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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeNov 21st 2022
   • (edited Nov 21st 2022)
   • PermaLink
   Author: Urs
   Format: MarkdownItexstarting a dedicated entry for the category of vector bundles with homomorphisms allowed to cover non-trivial base maps (while previously we only had [[VectBund(B)]] for fixed base $B$). For the moment the main point is to record the interesting cartesian- and tensor-monoidal structure (now [here](https://ncatlab.org/nlab/show/VectBund#ClosedMonoidalStructures)) <a href="https://ncatlab.org/nlab/revision/VectBund/1">v1</a>, <a href="https://ncatlab.org/nlab/show/VectBund">current</a>

   starting a dedicated entry for the category of vector bundles with homomorphisms allowed to cover non-trivial base maps (while previously we only had VectBund(B) for fixed base $B$).

   For the moment the main point is to record the interesting cartesian- and tensor-monoidal structure (now here)

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 5th 2023
   • (edited Apr 5th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded ([here](https://ncatlab.org/nlab/show/VectBund#DistributiveMonoidalStructure)) a section "Properties --- Distributive monoidal structure" which spells out the elementary argument that $(Vect_{Set}, \sqcup, \boxtimes)$ is distributive monoidal (a verbatim copy of the same few paragraphs which I just added at *[[distributive monoidal category]]*, announed [there](https://nforum.ncatlab.org/discussion/5102/distributive-monoidal-category/?Focus=108326#Comment_108326)) also added ([here](https://ncatlab.org/nlab/show/VectBund#AmalgamationOfMonoidalAndParameterStructures)) a further subsection "Properties --- Amalgamation of monoidal and parameter structures" which is meant to be experimental for the moment (I left a disclaimer "under construction"). The point of this last subsection I discuss in another thread: [here](https://nforum.ncatlab.org/discussion/16295/freedman-hastings-question/?Focus=108328#Comment_108328). <a href="https://ncatlab.org/nlab/revision/diff/VectBund/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/VectBund/3">v3</a>, <a href="https://ncatlab.org/nlab/show/VectBund">current</a>

   added (here) a section “Properties — Distributive monoidal structure” which spells out the elementary argument that $(Vect_{Set}, \sqcup, \boxtimes)$ is distributive monoidal (a verbatim copy of the same few paragraphs which I just added at distributive monoidal category, announed there)

   also added (here) a further subsection “Properties — Amalgamation of monoidal and parameter structures” which is meant to be experimental for the moment (I left a disclaimer “under construction”).

   The point of this last subsection I discuss in another thread: here.

   diff, v3, current

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 17th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded a line ([here](https://ncatlab.org/nlab/show/VectBund#BireflectiveInclusionOfZeroBundles)) on the bireflective inclusion of zero-bundles <a href="https://ncatlab.org/nlab/revision/diff/VectBund/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/VectBund/7">v7</a>, <a href="https://ncatlab.org/nlab/show/VectBund">current</a>

   added a line (here) on the bireflective inclusion of zero-bundles

   diff, v7, current

4. • CommentRowNumber4.
   • CommentAuthorperezl.alonso
   • CommentTimeDec 3rd 2024
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexWhen does this category have equalizers preserved by the tensor product? I imagine that's when the category of base spaces satisfies the same condition?

   When does this category have equalizers preserved by the tensor product? I imagine that’s when the category of base spaces satisfies the same condition?

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeDec 3rd 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexThat should be right. Abstractly one can appeal to the description of (co)limits in a Grothendieck construction given [there](https://ncatlab.org/nlab/show/Grothendieck%20construction#CoLimitsInAGrothendieckConstruction): One first forms the (co)limit in the base category, then pushes/pulls the remaining diagram of fibers to a diagram all over the base (co)cone tip and forms the remaining (co)limit of fibers there.

   That should be right. Abstractly one can appeal to the description of (co)limits in a Grothendieck construction given there: One first forms the (co)limit in the base category, then pushes/pulls the remaining diagram of fibers to a diagram all over the base (co)cone tip and forms the remaining (co)limit of fibers there.

6. • CommentRowNumber6.
   • CommentAuthorperezl.alonso
   • CommentTimeDec 3rd 2024
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexI see, so that means that if I have a principal $G$-bundle on the category on smooth manifolds expressed as an equalizer $G\times P \rightrightarrows P\to M$ this will allow me to construct principal $H$-bundles on $VectBun$ as an equalizer $H\otimes V \rightrightarrows V\to V'$ for $H$ a Hopf algebra object in $VectBun$, a vector bundle over $G$, and $V,V'$ coalgebra vector bundles over $P,M$, resp., right?

   I see, so that means that if I have a principal $G$-bundle on the category on smooth manifolds expressed as an equalizer $G\times P \rightrightarrows P\to M$ this will allow me to construct principal $H$-bundles on $VectBun$ as an equalizer $H\otimes V \rightrightarrows V\to V'$ for $H$ a Hopf algebra object in $VectBun$, a vector bundle over $G$, and $V,V'$ coalgebra vector bundles over $P,M$, resp., right?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeDec 3rd 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot sure if I am following this description. Incidentally, the first equalizer you mention is a coequalizer -- which is good if you are still interested in it being preserved by external tensor -- but I don't see where this enters in what you write now. Do you really mean to have a Hopf algebra object over a base object that is a group object -- this seems to lead to a clash of variances. So I don't quite see what you are after here.

   Not sure if I am following this description.

   Incidentally, the first equalizer you mention is a coequalizer – which is good if you are still interested in it being preserved by external tensor – but I don’t see where this enters in what you write now.

   Do you really mean to have a Hopf algebra object over a base object that is a group object – this seems to lead to a clash of variances.

   So I don’t quite see what you are after here.

8. • CommentRowNumber8.
   • CommentAuthorperezl.alonso
   • CommentTimeDec 3rd 2024
   • PermaLink
   Author: perezl.alonso
   Format: MarkdownItexRight. I'm trying to construct principal bundles in $VectBun$ along the lines of [Brzeziński 09](https://arxiv.org/abs/0912.0213). One regards the groups as Hopf objects given that every set is a comonoid.

   Right. I’m trying to construct principal bundles in $VectBun$ along the lines of Brzeziński 09. One regards the groups as Hopf objects given that every set is a comonoid.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeDec 3rd 2024
   • PermaLink
   Author: Urs
   Format: MarkdownItexI see. But do you want $H$ to co-act on $V$ or to act? In the first case the relation to the underlying $G$-action is lost, im the second case group structure is missing. In any, I may not have the leisure right now to sort through this.

   I see. But do you want $H$ to co-act on $V$ or to act? In the first case the relation to the underlying $G$-action is lost, im the second case group structure is missing.

   In any, I may not have the leisure right now to sort through this.

10. • CommentRowNumber10.
    • CommentAuthorperezl.alonso
    • CommentTimeDec 3rd 2024
    • PermaLink
    Author: perezl.alonso
    Format: MarkdownItexI'd want it to act, this would be the dual of the algebraic description of quantum group bundles expressed in terms of a Hopf coaction $B^{co H}\to B \rightrightarrows B\otimes H$. And no worries, this is pretty much early stage by now, but I think #5 is along the lines of what I would need, thanks!

    I’d want it to act, this would be the dual of the algebraic description of quantum group bundles expressed in terms of a Hopf coaction $B^{co H}\to B \rightrightarrows B\otimes H$.

    And no worries, this is pretty much early stage by now, but I think #5 is along the lines of what I would need, thanks!