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$\require{AMScd}$Are there references for a construction of the enriched slice category of $\mathcal A \in \mathcal{V}\text{-Cat}$? A reasonable definition should be

  1. Fix an object $a\in\mathcal A$ and let the objects of ${\cal A}/a$ be the set ${\cal V}(J, {\cal A}(x,a))$ for all $x\in\cal A$. $J$ is the monoidal unit of $\cal V$.

  2. Let ${\cal A}/a(p,q)$ be the $\cal V$-object resulting from the pullback $$ \begin{CD} P @>>> {\cal A}(x,y) \\ @VVV {}@VVq_*V\\ J @>>p> {\cal A}(x, a) \end{CD} $$ for $p : x\to a, q : y \to a$ and $J$ the monoidal unit.

Does it work for every $\cal V$?

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    $\begingroup$ It is not clear what do you mean by "work" - depends on what do you need it for. In my opinion generally this looks somehow strange unless $J$ is terminal. The general attitude I like is this. In the "cartesian" case, every object has a unique cocommutative comonoid structure, and the slice can be viewed as the category of coactions of this comonoid. So for a general $\mathcal V$ it is also natural to view coactions of a comonoid as slices, and they are then only defined for comonoids. Btw this also relates to the problem with $J$ not being terminal: it becomes terminal in comonoids. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 17, 2017 at 17:25
  • $\begingroup$ "Does it work"="do I obtain what I want?" :-) Thanks. So you're saying that for a general $\cal V$ there is only a subclass of arrows $x \to a$ that "do well" for a slice category structure? Or rather, that ${\cal A}/a$ is only defined when $a$ is a comonoid in $\cal A$? $\endgroup$
    – fosco
    Commented Jan 17, 2017 at 17:30
  • $\begingroup$ Mmm no I had neither in mind. I just found it a bit strange that you get enrichment in $\mathcal V/J$, not just in $\mathcal V$. But in fact this might be a feature rather than a bug. E. g. in vector spaces this I believe gives affine spaces. Btw I also wanted to note that the underlying "ordinary" category of your $\mathcal A/a$ is just the ordinary slice over $a$, so it could be better understood if you talk not about enriched slices but about extending enrichment from $\mathcal A$ to (ordinary) slices of it. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 17, 2017 at 18:15
  • $\begingroup$ To be more specific - take $\mathcal A=\mathcal V=$ vector spaces and let $a=0$. Then $\mathcal A/a$ is again vector spaces, but the enrichment is $J\times\mathcal V(x,y)$. $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Jan 17, 2017 at 18:25
  • $\begingroup$ mmmh. Thanks. I'll meditate on this $\endgroup$
    – fosco
    Commented Jan 17, 2017 at 22:36

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There is at least one sense in which this "works". Namely, it is the comma object in the 2-category $\mathcal{V}$-Cat of $\mathrm{Id} : \mathcal{A} \to \mathcal{A}$ over $[a] : \mathcal{J} \to \mathcal{A}$, where $\mathcal{J}$ is the unit $\mathcal{V}$-category with one object and $J$ as its hom-object. It's true that it doesn't capture as much information in the enriched case as it does in the ordinary case, but given that it has this universal property I don't know what else one might mean by "enriched slice category".

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  • $\begingroup$ Thanks, I agree with the last sentence. Has this description of enriched slices been used somewhere for some "practical" purpose? Does the construction in the opening post have the universal prop of the comma object $(\text{id}\downarrow a)$? $\endgroup$
    – fosco
    Commented Jan 18, 2017 at 9:09
  • $\begingroup$ I don't know of any use for this enriched slice. And yes, as I said, your construction is the comma object. $\endgroup$
    – Mike Shulman
    Commented Jan 19, 2017 at 12:45
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    $\begingroup$ My brain must have been turned off when I said I didn't know any use for this enriched slice. For $V=\rm sSet$ (or similarly $\rm Top$) this has plenty of uses in model category theory! And for $V=\rm Cat$ it is the right notion of strict slice 2-category. What I don't know (I think) is any uses for it when $V$ is not at least semicartesian (that is, its monoidal unit is terminal). $\endgroup$
    – Mike Shulman
    Commented Jan 22, 2020 at 5:22

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