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Let $\cal K$ be a 2-category, and $A$ an object therein. Is there a way to define the "slice object" $A/a$ in $\cal K$ formally (in such a way that if $K = Cat$ we recover precisely the notion of the slice category $A/a$)?

I suggest that if $\cal K$ has comma objects then $A/a = (1_A\downarrow a)$ where $a\colon 1\to A$ is an "element" of $A$ and $1_A$ is the identity.

Is this construction meaningful? If yes, there is a diagram $a\mapsto A/a$ from ${\cal K}(1,A)$ to $\cal K$. What's its colimit (supposing each hom-category is small and $\cal K$ admits this colimit)?

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Yes, that's the right way to define "slice objects" in a 2-category. But there's no way to say what the colimit of that diagram is for a general $K$ and $A$. For instance, $A$ might not admit any "global elements" $1\to A$, so that the colimit would be the initial object. Or, if $K$ is a 1-category regarded as a 2-category, then "$A/a$" is the terminal object for all $a$, so your colimit is the coproduct of one copy of the terminal object for each global element of $A$. You might get something more interesting by looking at generalized elements $X\to A$, or at the universal generalized element $1_A : A\to A$, but you'll probably still need some "exactness" conditions on $K$ to be able to say anything.

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  • $\begingroup$ This is what I expected, indeed. I just want to have an intuition for this construction, so it's not a problem to stick to more well-behaved $K$'s. So, what are some sufficient conditions on $K$ ensuring that the construction is non-trivial and interesting? $\endgroup$ – fosco Dec 14 '16 at 23:56
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    $\begingroup$ @FoscoLoregian I kind of feel that that's like asking for sufficient conditions on a 1-category that pullbacks (say) are nontrivial and interesting. Comma objects are a basic notion of 2-category theory; they're probably going to be interesting in any interesting and non-trivial 2-category. If the question is about when objects admit "enough" global elements, then I can't think of much to say except "sometimes"; that by itself is a condition on a 2-category, analogous to well-pointedness" for 1-categories. $\endgroup$ – Mike Shulman Dec 15 '16 at 11:30
  • $\begingroup$ Ok, I like the analogy $\endgroup$ – fosco Dec 15 '16 at 11:54

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