Geodesic
Internal choice holds in the discrete part of any cohesive topos satisfying stable connected codiscreteness
Theory and applications of categories, Tome 30 (2015), pp. 909-932 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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We introduce an apparent strengthening of Sufficient Cohesion that we call Stable Connected Codiscreteness (SCC) and show that if $p: E --> S$ is cohesive and satisfies SCC then the internal axiom of choice holds in $S$. Moreover, in this case, $p^!: S --> E$ is equivalent to the inclusion $E_{\neg\neg} --> E$.

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Classification : 18B25, 03G30, 18F99
Keywords: Axiomatic cohesion, Topos theory 
@article{TAC_2015_30_a25,
     author = {F. W. Lawvere and M. Menni},
     title = {Internal choice holds in the discrete part
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     journal = {Theory and applications of categories},
     pages = {909--932},
     year = {2015},
     volume = {30},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2015_30_a25/}
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F. W. Lawvere; M. Menni. Internal choice holds in the discrete part
of any cohesive topos satisfying stable connected codiscreteness. Theory and applications of categories, Tome 30 (2015), pp. 909-932. http://geodesic.mathdoc.fr/item/TAC_2015_30_a25/