply that any $\textrm{can}$-sheaf is obviously a $\textrm{fin}$-sheaf too, the reflector being given by $\textrm{can}$-sheafification of $\textrm{fin}$-sheaves. So in some sense $\beta X$ "is even simpler than $X$". | |
| Jan 30, 2010 at 3:14 | history | answered | François G. Dorais | CC BY-SA 2.5 |