show/hide this revision's text 4 Fixed typo in my previous edit

Here is a fairly polemical answer, in a similar spirit to Brian's:

Sheafification is not a painful process: you take a presheaf, and you think about how you need to change it so that the stalks are the same, but sections can be glued. It is very natural.

The inverse image is also naturally understood in the same kind of terms: you have a sheaf $\mathcal F$ on $X$, and you would like to make a sheaf on $Y$ whose stalk at $y$ is equal to the stalk of $\mathcal F$ at $f(y)$ (i.e. $(f^{-1}\mathcal F)_x = \mathcal F_{f(x)}$). If you ponder how you can make a rigorous construction with these properties, you will be led to the inverse image. (It's essentially taking the fibre product of $\mathcal F$ over $X$ with the map $f:Y \to X$, and indeed thinking about the inverse image is good practice for developing intuitions about fibre products in lots of other contexts.)

Using the crutch of affines and quasi-coherent sheaves discourages thinking about the (fairly simple and natural) local picture of a sheaf as a bunch of stalks glued together. A lot of the power of the geometric ideas in algebraic geometry comes from thinking geometrically, so one doesnt' want to be discouraging thinking about sheaves in this way; rather, you want to encourage it.

As for applications, Donu notes some in his answer.

Let me note another here: if $\mathcal I$ is an ideal sheaf on $X$, then $f^{-1}\mathcal I$ is naturally a subsheaf of $f^{-1} \mathcal O_Y$ O_X$ (because $f^{-1}$ is exact, as one sees immediately by looking on stalks and using the fact that $f^{-1}$ doesn't change stalks!), and one often wants to look at the ideal sheaf in $\mathcal O_Y$ generated by this. This is not the same (typically) as $f^*\mathcal O_Y$. (Just as, if $I$ is an ideal in $A$ and $B$ is an $A$-algebra, $B\otimes_A I$ is typically is not isomorphic to the ideal in $B$ generated by $I$.)

Now there are other ways to describe this ideal sheaf in $\mathcal O_Y$ (e.g. it is the image of the natural map $f^*\mathcal I \to \mathcal O_Y$), but the description of it in terms of $f^{-1}\mathcal I$ is convenient and very natural.
show/hide this revision's text 3 changed "Its" to "It's"

Here is a fairly polemical answer, in a similar spirit to Brian's:

Sheafification is not a painful process: you take a presheaf, and you think about how you need to change it so that the stalks are the same, but sections can be glued. It is very natural.

The inverse image is also naturally understood in the same kind of terms: you have a sheaf $\mathcal F$ on $X$, and you would like to make a sheaf on $Y$ whose stalk at $y$ is equal to the stalk of $\mathcal F$ at $f(y)$ (i.e. $(f^{-1}\mathcal F)_x = \mathcal F_{f(x)}$). If you ponder how you can make a rigorous construction with these properties, you will be led to the inverse image. (Its It's essentially taking the fibre product of $\mathcal F$ over $X$ with the map $f:Y \to X$, and indeed thinking about the inverse image is good practice for developing intuitions about fibre products in lots of other contexts.)

Using the crutch of affines and quasi-coherent sheaves discourages thinking about the (fairly simple and natural) local picture of a sheaf as a bunch of stalks glued together. A lot of the power of the geometric ideas in algebraic geometry comes from thinking geometrically, so one doesnt' want to be discouraging thinking about sheaves in this way; rather, you want to encourage it.

As for applications, Donu notes some in his answer.

Let me note another here: if $\mathcal I$ is an ideal sheaf on $X$, then $f^{-1}\mathcal I$ is naturally a subsheaf of $f^{-1} \mathcal O_Y$ (because $f^{-1}$ is exact, as one sees immediately by looking on stalks and using the fact that $f^{-1}$ doesn't change stalks!), and one often wants to look at the ideal sheaf in $\mathcal O_Y$ generated by this. This is not the same (typically) as $f^*\mathcal O_Y$. (Just as, if $I$ is an ideal in $A$ and $B$ is an $A$-algebra, $B\otimes_A I$ is typically is not isomorphic to the ideal in $B$ generated by $I$.)

Now there are other ways to describe this ideal sheaf in $\mathcal O_Y$ (e.g. it is the image of the natural map $f^*\mathcal I \to \mathcal O_Y$), but the description of it in terms of $f^{-1}\mathcal I$ is convenient and very natural.
show/hide this revision's text 2 Filled in a missing $f^{-1}$ in 2nd to last paragraph, line 1.

Here is a fairly polemical answer, in a similar spirit to Brian's:

Sheafification is not a painful process: you take a presheaf, and you think about how you need to change it so that the stalks are the same, but sections can be glued. It is very natural.

The inverse image is also naturally understood in the same kind of terms: you have a sheaf $\mathcal F$ on $X$, and you would like to make a sheaf on $Y$ whose stalk at $y$ is equal to the stalk of $\mathcal F$ at $f(y)$ (i.e. $(f^{-1}\mathcal F)_x = \mathcal F_{f(x)}$). If you ponder how you can make a rigorous construction with these properties, you will be led to the inverse image. (Its essentially taking the fibre product of $\mathcal F$ over $X$ with the map $f:Y \to X$, and indeed thinking about the inverse image is good practice for developing intuitions about fibre products in lots of other contexts.)

Using the crutch of affines and quasi-coherent sheaves discourages thinking about the (fairly simple and natural) local picture of a sheaf as a bunch of stalks glued together. A lot of the power of the geometric ideas in algebraic geometry comes from thinking geometrically, so one doesnt' want to be discouraging thinking about sheaves in this way; rather, you want to encourage it.

As for applications, Donu notes some in his answer.

Let me note another here: if $\mathcal I$ is an ideal sheaf on $X$, then $f^{-1}\mathcal I$ is naturally a subsheaf of $\mathcal f^{-1} \mathcal O_Y$ (because $f^{-1}$ is exact, as one sees immediately by looking on stalks and using the fact that $f^{-1}$ doesn't change stalks!), and one often wants to look at the ideal sheaf in $\mathcal O_Y$ generated by this. This is not the same (typically) as $f^*\mathcal O_Y$. (Just as, if $I$ is an ideal in $A$ and $B$ is an $A$-algebra, $B\otimes_A I$ is typically is not isomorphic to the ideal in $B$ generated by $I$.)

Now there are other ways to describe this ideal sheaf in $\mathcal O_Y$ (e.g. it is the image of the natural map $f^*\mathcal I \to \mathcal O_Y$), but the description of it in terms of $f^{-1}\mathcal I$ is convenient and very natural.
show/hide this revision's text 1