Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I feel like I should know the answer to this, but I don't think I do.

The Gray tensor product of 2-categories $C$ and $D$ is a "fattening up" of the cartesian product $C\times D$ in which the equality $(f,1)(1,g) = (1,g)(f,1)$ is replaced by a 2-morphism. Nowadays the term "Gray tensor product" more often seems to refer to the pseudo version in which these 2-morphisms are invertible, but Gray's original version was "lax" (or colax) and had them not necessarily invertible.

The Gray tensor product has a universal property

$$ \mathrm{Fun}_{\mathrm{strict}}(C\otimes_w D, E) \cong \mathrm{Fun}_{\mathrm{strict}}(C, \mathrm{Fun}_w(D,E)) $$

where $\mathrm{Fun}_w$ denotes the 2-category of strict 2-functors and $w$-natural transformations. One can also spell out explicitly the morphisms which are represented by $C\otimes_w D$ as a sort of "$w$-cubical functor"; these can be identified with a certain class of $w$-functors $C\times D\to E$ which are strict in certain ways.

In sum, the Gray tensor product is a beautiful thing for talking about strict 2-functors and all sorts of weak natural transformations. My question is, what happens when we move to pseudo 2-functors? I'm happy to keep my 2-categories strict and not to worry about lax or oplax 2-functors. Is there an equivalence of bicategories

$$ \mathrm{PsFun}_{\mathrm{pseudo}}(C\otimes_w D, E) \simeq \mathrm{PsFun}_{\mathrm{pseudo}}(C, \mathrm{PsFun}_w(D,E)) $$

where ${\mathrm{PsFun}_w (-,-)}$ denotes the 2-category of pseudofunctors and $w$-natural transformations? This is true when $w=$ pseudo, since in that case $C\otimes D$ is equivalent to $C\times D$ as a bicategory, and the tricategory of bicategories is cartesian closed with internal-hom $\mathrm{PsFun}_{\mathrm{pseudo}}$. But what about when $w=$ lax?

share|improve this question
    
See JW Gray Formal Category Theory: Adjointness for 2-Categories (Lnm 391) p. 86. I guess that this has a formulation in terms of tricategories, and $lax_{(3)}$-funtors –  Buschi Sergio May 11 '12 at 11:42
    
@Buschi: Gray uses "pseudo-functor" to mean what nowadays is called a "lax functor". As far as I can tell, his discussion of why the tensor product doesn't work for lax functors has no bearing on pseudo-functors. –  Mike Shulman May 11 '12 at 18:28
    
@Mike: Yup, this is basically true and can be proved using the folk model structure and showing that the lax tensor product is left-derivable in its second component. –  Harry Gindi Jun 23 '12 at 9:36
    
@Harry: Sounds like a plausible approach... but you sound very confident, have you worked out the details? –  Mike Shulman Jun 25 '12 at 15:46

1 Answer 1

This wants to be a track, I have not checked the details in their entirety

bibliography:

[G] J.W Gray Formal Category Theory: Adjointness for 2-Categories (Lnm 391).

Consider at first normal pseudo-funtors (on 2-cetegories) we call a pseudo.functor $F: \mathcal{A}\to\mathcal{B}$ normal if for any $A$: $F(1_A)=1_{FA}$ and the canonical isomorphism is the identity. Let $Fun_{np}(\mathcal{A}, \mathcal{B})$ the category of normal pseudofuntors and lax-transformations (with modifications too, is a 2-category). Now a normal pseudfunctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ give a family of normal pseudofunctors

$(F(A, -): \mathcal{B}\to \mathcal{C})_{A\in \mathcal{A}}$

$(F(?, B): \mathcal{A}\to \mathcal{C})_{B\in \mathcal{B}}$

such that $F(A, -)(B)= F(?, B)(A)$ and for $f: A\to A',\ g: B\to B'$ a 2-cell $\gamma_{f, g}$ (the $g$-component of the lax transformation $(F(f): F(A, -)\rightarrow F(A', -) $)

as in the diagram of [G] p. 57, which verify the properties $QF_21, QF_22,\ QF_23$ of of [G] p. 57. We call this "data" a normal quasi-pseudo-funtor. Similarly a 2-cell to induce what is called a lax transformation between normal quasi-pseudo-funtor.

mutually this data describe exactly a normal pseudofuctor $F:\mathcal{A}\to Fun_{np}(\mathcal{B}, \mathcal{C})$ (this is sketched in [G] p. 60 for 2-functors) and this involve also lax-transformation, then we have a (isomorphism):

$Fun_{np}(\mathcal{A}, Fun_{np}(\mathcal{B}, \mathcal{C}))\cong n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})$

where the right member is the category (2-category) of normal quasi-pseudo-funtors and lax transformation (and modifications).

Now I think that exist a natural the isomorphism: $n.q.p-Fun(\mathcal{A}\times \mathcal{B}, \mathcal{C})\cong Fun_{np}(\mathcal{A}\otimes_w \mathcal{B}, \mathcal{C})$.

This follow as in [G] p.73,74,75, 76,77 (using also coherence criterion for pseudofunctor)

EDIT: the part about general pseudofunctors (no normal) , I'm working about...

Now consider general pseudo functors.

Let

[B] Introduction to Bicategories , J. Benabou.

Its enough show a natural equivalence $Fun_{pn}(\mathcal{A},\mathcal{B})\simeq Fun_{p}(\mathcal{A},\mathcal{B})$ where the latter member is the category of pseudo-functors and lax transformations.

These is a full inclusion $Fun_{pn}(\mathcal{A},\mathcal{B})\subset Fun_{p}(\mathcal{A},\mathcal{B})$ For $(F, \phi)\in Fun_{p}(\mathcal{A},\mathcal{B})$ give a natural construction of a 2-isomorphism (a lax transformation with isomorphisms components) $\eta_F: (F, \phi)\to N((F, \phi))$.

We have that $(F, \phi)$ consist of

a family of functors $F_{A,B}: \mathcal{A}(A, B)\to \mathcal{B}(FA, FB)$

a family of isomorphisms $\phi_A: I_{FA}\to F(I_A)$

a family of 2-isomorphisms $\phi_{f, g}: F(g)\circ F(f)\rightarrow F(g\circ f)$

with the usual coherence conditions M1, M2 p. 30 of [B].

Then define the normal $N((F, \phi))$ as $(F', \phi')$ where:

$F'(A):=F(A)$ for each object $A$ of $\mathcal{A}$ different form any $I_B$, and $F'(I_A):= I_{FA}$.

Let $F_{A, B}:= F_{A,B}$ for each couple of object $A,\ B$ of $\mathcal{A}$ both different form any $I_B$.

let $F'_{A, I_B}: \mathcal{A}(A, I_B)\xrightarrow{F_{A,B}}\mathcal{B}(F(A), F(I_B))\xrightarrow{(1, \phi_A^{-1})}\mathcal{B}(F(A), I_{F(B)}) $ (for $A$ different form any $I_B$)

Similarly we define $F'_{I_A, B}$ and $F'_{I_A, I_B}$.

Then let $\phi'_ A:=1: I_{F'A}\to F'(I_A)$

and define $\phi'_ {f, g}:= \phi_{f, g}$ if $g: B\to C$, $f: A\to B$ and each $A,\ B,\ C$ different form any $I_B$.

if for example only the codomain of $g$ is not of this type i.e. $g: B\to I_C$ then let

$\phi'_{f, g}: F'(g)\circ F(f)=\phi_C^{-1}\circ F(g)\circ F(f)\xrightarrow{\phi_C^{-1}\circ\phi_{f,g}} \phi_C^{-1}\circ F(g\circ f)=F'(g\circ f)$

Similarly we define $\phi'_{f, g}$ also if other of the objects $A,\ B, C$ are of type $I_D$ for some object $D$.

remains the verification of the conditions of consistency, but this follow from the general criterion of coherency for pseudo-functors (S. MacLane, R. Paré, "Coherence for bicategories and indexed categories") or for direct verification

share|improve this answer
    
I don't understand your edit: do you mean it doesn't work in the non-normal case? –  Mike Shulman Jun 22 '12 at 20:27
    
THe edit mean that I'm working about non-normal case I actually missed the problem because it seemed that nobody cared. I know how to go forward, if you are interested I'll go ahead. (sorry for my English) –  Buschi Sergio Jun 23 '12 at 5:57

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.