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1. • CommentRowNumber1.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 12th 2016
   • (edited Aug 12th 2016)
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   Author: David_Corfield
   Format: MarkdownItexThere's an entry [[enriched homotopy theory]] which hasn't been touched for many years. I came across it from a Cafe [conversation](https://golem.ph.utexas.edu/category/2009/01/nlab_general_discussion.html#c021228) which has Mike saying: >Enriched topos theory is actually a deep and (at least, to me) unclear subject. So much of topos theory seems to depend on cartesianness. I’ve occasionally thought about what an enriched topos might be, but never really come up with anything really satisfactory. If the enrichment isn’t cartesian monoidal, then the internal logic of an enriched topos would probably be linear logic, but how to interpret linear logic internally in some category is also not an obvious question. One expects that perhaps “quantales” (closed monoidal suplattices) will play a role. I found this in turn after a [conversation](https://golem.ph.utexas.edu/category/2016/08/a_survey_of_magnitude.html#c050941) with Tom Leinster. Anyone here able to shed some light on whether there's a cohomology for enriched settings, or to update us on other matters enriched?

   There’s an entry enriched homotopy theory which hasn’t been touched for many years. I came across it from a Cafe conversation which has Mike saying:

   Enriched topos theory is actually a deep and (at least, to me) unclear subject. So much of topos theory seems to depend on cartesianness. I’ve occasionally thought about what an enriched topos might be, but never really come up with anything really satisfactory. If the enrichment isn’t cartesian monoidal, then the internal logic of an enriched topos would probably be linear logic, but how to interpret linear logic internally in some category is also not an obvious question. One expects that perhaps “quantales” (closed monoidal suplattices) will play a role.

   I found this in turn after a conversation with Tom Leinster.

   Anyone here able to shed some light on whether there’s a cohomology for enriched settings, or to update us on other matters enriched?

2. • CommentRowNumber2.
   • CommentAuthorTim_Porter
   • CommentTimeAug 13th 2016
   • (edited Aug 13th 2016)
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   Author: Tim_Porter
   Format: MarkdownItexDavid: What sort of cohomology do you want or hope for and in what context i.e. a general enriched category in which a viable homotopy theory exists or what?

   David: What sort of cohomology do you want or hope for and in what context i.e. a general enriched category in which a viable homotopy theory exists or what?

3. • CommentRowNumber3.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 13th 2016
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   Author: David_Corfield
   Format: MarkdownItexAs in the comments after Tom's [post](https://golem.ph.utexas.edu/category/2016/08/a_survey_of_magnitude.html), there was a [question](https://golem.ph.utexas.edu/category/2016/08/a_survey_of_magnitude.html#c050940) about finding a homology theory for enriched categories other than graphs. I was just wondering then what there is known about enriched cohomology and homotopy, and so came across some old material. In the meantime, it seems that Mike has a good suggestion for the homology of categories enriched in semicartesian monoidal posets, and even for semicartesian monoidal categories.

   As in the comments after Tom’s post, there was a question about finding a homology theory for enriched categories other than graphs.

   I was just wondering then what there is known about enriched cohomology and homotopy, and so came across some old material.

   In the meantime, it seems that Mike has a good suggestion for the homology of categories enriched in semicartesian monoidal posets, and even for semicartesian monoidal categories.

4. • CommentRowNumber4.
   • CommentAuthorTim_Porter
   • CommentTimeAug 13th 2016
   • (edited Aug 13th 2016)
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   Author: Tim_Porter
   Format: MarkdownItexThere is some recent work in directed homotopy that may be related to this. Goubault and Mimram have been looking at generalised metric spaces and their relation with directed homotopy and its applications in concurrency. I have seen references to a paper `in preparation'. Another point might be to think of what such a cohomology theory should classify and then try to finding a classification of those objects may lead one back to a formulation of cohomology. I did something like that in discussions with Philippe Malbos and Yves Guiraud in an attempt to classify group objects in the category of categories over a given (fixed) category. Understanding the objects led me back to the cohomology of categories which was in Charles Wells unpublished note (eventually published in TAC I think.) This way the factorisation category came out quite naturally together with a non-Abelian version of Baues-Wirsching cohomology already in Wells' note.

   There is some recent work in directed homotopy that may be related to this. Goubault and Mimram have been looking at generalised metric spaces and their relation with directed homotopy and its applications in concurrency. I have seen references to a paper ‘in preparation’.

   Another point might be to think of what such a cohomology theory should classify and then try to finding a classification of those objects may lead one back to a formulation of cohomology. I did something like that in discussions with Philippe Malbos and Yves Guiraud in an attempt to classify group objects in the category of categories over a given (fixed) category. Understanding the objects led me back to the cohomology of categories which was in Charles Wells unpublished note (eventually published in TAC I think.) This way the factorisation category came out quite naturally together with a non-Abelian version of Baues-Wirsching cohomology already in Wells’ note.

5. • CommentRowNumber5.
   • CommentAuthorJohn Baez
   • CommentTimeNov 20th 2017
   • (edited Nov 20th 2017)
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   Author: John Baez
   Format: MarkdownItexMike wrote: > Enriched topos theory is actually a deep and (at least, to me) unclear subject. So much of topos theory seems to depend on cartesianness. I’ve occasionally thought about what an enriched topos might be, but never really come up with anything really satisfactory. If the enrichment isn’t cartesian monoidal, What if you've got a category $C$ enriched over $V$ where $V$ _is_ cartesian, perhaps also other nice properties? Does anyone know what it means to say `$C$ is a $V$-enriched topos' in _this_ case? For example suppose $V$ is a topos: do we know what a `$V$-enriched topos' is, or should be?

   Mike wrote:

   Enriched topos theory is actually a deep and (at least, to me) unclear subject. So much of topos theory seems to depend on cartesianness. I’ve occasionally thought about what an enriched topos might be, but never really come up with anything really satisfactory. If the enrichment isn’t cartesian monoidal,

   What if you’ve got a category $C$ enriched over $V$ where $V$ is cartesian, perhaps also other nice properties? Does anyone know what it means to say ‘$C$ is a $V$-enriched topos’ in this case?

   For example suppose $V$ is a topos: do we know what a ‘$V$-enriched topos’ is, or should be?

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeNov 21st 2017
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   Author: Mike Shulman
   Format: MarkdownItexWell, it depends a bit on what you mean by "topos". If you mean *Grothendieck* topos, then there is a clear candidate for a notion of $V$-enriched topos (which even makes sense if $V$ is not cartesian): a reflective sub-$V$-category of a presheaf $V$-category with left exact reflector. Such a category will inherit all the "exactness properties" possessed by $V$ that relate to finite limits and arbitrary colimits, and one could then start asking what information on a small $V$-category $A$ induces such a reflective subcategory of the presheaf $V$-category $[A^{op},V]$ to obtain a notion of $V$-site. If you mean *elementary* topos then I think the situation is rather murkier even when $V$ is cartesian. For instance, a subobject classifier should have some kind of universal property relating some meaning of "$Sub(X)$" to $Hom(X,\Omega)$... but the former is a $V$-category (a subcategory of the slice over $X$) while the latter is an object of $V$. It just so happens when $V=Set$ that the core of a posetal $V$-category can be identified with an object of $V$ (and when $V=\infty Gpd$ that the core of *any* $V$-category can be identified with an object of $V$, which is what leads us to *object* classifiers and $\infty$-topoi). But for a general $V$, even a cartesian one, I don't know what to say here. Which is to say, I'm not sure I would express the problem quite the same way I did 8 years ago.

   Well, it depends a bit on what you mean by “topos”. If you mean Grothendieck topos, then there is a clear candidate for a notion of $V$-enriched topos (which even makes sense if $V$ is not cartesian): a reflective sub-$V$-category of a presheaf $V$-category with left exact reflector. Such a category will inherit all the “exactness properties” possessed by $V$ that relate to finite limits and arbitrary colimits, and one could then start asking what information on a small $V$-category $A$ induces such a reflective subcategory of the presheaf $V$-category $[A^{op},V]$ to obtain a notion of $V$-site.

   If you mean elementary topos then I think the situation is rather murkier even when $V$ is cartesian. For instance, a subobject classifier should have some kind of universal property relating some meaning of “$Sub(X)$” to $Hom(X,\Omega)$… but the former is a $V$-category (a subcategory of the slice over $X$) while the latter is an object of $V$. It just so happens when $V=Set$ that the core of a posetal $V$-category can be identified with an object of $V$ (and when $V=\infty Gpd$ that the core of any $V$-category can be identified with an object of $V$, which is what leads us to object classifiers and $\infty$-topoi). But for a general $V$, even a cartesian one, I don’t know what to say here.

   Which is to say, I’m not sure I would express the problem quite the same way I did 8 years ago.

7. • CommentRowNumber7.
   • CommentAuthorDavid_Corfield
   • CommentTimeAug 1st 2019
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   Author: David_Corfield
   Format: MarkdownItexCould we run through the thinking of #6 in the case of Cho's [Categorical semantics of metric spaces and continuous logic](https://arxiv.org/abs/1901.09077)? He's dealing with some category of pseudometric spaces. (Symmetric ones? That would make sense in view of the 'truth value: real interval = set: symmetric metric space' analogy. Then $Met$ ought to display topos-like characteristics.) So he has $[0, 1]$ as his continuous subobject classifier for pseudometric spaces, based on some correspondence between maps $(X, [0,1])$ and 'continuous' subobjects of $X$. The concern from #6 then is that the former is a metric space and the latter is a posetal $Met$-category, and we can't now use the trick of identifying the core of the latter as a metric space? But if forming the core is something like taking the symmetric part (Remark 1.2 of [[core]]), can't we do some similar symmetrising?

   Could we run through the thinking of #6 in the case of Cho’s Categorical semantics of metric spaces and continuous logic? He’s dealing with some category of pseudometric spaces. (Symmetric ones? That would make sense in view of the ’truth value: real interval = set: symmetric metric space’ analogy. Then $Met$ ought to display topos-like characteristics.)

   So he has $[0, 1]$ as his continuous subobject classifier for pseudometric spaces, based on some correspondence between maps $(X, [0,1])$ and ’continuous’ subobjects of $X$. The concern from #6 then is that the former is a metric space and the latter is a posetal $Met$-category, and we can’t now use the trick of identifying the core of the latter as a metric space?

   But if forming the core is something like taking the symmetric part (Remark 1.2 of core), can’t we do some similar symmetrising?