In a presheaf, the restriction maps are often not injective, and often not surjective. For example, consider the sheaf on the real line, where is the set of continuous functions on the open set Here restriction is neither injective nor surjective, in general.

]]>Thanks! You’re right about these typos. I’ve tried to fix both of them.

By the way, to get LaTeX to work here, follow the instructions right above the box in which you type your comment.

]]>you said “We just need to put a topology on $\Lambda(X)$”–I think you meant $\Lambda(F)$;

you said “We take the collection of all these germs to be an open neighbourhood of $x$”–I think instead of $x$ you meant the unnamed point introduced in a previous sentence “More specifically, any point in $\Lambda(F)$ is in some set $\Lambda(F)_x$…”.

Thanks a lot for these posts JB!

]]>Thank you for your comment.

You and John helped me a lot to think deeper to this subject.

I think that now I’m a bit more closer to the question of the germ. Let’s see if I’m loosing something else!

What I’m not able to do is to be certain of *why* the restrictions produce equivalent classes.

When I think to the case in which the open subsets are finite sets, then I see how the equivalent classes emerge.

E.g., let and ; and let the functions picked by the presheaf be functions with codomain the set , then:

has functions

has functions

A *restriction* should be a *surjective function*, partitioning in 4 parts, where each part contains 2 function.

In summary, the equivalence classes here are due to the surjectivity, surjectivity which is motivated by the fact that is built to get *all functions* of type , hence in term of cardinality .

Considering continuous functions of type , I think that the intuition is similar: we need to prove that the restrictions are surjective morphisms; but I’m not able to prove it, because the simple cardinality argument discussed so far doesn’t work anymore.

From a categorical point of view, the surjectivity in the Set category is identified by the *epimorphisms*: given any , then a function is an epimorphism if and only if .

But in the category there is at most one single morphism between any two open subsets of . This means that *any* morphism in this category is an epimorphism.

The last passage is the following. Using the presheaf , the *subcategory* of Set has the same property: no more than one morphism between any two object. Hence any of their morphism is an epimorphism.

Should this imply that these morphisms are surjective?

]]>Thank you for pointing it out! I was totally blind to the possibility that a smallest open subset could not exist in some topology.

]]>As John said, usually there is no smallest open set owning (that is, there is no smallest *neighbourhood* of ). But let's consider when there is. Then as you said, can simply be , a set of functions on . But *this* set is not meant to be a germ; it’s an *individual* element of that's meant to be a germ. And that works, because even though a germ is normally taken to be a set of functions (an equivalence class), in this case where is the smallest neighbourhood of , each equivalence class consists of a single function on . (Part of the reason is that in the definition of a germ, two functions are only equivalent if there is a *neighbourhood* of on which they are equal.)

Indeed, if every point in every topological space had a smallest neighbourhood, then there would be no need to define a germ as an equivalence class! It’s precisely because there usually is no smallest neighbourhood that we instead have to define a germ in a relatively complicated way. A germ at is what *would* be a function on the smallest neighbourhood of if there were such a thing. (In nonstandard analysis, you can talk about *infinitesimal* neighbourhoods, and then a germ at can simply be defined as a standard function on an infinitesimal neighbourhood of .)

Kvantumo wrote:

We defined as the

colimitof for all .

Right.

Then is a set of functions defined on the

smallestopen subset (that is, and and not ).

That’s true if there exists a smallest open set containing But in most examples that’s not true! For example if is the real line, or the complex plane, there is no smallest open set containing a point.

So, you need to stop thinking about the case where there exists a smallest open set containing the point and start thinking about these other examples. Then you’ll see that a germ at defined as I defined it, is indeed an equivalence class of functions defined on open sets containing where two functions are equivalent if they become equal when restricted to a small enough open set. This is how colimits work in this example.

]]>We defined as the *colimit* of for all .

Then is a set of functions defined on the *smallest* open subset (that is, and and not ).

Then you consider the elements of **germs** at .

This is what I don’t get. Because the definition which I remember for a germ of function at is a *class of equivalence* of functions, for which there is a subset containing $x$ where they are *equal*.

But the restricted functions in are not necessarily equal in that smallest .

So how come that they are called germs?

]]>We say it’s an **etale space** over if it has a special property: each point has an open neighborhood such that restricted to this neighborhood is a homeomorphism from this neighborhood to an open subset of In Part 2 we defined the category of bundles over which is called and the full subcategory of this whose objects are etale spaces, called […]