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Hi,

I would like to know whether there is a standard name for the following "slice" category: Let $\mathcal{C}$ be a 2-category and $c \in \mathcal{C}$ an object of $\mathcal{C}$. We can form the category where an object $(d,f)$ is a pair of an object $d\in\mathcal{C}$ and an arrow $f : d\to c$. A morphism $(h, \alpha)$ from $(d,f)$ to $(e,g)$ is given by $h : d \to e$ and a 2-cell $\alpha : f \Rightarrow g \circ h$.

Thanks a lot, ben

[Edit: fixed typo mentioned by Martin]

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  • $\begingroup$ I expect that you mean $h : d \to e$. Well I think that this is just the correct notion of slice category in the context of $2$-categories. I would call it (and actually have called it in one of my texts) slice category and denote it by $C / c$. $\endgroup$
    – Martin Brandenburg
    Commented Jun 24, 2011 at 12:47

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See this answer to much the same question. I would call this the 'lax' slice category, although it's not so common a notion that everyone would know what you meant, so maybe you should keep the scare quotes around 'lax'.

A propos of Martin's comment, the correct notion of slice 2-category depends on what you're doing -- you might want the strict version, with strictly commuting triangles, or the pseudo version, with invertible 2-cells (this is the strictest one that makes sense for non-strict 2-categories), or this lax version. Or you might want to restrict to (discrete) (op)fibrations as objects.

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  • $\begingroup$ You're right, there is some ambiguity. But instead of introducing an extra notion for the case where everything invertible, I would just use $(C/c)_{cart}$ with the above definition of $C/c$. $\endgroup$
    – Martin Brandenburg
    Commented Jun 24, 2011 at 15:56
  • $\begingroup$ I don't think there's any real ambiguity -- what I meant is that there's more than one sensible notion of 'slice 2-category', and which one is correct depends on what you want it for. But yes, they're all contained in the lax slice, so you could take that as fundamental. $\endgroup$
    – Finn Lawler
    Commented Jun 24, 2011 at 18:37
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    $\begingroup$ I feel fairly strongly that the unqualified term "slice 2-category" should refer to the pseudo version. Lax objects are generally less well-behaved and less fundamental than pseudo ones, and while they have specialized uses, the central role in the theory is usually played by the pseudo version. $\endgroup$
    – Mike Shulman
    Commented Jun 24, 2011 at 21:32

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