Sheaves in mathematics get their name from the agricultural term “sheaf” – a collection of stalks of grain or cereal all bundled together. This is like how properties of sheaves are determined by their stalks from the gluing axioms.

This time, as promised, we will discuss **sheafification** and show it is the left-adjoint of the inclusion functor .

**Defintion 1**: Let be a presheaf (of sets, say) on a topological space . The **sheafification** of is the presheaf where is the set of functions such that for each , (i) and (ii) there exists an open neighbourhood of and a section such that for all we have . If then the restriction map is the restriction of each of these functions to .

So the sheafification of a presheaf basically bundles all the sections of the presheaf into “compatible stalks” in the sense that a section of the sheafification is the collection of germs of a particular section of in all its stalks. As you’d expect, the sheafification of a presheaf is actually a sheaf:

**Theorem 1:** The sheafification is a sheaf (of sets, but this will generalise to other categories).

**Proof: **We know that is a presheaf so we need to just check the identity and gluability axioms. Let be an open subset of and let be an open cover.

First, let’s check the identity axiom. Let be sections over such that their restrictions are equal for all open sets in the cover. Then for any there exists some containing and on which , so . This holds for all , so .

Now let’s check the gluing axiom. Let be sections such that for each the restrictions of and to are equal. Now define a function by for any containing . This is well-defined because the sections agree on overlaps. So now we just need to check that i.e. that it satisfies the two conditions in the definition. The fact that the ‘s all satisfy give that . Now for each there exists an open set and a section such that for all , . This means that this condition holds for too – just take any of the containing . So we actually have . This completes the proof.

So given any presheaf we can construct a sheaf from its stalks. There is a natural morphism of presheaves whose components are given by

For any morphism of presheaves, the universal property of the colimit means we get a unique morphism of stalks for each stalk (which means taking the stalk is a functor!). It follows that the morphism induces a unique morphism on stalks . But we also obtain a unique morphism for each as follows:

Take and for each let be the section (defined above) such that for all points . Then we define a collection of morphisms in our target category (one for each open set containing ) by . These are clearly compatible with restriction and so we obtain a cocone over . By the universal property of the colimit , we obtain a unique morphism such that the maps factor as the compositions of and the inclusions into the colimit.

So now we have unique morphisms in our target category for each point . It follows that these morphisms compose to give the identity morphisms, and therefore ** and its sheafification have isomorphic stalks.** This is a hugely useful fact which we will use again. For now, note that if is already a sheaf then and its sheafification are isomorphic **as sheaves**, because morphisms of sheaves are determined by their stalks!

Now let be another sheaf on and let be a morphism (here is a presheaf). Then there is a morphism of sheaves whose components are defined as follows: for the open neighbourhoods containing each point each have a section such that for all . This condition ensures that the restrictions of and to are equal, and therefore their images and are also equal on the overlaps . Therefore since is a sheaf and is an open cover, the sections all glue together to give a section . We then set . This clearly commutes with restrictions so it is actually a sheaf morphism . Furthermore you can check that it also satisfies , so the diagram below commutes:

Now suppose that is another sheaf morphism that satisfies . We have induced maps on stalks: . But since sheaf morphisms are determined by their stalks and as we saw above, it follows that and induce the same map on every stalk and therefore they are equal. So the map is uniquely determined by .

Now if is a morphism of *presheaves* on then the composition (hopefully you can work out the notation from above!) gives a uniquely defined morphism of *sheaves* on . In this way, we see that sheafification is a functor . Furthermore, the discussion above shows that in fact is the left-adjoint of the inclusion functor from sheaves to presheaves on .

A subcategory of a category is called *reflective* if its inclusion functor has a left adjoint, which we call the *reflection*. One immediate thing to note is that since is a left adjoint, it preserves colimits. Some general theorems which you can look up in the *Handbook of Categorical Algebra *(section 3.5 on reflective subcategories) or on the nLab page for reflective subcategories and the nLab page for presheaves tell us that if is a reflective subcategory of a (co)complete category (i.e. every diagram in has a (co)limit) then is itself (co)complete. The nLab page says the category of presheaves on any space with values in the category of sets (or I would imagine in *any* complete and cocomplete category) is (co)complete, and so the category of sheaves valued in that category is (co)complete because of the above.

This means we can compute any limits we like in most of the frequently-used categories of sheaves (e.g. with values in or ). Moreover, since the inclusion functor from sheaves to presheaves is the right adjoint of sheafification (as we have just seen) it follows that these limits are preserved in presheaves. And by section 2.15 of the Handbook since the presheaf category is just a functor category we just compute limits pointwise, in the sense that .

Thus if have a morphism of sheaves of abelian groups (or any abelian category) we can compute the kernel (which exists as a sheaf since the category of sheaves is complete and kernel is a limit – it’s an equaliser), and we know by the above argument that it is computed “pointwise”, so . This is a nice application of how the abstract functorial machinery can be used to deduce (relatively – we’re still dealing with sheaves!!!) concrete things about how to calculate natural constructions relating to sheaves.

The same result does not hold for cokernels, which are colimits (coequalisers) and generally do not commute with the inclusion functor. To define the cokernel of a sheaf morphism we need to sheafify, so is the sheaf cokernel. This is constructed in two steps – first take the pointwise cokernel, which is a presheaf colimit. Then sheafify and note that since sheafification is a left adjoint it preserves colimits, so this really is a colimit (and has the cokernel UMP).

I think I understand the main points of the post, but I am still left scratching my head at the grammar in the definition of sheafification: what exactly is $s_y$? The notation from the previous post suggests that it might mean a section of $F(W)$ for some open neighborhood $W\subset V_x$ of $y$. But this is a problem because $f(y)$, being a germ, has no hope of being equal to a section; perhaps it should be that $(s_y,W)\in f(y)$?

I tried to look up alternative definitions and compare, but because you simplified it to concrete $\mathcal C$ I got lost in the translation.

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Sorry for the delay in getting back to you, things have been pretty busy here! Your comments are helpful because they’ve made me realise there’s lots of notation I haven’t defined (partly as I am not trying to write another textbook here, but obviously some of these things are pretty important!).

Here is the image of the section in the stalk at – so it is actually a germ. This follows the convention that people often write for the restriction of a section to the open set , and so when you take the colimit over all opens containing you just add a subcript instead.

I’m actually not at all confident with the general categorical definition with sheafification, but this “set-like” one is easily generalised to concrete categories.

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Okay, I think I know what you are talking about; you mean that since $V_x$ contains $y$, there is a morphism $\psi: F(V_x)\to F_y$ by the definition of a colimit, and $s_y=\psi(s)$.

(The “the image of the section” phrasing threw me off for a good while; I was trying to understand how a section could have an image I’m not sure what I’d say to clear things up for me an hour ago; maybe just flip “of the section $s$” and “in the stalk at $y$”. OTOH this is likely me being a novice at categories and not really internalizing the idea that, say, colimits include the morphisms.)

Also, I finally understand the “stalk” imagery, yay =D

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Exactly right! This stuff definitely took me a while to get to grips with too, especially the whole business of the stalk at a point as a colimit. I think that the thing which helped (and which initially confused you) is really treating the restriction morphisms explicitly as categorical morphisms rather than just thinking “restrict that section to this open set”; in that way, when you pass to the colimit – the stalk – you don’t need to think about germs as equivalence classes of pairs – you just use your universal morphisms to the colimit and then it makes perfect sense to take the image of a section .

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Btw just seen your post on One Thousand Adventures in Maths about my blog! Thanks for writing it, I was wondering why I was getting more traffic in the last few days 🙂 Getting my tumblr set up so next time I can contact you about your blog on your actual blog haha

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Yeah, hehe. I know that tumblr is not where all the big math blogs live, but… honestly the fonts are a deal-breaker to me for WordPress. LaTeX looks strange to me sitting next to sans-serif plaintext. I don’t mind reading it too much, but writing in it sets me on edge. Tumblr doesn’t display latex on the dashboard but at least it makes it pretty on the actual website 🙂

Anyway, I’m almost done with the scheme post (need to go through the $\mathcal B$-sheaf proof), write more pl0x 😛

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