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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeNov 29th 2019
   • PermaLink
   Author: Dmitri Pavlov
   Format: MarkdownItexAdded redirects. <a href="https://ncatlab.org/nlab/revision/diff/arrow+category/21">diff</a>, <a href="https://ncatlab.org/nlab/revision/arrow+category/21">v21</a>, <a href="https://ncatlab.org/nlab/show/arrow+category">current</a>

   Added redirects.

   diff, v21, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeApr 1st 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexadded remark ([here](https://ncatlab.org/nlab/show/arrow+category#SliceCategoriesAndArrowCategory)) that the arrow category is the Grothendieck construction on the slice categories (copied over just with mild adjustments of notation from the same example that I just added to *[[Grothendieck construction]]*) <a href="https://ncatlab.org/nlab/revision/diff/arrow+category/23">diff</a>, <a href="https://ncatlab.org/nlab/revision/arrow+category/23">v23</a>, <a href="https://ncatlab.org/nlab/show/arrow+category">current</a>

   added remark (here) that the arrow category is the Grothendieck construction on the slice categories (copied over just with mild adjustments of notation from the same example that I just added to Grothendieck construction)

   diff, v23, current

3. • CommentRowNumber3.
   • CommentAuthorperezl.alonso
   • CommentTimeMar 7th 2025
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   Author: perezl.alonso
   Format: MarkdownItexIs there a specific use or characterization of the arrow category of a category of representations $Rep(A)$ of some algebra $A$? I _think_ I've seen at some point someone discuss arrow categories of representations of quivers but perhaps the case of algebra reps is more familiar?

   Is there a specific use or characterization of the arrow category of a category of representations of some algebra ? I think I’ve seen at some point someone discuss arrow categories of representations of quivers but perhaps the case of algebra reps is more familiar?

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2025
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   Author: Urs
   Format: MarkdownItexThere is a somewhat tautological "meaning" of the arrow category of a category of representations: It's the category of all "intertwiners"/"natural transformations" without restriction on the (co)domains being intertwined. Maybe it helps to unwind a little: Let $\mathcal{C}$ be a small category, which may be a [[delooping groupoid]] if we are talking about group representations, or the [[free category]] of a graph if we are talking about quiver representations, etc. Then for some coefficient category $\mathcal{D}$ (like vector spaces), the representations of $\mathcal{C}$ in $\mathcal{D}$ are the functors $\mathcal{C} \to \mathcal{D}$, and the representation category is the functor category $Rep(\mathcal{C};\mathcal{D}) \equiv Func(\mathcal{C}, \mathcal{D})$. Now an arrow category, of course, is a functor category out of an [[interval category]] $I$, and since $Cat$ [is cartesian closed](Cat#CartesianClosedStructure) we have $$ Arr\big( Rep(\mathcal{C}; \mathcal{D}) \big) \;\equiv\; Arr\big( Func(\mathcal{C}, \mathcal{D}) \big) \;\equiv\; Func\Big( I,\,\big( Func(\mathcal{C}, \mathcal{D}) \big) \Big) \;\simeq\; Func(I \times \mathcal{C}, \mathcal{D}) \;\equiv\; Rep(I \times \mathcal{C}; \mathcal{D}) \,. $$ On the right $I \times \mathcal{C}$ is the "directed cylinder diagram" over $\mathcal{C}$ (two copies of $\mathcal{C}$, all directed pairs of copies of objects connected by a morphism, such that all resulting squares commute). Images of $I \times \mathcal{C}$ in $\mathcal{D}$ are pairs of functors $\mathcal{C} \to \mathcal{D}$ with components of a natural transformation between them. In particular, if $\mathcal{C}$ is a free category on a graph, then so is $I \times \mathcal{C}$.

   There is a somewhat tautological “meaning” of the arrow category of a category of representations: It’s the category of all “intertwiners”/”natural transformations” without restriction on the (co)domains being intertwined.

   Maybe it helps to unwind a little:

   Let be a small category, which may be a delooping groupoid if we are talking about group representations, or the free category of a graph if we are talking about quiver representations, etc.

   Then for some coefficient category (like vector spaces), the representations of in are the functors , and the representation category is the functor category .

   Now an arrow category, of course, is a functor category out of an interval category , and since is cartesian closed we have

   On the right is the “directed cylinder diagram” over (two copies of , all directed pairs of copies of objects connected by a morphism, such that all resulting squares commute). Images of in are pairs of functors with components of a natural transformation between them.

   In particular, if is a free category on a graph, then so is .

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2025
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   Author: Urs
   Format: MarkdownItexP. S. The same argument applies to representations of algebras, if we regard these are $Vect$-enriched categories and speak about $Vect$-[[enriched functor categories]], etc.

   P. S. The same argument applies to representations of algebras, if we regard these are -enriched categories and speak about -enriched functor categories, etc.

6. • CommentRowNumber6.
   • CommentAuthorperezl.alonso
   • CommentTimeMar 8th 2025
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   Author: perezl.alonso
   Format: MarkdownItexHmm, so in the case of algebras is it something more familiar like $Mod(End(A))$ (though with which algebra structure?)? For example, if $\mathcal{C}= Mod(A)$ for $A$ an algebra, then the monoidal category of right exact endofunctors of $Mod(A)$ is equivalent to $Bimod(A)$. Do you see a somewhat more familiar characterization of these intertwiners?

   Hmm, so in the case of algebras is it something more familiar like (though with which algebra structure?)? For example, if for an algebra, then the monoidal category of right exact endofunctors of is equivalent to . Do you see a somewhat more familiar characterization of these intertwiners?

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2025
   • (edited Mar 8th 2025)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNot sure if I understand this question now. Let me highlight that in the language I used above, we have $$ Mod(A) \simeq Func(\mathbf{B}A, Vect) \,, $$ with $\mathbf{B}A$ the one-object $Vect$-enriched category with endomorphism of that one object being $A$, and with the $Vect$-enriched functor category on the right.

   Not sure if I understand this question now.

   Let me highlight that in the language I used above, we have

   with the one-object -enriched category with endomorphism of that one object being , and with the -enriched functor category on the right.

8. • CommentRowNumber8.
   • CommentAuthorperezl.alonso
   • CommentTimeMar 8th 2025
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   Author: perezl.alonso
   Format: MarkdownItexRIght, and then the arrow category is the representations of the algebroid $( \mathbf{B} A \to \mathbf{B} A)$, but does that algebroid have a name? For example, if I take a groupoid $\mathcal{G}$ and do $[\mathcal{G}, \text{Vect}]$, these are the representations of a weak Hopf algebra, the groupoid algebra $k[\mathcal{G}]$.

   RIght, and then the arrow category is the representations of the algebroid , but does that algebroid have a name? For example, if I take a groupoid and do , these are the representations of a weak Hopf algebra, the groupoid algebra .

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2025
   • PermaLink
   Author: Urs
   Format: MarkdownItexI would call this the "directed [[cylinder object|cylinder]]" over $\mathbf{B}A$. Don't know if in other areas this goes by another name.

   I would call this the “directed cylinder” over . Don’t know if in other areas this goes by another name.