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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexexpanded the Idea-section and added a reference: * [[Francis Borceux]], [[George Janelidze]], [[Gregory Maxwell Kelly]], p. 8 of: _Internal object actions_, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 ([dml:249553](https://eudml.org/doc/249553)) <a href="https://ncatlab.org/nlab/revision/diff/module+object/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/6">v6</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   expanded the Idea-section and added a reference:

   • Francis Borceux, George Janelidze, Gregory Maxwell Kelly, p. 8 of: Internal object actions, Commentationes Mathematicae Universitatis Carolinae (2005) Volume: 46, Issue: 2, page 235-255 (dml:249553)

   diff, v6, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded the example of [[module spectra]] (which was the context in which I had originally created this entry here, but apparently I forgot to cross-link) <a href="https://ncatlab.org/nlab/revision/diff/module+object/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/6">v6</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   added the example of module spectra (which was the context in which I had originally created this entry here, but apparently I forgot to cross-link)

   diff, v6, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2021
   • (edited Mar 22nd 2021)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded this pointer, for what it's worth: * Tao Lu, Zhenzhen Zhu, _The Action of Group Object in A Topos_, Proceedings of: _2015 International Conference on Management, Education, Information and Control_ ([doi:10.2991/meici-15.2015.312](https://doi.org/10.2991/meici-15.2015.312)) [Edit: removed it again -- too incoherent ] <a href="https://ncatlab.org/nlab/revision/diff/module+object/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/6">v6</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   added this pointer, for what it’s worth:

   • Tao Lu, Zhenzhen Zhu, The Action of Group Object in A Topos, Proceedings of: 2015 International Conference on Management, Education, Information and Control (doi:10.2991/meici-15.2015.312)

   [Edit: removed it again – too incoherent ]

   diff, v6, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 22nd 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Saunders Mac Lane]], [[Ieke Moerdijk]], Section V.6 of: _[[Sheaves in Geometry and Logic]]_, Springer 1992 ([doi:10.1007/978-1-4612-0927-0](https://link.springer.com/book/10.1007/978-1-4612-0927-0)) <a href="https://ncatlab.org/nlab/revision/diff/module+object/6">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/6">v6</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   added pointer to:

   • Saunders Mac Lane, Ieke Moerdijk, Section V.6 of: Sheaves in Geometry and Logic, Springer 1992 (doi:10.1007/978-1-4612-0927-0)

   diff, v6, current

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeJun 18th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded pointer to: * [[Alexander Grothendieck]], p. 104 (7 of 21) in: _Techniques de construction et théorèmes d'existence en géométrie algébrique III: préschémas quotients_, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, ([numdam:SB_1960-1961__6__99_0](http://www.numdam.org/item/?id=SB_1960-1961__6__99_0), [pdf](http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf)) <a href="https://ncatlab.org/nlab/revision/diff/module+object/8">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/8">v8</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   added pointer to:

   • Alexander Grothendieck, p. 104 (7 of 21) in: Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (numdam:SB_1960-1961__6__99_0, pdf)

   diff, v8, current

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeJul 4th 2021
   • PermaLink
   Author: Urs
   Format: MarkdownItexAdded to the list of examples a pointer to *[[simplicial group action]]* (to be created in a moment...) <a href="https://ncatlab.org/nlab/revision/diff/module+object/10">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/10">v10</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   Added to the list of examples a pointer to simplicial group action (to be created in a moment…)

   diff, v10, current

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeDec 9th 2022
   • (edited Dec 9th 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded mentioning of the example of coherent action objects in $Cat$ ([here](https://ncatlab.org/nlab/show/module+object#ActionObjectsInCat)) <a href="https://ncatlab.org/nlab/revision/diff/module+object/14">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/14">v14</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   added mentioning of the example of coherent action objects in $Cat$ (here)

   diff, v14, current

8. • CommentRowNumber8.
   • CommentAuthorUrs
   • CommentTimeSep 6th 2023
   • (edited Sep 6th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexHere and in the related entries on ([[commutative monoid in a symmetric monoidal category|commutative]]) [[monoid objects]], I have added more pronounced pointer to * [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], pp. 15 in: *Symmetric spectra*, J. Amer. Math. Soc. **13** (2000) 149-208 &lbrack;[arXiv:math/9801077](http://arxiv.org/abs/math/9801077), [doi:10.1090/S0894-0347-99-00320-3](https://doi.org/10.1090/S0894-0347-99-00320-3)&rbrack; where citable direct proofs are given for the symmetric monoidal closure of categories of modules. The entries used to point more prominently to the MO question requesting such a reference * [MO:180673](http://mathoverflow.net/questions/180673/category-of-modules-over-commutative-monoid-in-symmetric-monoidal-category). As noticed there, a detailed direct proof is also laid out in * [[Florian Marty]], Prop. 1.2.14, 1.2.16, 1.2.17 in: _Des Ouverts Zariski et des Morphismes Lisses en G&#233;om&#233;trie Relative_, Ph.D. Toulouse (2009) &lbrack;[theses:2009TOU30071](https://www.theses.fr/2009TOU30071), [pdf](https://ncatlab.org/nlab/files/Marty-DesOuverts.pdf)&rbrack; but this part does not seem to have made it into a publication. <a href="https://ncatlab.org/nlab/revision/diff/module+object/20">diff</a>, <a href="https://ncatlab.org/nlab/revision/module+object/20">v20</a>, <a href="https://ncatlab.org/nlab/show/module+object">current</a>

   Here and in the related entries on (commutative) monoid objects, I have added more pronounced pointer to

   • Mark Hovey, Brooke Shipley, Jeff Smith, pp. 15 in: Symmetric spectra, J. Amer. Math. Soc. 13 (2000) 149-208 [arXiv:math/9801077, doi:10.1090/S0894-0347-99-00320-3]

   where citable direct proofs are given for the symmetric monoidal closure of categories of modules.

   The entries used to point more prominently to the MO question requesting such a reference

   • MO:180673.

   As noticed there, a detailed direct proof is also laid out in

   • Florian Marty, Prop. 1.2.14, 1.2.16, 1.2.17 in: Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative, Ph.D. Toulouse (2009) [theses:2009TOU30071, pdf]

   but this part does not seem to have made it into a publication.

   diff, v20, current