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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeMay 19th 2022
   • PermaLink
   Author: Urs
   Format: MarkdownItexI just discovered that, all along, the term "quiver representation" was just redirecting to *[[representation]]*. Have started this dedicated page now, with the bare minimum <a href="https://ncatlab.org/nlab/revision/quiver+representation/1">v1</a>, <a href="https://ncatlab.org/nlab/show/quiver+representation">current</a>

   I just discovered that, all along, the term “quiver representation” was just redirecting to representation. Have started this dedicated page now, with the bare minimum

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorAli Caglayan
   • CommentTimeApr 25th 2024
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   Author: Ali Caglayan
   Format: MarkdownItexIf a quiver representation is a functor from $FrCat(Q) \to \mathrm{Vect}$ then doesn't this mean it is equivalently a graph map $Q \to \mathrm{Vect}$? I.e. you associate to each node in the quiver a vector space and to each edge a linear map. I couldn't workout what a morphism of quiver representations would be from this point of view however.

   If a quiver representation is a functor from $FrCat(Q) \to \mathrm{Vect}$ then doesn’t this mean it is equivalently a graph map $Q \to \mathrm{Vect}$? I.e. you associate to each node in the quiver a vector space and to each edge a linear map. I couldn’t workout what a morphism of quiver representations would be from this point of view however.

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeApr 25th 2024
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   Author: Urs
   Format: MarkdownItexYes, that's what it means for $FrCat(-)$ to be the free category on a graph. The general mechanism is explained at *[[free functor]]*.

   Yes, that’s what it means for $FrCat(-)$ to be the free category on a graph. The general mechanism is explained at free functor.

4. • CommentRowNumber4.
   • CommentAuthorAli Caglayan
   • CommentTimeApr 25th 2024
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   Author: Ali Caglayan
   Format: MarkdownItexRight. So why do representation theorists present quiver representations using the free category if this point of view is simpler? The mechanism explained in the "free functor" article explains the 1-categorical case well, but we are working in a 2-category Cat here. We would need a biadjunction (really a strict 2-adjunction) to transfer morphisms across the view point. It's not immediately obvious to me that the free category is biadjoint to the forgetful functor. I don't think the 1-category of quivers has enough info to get morphisms of quiver representations. I would assume there is some 2-category of quivers to do that, but I have no idea what the 2-cells ought to be.

   Right. So why do representation theorists present quiver representations using the free category if this point of view is simpler?

   The mechanism explained in the “free functor” article explains the 1-categorical case well, but we are working in a 2-category Cat here. We would need a biadjunction (really a strict 2-adjunction) to transfer morphisms across the view point. It’s not immediately obvious to me that the free category is biadjoint to the forgetful functor.

   I don’t think the 1-category of quivers has enough info to get morphisms of quiver representations. I would assume there is some 2-category of quivers to do that, but I have no idea what the 2-cells ought to be.

5. • CommentRowNumber5.
   • CommentAuthorAli Caglayan
   • CommentTimeApr 25th 2024
   • (edited Apr 25th 2024)
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   Author: Ali Caglayan
   Format: MarkdownItexAnd just in case it wasn't clear, my point is that there can be multiple morphisms of quiver representations since we are asking for natural transformations not equivalences, but not multiple 2-cells of graph maps. It seems a better 2-category (probably bicategory) of graphs is needed to state this POV correctly.

   And just in case it wasn’t clear, my point is that there can be multiple morphisms of quiver representations since we are asking for natural transformations not equivalences, but not multiple 2-cells of graph maps. It seems a better 2-category (probably bicategory) of graphs is needed to state this POV correctly.

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeApr 26th 2024
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   Author: Urs
   Format: MarkdownItexNo, it's just the 1-category of categories and functors between them that is used. There is nothing subtle going on here.

   No, it’s just the 1-category of categories and functors between them that is used. There is nothing subtle going on here.