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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeJun 16th 2018
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have spelled out an argument ([here](https://ncatlab.org/nlab/show/concrete+object#ConcretificationFactorization)) for the statement that in a local topos the full subcategory of concrete objects provides a factorization of $\Gamma \dashv coDisc$ as $$ \Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AAA}}{\hookleftarrow} } Set $$ <a href="https://ncatlab.org/nlab/revision/diff/concrete+object/5">diff</a>, <a href="https://ncatlab.org/nlab/revision/concrete+object/5">v5</a>, <a href="https://ncatlab.org/nlab/show/concrete+object">current</a>

   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

   $$\Gamma \;\dashv\; coDisc \;\;\colon\;\; \mathbf{H} \array{ \overset{\phantom{AA} conc \phantom{AA}}{\longrightarrow} \\ \overset{\phantom{AA} \iota_{conc} \phantom{AA}}{\hookleftarrow} } \mathbf{H}_{conc} \array{ \overset{\phantom{AAA}}{\longrightarrow} \\ \overset{\phantom{AAA}}{\hookleftarrow} } Set$$

   diff, v5, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJun 22nd 2018
   • (edited Jun 22nd 2018)
   • PermaLink
   Author: Urs
   Format: MarkdownItexthe proof of that Proposition (still [here](https://ncatlab.org/nlab/show/concrete+object#ConcretificationFactorization)) was lacking the argument that $im( \eta^{\sharp}_X )$ is indeed concrete. Have added that now. <a href="https://ncatlab.org/nlab/revision/diff/concrete+object/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/concrete+object/7">v7</a>, <a href="https://ncatlab.org/nlab/show/concrete+object">current</a>

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

   diff, v7, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 6th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexI have added ([here](https://ncatlab.org/nlab/show/concrete+object#CohesiveMapsToConcreteObjectsGlue)) 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. <a href="https://ncatlab.org/nlab/revision/diff/concrete+object/11">diff</a>, <a href="https://ncatlab.org/nlab/revision/concrete+object/11">v11</a>, <a href="https://ncatlab.org/nlab/show/concrete+object">current</a>

   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.

   diff, v11, current