Timeline for When does the direct image functor nicely push past the power/exists functor?
Current License: CC BY-SA 3.0
6 events
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| Mar 18, 2013 at 15:55 | history | edited | David Spivak | CC BY-SA 3.0 |
fixed typo, added space.
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| Mar 18, 2013 at 4:07 | comment | added | David Spivak |
For posterity, let $D$ be the topos of cospans in ${\bf Set}$ and let $E$ be the topos ${\bf Set}$ of sets. The unique geometric morphism $f_{\ast}\colon D\to E\ $ sends each cospan to its fiber product. It does not preserve epis, indeed let $X$ be the cospan $\{1\}\to\{1,2\}\leftarrow\{2\}\ $ and let $Y$ be the terminal cospan. The unique morphism $p\colon X\to Y\ $ is epi but $f_{\ast}(X)=\emptyset$ whereas $f_{\ast}(Y)=1\ $. One can check that the components given by $\theta$ constructed above do not form a naturality square for $p$.
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| Mar 18, 2013 at 0:05 | vote | accept | David Spivak | ||
| Mar 17, 2013 at 14:21 | history | edited | David Spivak | CC BY-SA 3.0 |
last edit caused weird mathjax.
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| Mar 17, 2013 at 1:38 | comment | added | Todd Trimble | That $f_\ast$ should preserve epis was the conclusion of my analysis as well (but when I came here to write it out, your answer was here already (-:). | |
| Mar 17, 2013 at 1:26 | history | answered | Zhen Lin | CC BY-SA 3.0 |