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I'm not sure if this is the right place to ask this question, but I'll ask it anyway, in the hope that some kindly Australian (true or honorary) is passing by and takes pity on me...

In Fibrations in bicategories, Street shows that V-profunctors are exactly the codiscrete cofibrations in the 2-category V-Cat (i.e. the discrete 2-sided fibrations in the opposite of V-Cat). Rosebrugh and Wood later generalized this to well-behaved proarrow equipments.

When V=Set, so that V-Cat is Cat, then, codiscrete cofibrations turn out to be essentially the same thing as discrete fibrations. My question is

Why is this true? That is, for which bicategories K is DFib(B,A) equivalent to CodCofib(B,A) for all objects A and B?

I ask because (aside from curiosity) I'd like to know whether I can expect a 'biprofunctor' $L^{\mathrm{op}} \times K \to \mathrm{Cat}$ to be the same as a discrete fibration in Bicat, or even whether this is true in the strictly Cat-enriched case. In my specific case L and K are locally discrete, if that makes a difference.

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A few thoughts related to defining fibrations of 2-categories internally to Bicat can be found here: – Mike Shulman May 13 '11 at 13:23
Very interesting ideas, but for now my brain starts to hurt somewhere between n=2 and n=3... Luckily I think that in the case I'm considering here, because the base bicategories are locally discrete, I can get away with fibrations in Cat. – Finn Lawler May 13 '11 at 19:22
up vote 2 down vote accepted

The fact that codiscrete cofibrations and discrete fibrations are equivalent is a very special exactness property of the 2-category Cat. I can't, off the top of my head, think of any other interesting examples. It's not true, for example in the 2-category 2-Cat of 2-categories, 2-functors, and 2-natural transformations you refer to, or in most (all?) other 2-categories of the form V-Cat. Consider, for example, the case where V=Cat and A=B=1. Then CodCofib(1,1) is just (the underlying ordinary category of) Cat. On the other hand, Fib(1,1) is 2-Cat, but discreteness of a fibration says that the 2-category A has an underlying ordinary category which is discrete.

If you want to move from 2-Cat to Bicat, you'd first have to decide how you want to make Bicat into a 2-category (or decide what you mean by internal discrete fibrations or codiscrete cofibrations in a tricategory). But I wouldn't hold out too much hope ....

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OK, thanks. So the correspondence in the case of Cat really is a red herring. When you say 'the 2-category A has an underlying ordinary category which is discrete', do you mean by A the total 2-category of a fibration from 1 to 1? Also, (I know it's unlikely but) are there, to your knowledge, any references that compare the two notions in detail? – Finn Lawler May 11 '11 at 17:51
First question: yes (sorry, that was unclear). Second question: no (but if I think of one I'll let you know). – Steve Lack May 11 '11 at 23:35
What about 2-categories of Cat-valued (2-)presheaves? Since all these notions can be defined in terms of limits and colimits, I would expect them to carry over pointwise. And if that works, then what about Grothendieck 2-topoi (left exact reflective sub-2/bi-categories of Cat-valued presheaf categories)? – Mike Shulman May 13 '11 at 13:20
Yes, Mike, quite right - the equivalence of discrete fibrations and codiscrete cofibrations is an exactness condition which will carry over to Cat-valued presheaves. And I'd guess Grothendieck 2-toposes should work as well, although I haven't checked the details. – Steve Lack May 13 '11 at 23:42

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