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• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJun 16th 2018

I have spelled out an argument (here) for the statement that in a local topos the full subcategory of concrete objects provides a factorization of $\Gamma \dashv coDisc$ as

$\Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AAA}}{\hookleftarrow} } Set$
• CommentRowNumber2.
• CommentAuthorUrs
• CommentTimeJun 22nd 2018
• (edited Jun 22nd 2018)

the proof of that Proposition (still here) was lacking the argument that $im( \eta^{\sharp}_X )$ is indeed concrete. Have added that now.

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeOct 6th 2021

I have added (here) an observation that in a cohesive $\infty$-topos concrete objects are such that cohesive maps to them “glue”, in that every cohesive map out of a cover whose map of underlying $\infty$-groupoids descends down the cover also descends as a cohesive map.

This is an immediate consequence of the definitions and the (-1)-connected/(-1)-truncated orthogonality. The point is just to observe that this lifting problem does have this interpretation for concrete objects.