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1. • CommentRowNumber1.
   • CommentAuthorDmitri Pavlov
   • CommentTimeNov 29th 2019
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   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 $Rep(A)$ 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?

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 $\mathcal{C}$ 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 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 $I \times \mathcal{C}$.

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 $Vect$-enriched categories and speak about $Vect$-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 $Mod(End(A))$ (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?

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

   $$Mod(A) \simeq Func(\mathbf{B}A, Vect) \,,$$

   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.

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 $( \mathbf{B} A \to \mathbf{B} A)$, 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 $k[\mathcal{G}]$.

9. • CommentRowNumber9.
   • CommentAuthorUrs
   • CommentTimeMar 8th 2025
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   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 $\mathbf{B}A$. Don’t know if in other areas this goes by another name.