## Name of “slice” category with 2-cells as morphisms ?

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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 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 $C / c$. – Martin Brandenburg Jun 24 2011 at 12:47

 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 $C/c$. – Martin Brandenburg Jun 24 2011 at 15:56 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. – Finn Lawler Jun 24 2011 at 18:37 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. – Mike Shulman Jun 24 2011 at 21:32