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Let A be a bicategory. Can I construct in some "natural" way a bicategory L(A) such that for every bicategory B, the bicategory of lax functors Lax(A, B) is (bi)equivalent to the bicategory of pseudofunctors Pseudo(L(A), B)? (Choose the morphisms in these functor categories as you wish. Ideally the same construction L would work for any choice.)

For example, when A = •, I believe we may take L(A) = BΔ+, the delooping of the augmented simplex category with monoidal structure given by ordinal sum. In general I imagine L(A) as being formed as something like the free category on the objects, 1-morphisms and 2-morphisms of A--denote the generator corresponding to a 1-morphism f of A by [f]--as well as 2-morphisms id → [id] and [f] o [g] → [f o g] for every composable f and g in A.

(By Chris's question here, L(A) cannot be an equivalence invariant of A.)

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Reid, I think there's a typo. You should have "Pseudo(L(A), B)" in the second sentence. – Tom Leinster Dec 4 '09 at 12:44
Yup, thanks. Fixed. – Reid Barton Dec 4 '09 at 15:47

Yes, there is. A relevant general framework is the following: for any 2-monad T, we can define notions of pseudo and lax morphism between T-algebras, and there is a forgetful functor from the 2-category of T-algebras and pseudo morphisms to the 2-category of T-algebras and lax morphisms. If T is well-behaved, this forgetful functor has a left adjoint; see for instance this paper.

There is a 2-monad on the 2-category of Cat-graphs whose algebras are bicategories, whose pseudo morphisms are pseudofunctors, and whose lax morphisms are lax functors. Therefore, the above applies to bicategories. If you trace through the construction, you'll see that it is given essentially by the recipe you proposed. (This case of the construction can probably be found elsewhere in the literature as well, in more explicit form, but this is the way I prefer to think about it.)

The caveat is that the 2-cells in the 2-categories defined above are not any of the the usual sort of transformations between bicategories, only the icons. (This is what allows you to have a 2-category containing lax functors.) However, the usual sorts of transformations are "corepresentable," that is, for any bicategory D there is a bicategory Cyl(D) such that pseudo or lax functors into Cyl(D) are the same as pairs of pseudo or lax functors into D and a pseudo (or lax, with a different definition of Cyl) natural transformation between them, and likewise we have 2Cyl(D) for modifications. I believe one can use this to show that in this case, the construction coming from 2-monad theory does have the property you want.

Of course, by Chris' question, it seems that this version of L cannot itself be described as a left adjoint, since there is no 2- or 3-category containing lax functors and arbitrary pseudo/lax transformations.

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For 2-categories there is a proof on "FOrmal CAtegory Theory I" of J. W. Gray LNM 391 (see I,4.23 pag. 92) . There is cited that J. Benabou did a general proof for bicategories.

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