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Stream: theory: category theory

Topic: "untwisted" tabulation - what is this?

Christian Williams (Jun 05 2022 at 00:40):

Hello, I've come to need a certain category which I want to understand better, by characterizing it abstractly. Any thoughts are appreciated.

Let $A,B$ be categories, and $R:A\to B$ a profunctor between them. Define the "arrow category" of $[R]$ as follows: objects are tuples $(a,b, r:R(a,b))$ and morphisms are $(f:a\to a', g:b\to b')$ such that $m'\cdot f = g\cdot m$.

This is the full subcategory of $[2,\overline{R}]$, the arrow category of the collage, determined by the elements of $R$. Does this category have a universal property? I was trying to construct it as a pullback, but it's a bit subtle because the "inclusion of R" into the collage is a transformation, not a functor.

Mike Shulman (Jun 05 2022 at 00:44):

It's the apex of the two-sided discrete fibration representation of $R$, and the comma category of the two inclusions of $A$ and $B$ into the collage of $R$.

Christian Williams (Jun 05 2022 at 00:44):

ah! yes, I agree with the second part.

Christian Williams (Jun 05 2022 at 00:45):

but the first, wouldn't that be the usual "twisted" tabulation?

Christian Williams (Jun 05 2022 at 00:46):

i.e. morphisms would be ternary factorizations, rather than commuting squares

Mike Shulman (Jun 05 2022 at 00:49):

I made that mistake at first too. The twisted version is what you get by taking the category of elements of $R$ regarded as a functor $A\times B^{\rm op}\to Set$. But the two-sided fibration is different than that; in particular it comes with a functor to $A\times B$, not to $A\times B^{\rm op}$.

Christian Williams (Jun 05 2022 at 00:51):

oh wow, because the variance is in the distinction of fibration and opfibration, so you don't need "op". awesome

Christian Williams (Jun 05 2022 at 00:51):

then how is it constructed? since it's different from the usual collage

Mike Shulman (Jun 05 2022 at 00:59):

What sort of answer do you want?

Christian Williams (Jun 05 2022 at 01:11):

if R:Aop x B to Set and the collage gives the twisted category, what is the construction which gives the category in question? (directly, rather than the two-step above)

Christian Williams (Jun 05 2022 at 01:30):

it's some "two-sided" Grothendieck construction. I think I can work it out, but it might be nice to find a reference

Christian Williams (Jun 05 2022 at 02:02):

the basic construction is the collage of (possibly lax) functors to Prof; and if the profunctors are companions or conjoints, we get the two variants of the "Grothendieck construction".

I think a functor Aop x B to Set is seen more clearly with codomain Rel, and morphisms in A give conjoints (cofunctions) while those in B give companions (functions). then I believe the collage there should give the two-sided construction.

Mike Shulman (Jun 05 2022 at 02:27):

You keep saying "collage" when I think you mean "Grothendieck construction".

Mike Shulman (Jun 05 2022 at 02:28):

The collage of a profunctor $R : A \nrightarrow B$ is the category with objects ${\rm ob}(A) +{\rm ob}(B)$ and morphisms coming from $A$ and $B$ and $R$, which is the lax colimit of $R$ considered as a morphism in $\rm Prof$.

Mike Shulman (Jun 05 2022 at 02:30):

The various kinds of Grothendieck construction are a different thing. As you know, the most general one-variable form involves a normal lax functor to $\rm Prof$, which includes the common cases of pseudofunctors to $\rm Cat$ and pseudofunctors to $\rm Cat^{\rm op}$. I don't think the two-variable Grothendieck construction (the one that gives rise to two-sided fibrations) can be obtained as a special case of that, but I could be wrong.

Mike Shulman (Jun 05 2022 at 02:31):

I'm still not sure what you mean by "what is the construction". It's the two-sided Grothendieck construction: that's what it is.

Christian Williams (Jun 05 2022 at 02:40):

The various kinds of Grothendieck construction are a different thing.

aren't they both lax colimits? one has the diagram of a single arrow, while the other is indexed by a whole category.

Christian Williams (Jun 05 2022 at 02:40):

maybe I still don't quite get the distinction between collages and lax colimits.

Mike Shulman (Jun 05 2022 at 14:38):

A lax colimit, like any kind of (co)limit, is something that makes sense for any shape diagram in any bicategory. The Grothendieck construction of a functor $A\to \rm Cat$ is its lax colimit considered as an $A$-shaped diagram in Cat. I think the Grothendieck construction of a normal lax functor $A \to \rm Prof$ is also its lax colimit as an $A$-shaped diagram in Prof. But the collage of a profunctor $R : A \nrightarrow B$ is its lax colimit considered as a $\mathbf{2}$-shaped diagram (i.e. an arrow) in $\rm Prof$.

Mike Shulman (Jun 05 2022 at 14:43):

I think in some paper Street (?) used the word "collage" for an arbitrary lax colimit in Prof or in a Prof-like bicategory.

Mike Shulman (Jun 05 2022 at 14:44):

But the point is that the lax colimit of a profunctor $R : A \nrightarrow B$ qua diagram $\mathbf{2}\to \rm Prof$ is very different from the lax colimit of any kind of diagram with domain $A\times B^{op}$.

Mike Shulman (Jun 05 2022 at 14:45):

I don't know of any way to view the 2-sided Grothendieck construction as a lax colimit.

Nathanael Arkor (Jun 06 2022 at 14:01):

I think in some paper Street (?) used the word "collage" for an arbitrary lax colimit in Prof or in a Prof-like bicategory.

Street uses that convention in Cauchy characterization of enriched categories, as one example.