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Stream: theory: category theory

Topic: sheafification as co/end

Matteo Capucci (he/him) (Nov 03 2021 at 17:48):

[[Sheafification]] is the left adjoint to inclusion of sheaves into presheaves $sh_J \dashv \iota : \mathrm{Sh}(X,J) \to \mathrm{Psh} X$.
Now left adjoints can be computed as right [[Kan extensions]], in particular $sh_J \cong \mathrm{Ran}_{\iota} 1_{Sh(X,J)}$. Moreover, Kan extensions admit an end formula, so one might define

$$sh_J(P) = \int_{S : \mathrm{Sh}(X,J)} S^{\mathrm{Psh}(P, S)}$$

where the power is meant to be interpreted as repeated product, not as internal hom.

Q1: do you see how one could interpret this formula geometrically?
Q2: is there a coend expression? Usually sheafification is a colimit, while ends 'are' limits.

Patrick Nicodemus (Nov 03 2021 at 19:31):

Matteo, the way I prefer to think about the sheafification is that one has an adjunction between presheaves over $X$ and "bundles" over $X$, i.e. the category $\mathbf{Top}/X$. This adjunction is of "nerve-realization" type. Writing $\mathcal{T}(X)$ for the poset category of opens of $X$, write $F : \mathcal{T}(X)\to \mathbf{Top}/X$ for the functor sending the open $U$ to the inclusion $i_U : U\to X$.

The "nerve" of $F$ here, it is not hard to see, is the sheaf-of-sections functor, and the left adjoint ("realization") associates to each presheaf its "espace etale".

One can compute the "realization" etale space of a presheaf by the usual coend formula for the left Kan extension, or for a weighted colimit, however you prefer to say it.

Imo this is more or less the same as computing the sheafification as the sections of this etale space are the sections of the sheafifcation of $P$.

Matteo Capucci (he/him) (Nov 03 2021 at 19:56):

Hi @Patrick Nicodemus, thanks for replying! Yeah I'm familiar with sheafification, although this nerve-realization pov is new for me.
I'm playing with ends and coends (and Tambara theory) and I was musing this way of writing it down. It looks unhelpful to me but maybe it can be interpreted in an interesting way.

Patrick Nicodemus (Nov 03 2021 at 20:00):

Well imo the etale space is easiest to describe by the coend, and this can be turned into something that you can visualize by a bit of computation. I have seen other ways of constructing it, for example as a topology on the set of all germs of all stalks, but they are not as enlightening.

Mike Shulman (Nov 03 2021 at 20:03):

Note that this is a "large" end, over the large category ${\rm Sh}(X,J)$. This is one reason it is not very useful for computations.

Mike Shulman (Nov 03 2021 at 20:05):

The coend version of the formula is the usual one for left adjoints out of presheaf categories. But it's a coend in ${\rm Sh}(X,J)$, and colimits in that category are computed by taking the colimit of presheaves and sheafifying. So expressing sheafification in this way is also not particularly helpful. (-:

Mike Shulman (Nov 03 2021 at 20:06):

If you're looking for a formula involving coends of presheaves, you may be interested in this paper, although you'll have to de-$\infty$-ify it.