I've read in a few different places that the standard fact $\text{Nat}\,(F,G) \cong \int_x \text{Hom}\,(Fx,Gx)$ can be upgraded to $\textbf{LaxNat}\,(F,G) \cong \oint_x\textbf{Hom}\,(Fx,Gx)$. where the left hand side is the category of lax natural transformations and modifications, and the right hand side is a lax end.

I am looking for a reference that gives the definition of lax end and proves this equivalence. I do know of the reference, S. Bozapalides, Théorie formelle des bicatégories, but I can't read French and I also can't find a copy. If someone can link me to the Bozapalides reference would be great. Or even better would be if there is a reference in English. Thanks!

  • $\begingroup$ I've added the ct.category-theory tag. $\endgroup$ Jun 7 '11 at 20:39
  • $\begingroup$ I guess there's an obvious candidate for what a "lax wedge" $c\Rightarrow F$ should be. Maybe the rest is straightforward too... $\endgroup$ Jun 8 '11 at 0:00
  • $\begingroup$ Yes as long as one sticks to strict 2-categories and strict functors, the details in proving the lax transformation identity are not too terrible using the obvious definition of lax wedge/lax end. I'll take that as evidence that obvious is right in this case. Still, a reference would be nice. $\endgroup$ Jun 8 '11 at 1:49
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    $\begingroup$ BOZAPALIDES, S., Les fins cartésiennes, C. R. A. S. Paris 281 ( 1975) and dml.cz/bitstream/handle/10338.dmlcz/106961/… $\endgroup$ May 16 '12 at 18:57
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    $\begingroup$ Bozapalides 'Some remarks on lax-presheafs' projecteuclid.org/euclid.ijm/1256047482 $\endgroup$
    – Ma Ming
    Apr 4 '14 at 23:06

This page says that you may be able to get a copy by emailing Andrée Ehresmann.

I don't know the exact answer to your question, but if you can't find a reference then it may be worth recalling that:

  • For Cat-valued F and G, $\mathrm{Nat}(F,G) \simeq \{F,G\}$, the limit of G weighted by F,
  • ends are $\hom$-weighted limits, and
  • there are lax morphism classifiers for 2-functors, meaning that $\mathrm{Lax}(F,G) \simeq \mathrm{Nat}(QF,G)$ for another 2-functor $QF$.

So if you define the lax end $\oint_x T(x,x)$ to be the representative of $\mathrm{Lax}(\hom_K, L(1,T))$, then you get $\oint_x [F x, G x] \simeq \mathrm{Lax}(\hom_K, [F-, G-])$, which is not quite what you want, but it's close.

Hope that helps.

  • $\begingroup$ Thanks, Finn. I'm still holding out hope for a reference in English, but all of the above is helpful. $\endgroup$ Jun 6 '11 at 22:55

A reference is Section 7.1 of Fosco Loregian's very nice book on coends, which treats co/lax co/ends. In particular, see Example 7.1.9 for a proof of the formula $$\mathrm{Nat}_\mathrm{lax}(F,G)=\int_{A\in\mathcal{C}}\mkern-2.05em\square\mkern+1.0em\mathsf{Hom}_{\mathcal{D}}(F(A),G(A)).$$ Another reference for bicategorical coends is Chapter 2 of Alexander Corner's thesis (PDF) or its arXived version, arXiv:1709.01332 [math.CT].

  • $\begingroup$ P.S. Bozapalides's PhD thesis is available on Libgen. $\endgroup$
    – Théo
    May 24 '20 at 6:33

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