I am reading these notes on topos theory, and I have a small confusion about Proposition 1.16 on page 12.
What is the difference between a "pointwise equalizer" ($K$ in proposition 1.16) and the standard notion of equalizer in category theory? Is the only difference that, in the pointwise case, we are dealing with morphisms in a functor space?
A great answer would clarify the idea of "pointwise computation of limits" and the definition of a pointwise equalizer.
Thank you!
If you have a small category $C$ and a category $D$, you can consider the functor category $D^C$.
In this category you can wonder what limits look like. The basic toy example is the product : given two functors $F,G: C\to D$, what might their product look like ?
Well it turns out that if $D$ has products, then $F\times G$ is computed pointwise (where "pointwise" refers to the objects of $C$ as points); that is for any object $c\in C, (F\times G)(c) = F(c)\times G(c)$ (this second product being in $D$); and it's actually easy to see that this new functor (with the obvious definition on arrows) is actually the product of $F$ and $G$.
We say "pointwise", just like, say, for functions $f,g: X\to \mathbb{R}$ we'd say that their "pointwise product" is $x\mapsto f(x)g(x)$.
Now of course this generalizes to any shape of diagram : if $I$ is a small category such that $D$ has all limits of diagrams of shape $I$, then $D^C$ does as well, and those limits are "computed pointwise". More precisely, if $G: I\to D^C$ is such a diagram, then for $c$ an object of $C$, $(\mathrm{Lim} G)(c) =\mathrm{Lim} G_c$ where $G_c$ is the functor $I\to D$ defined on objects by $G_c(i) = G(i)(c)$ (and obviously on arrows)
Applying this to the category $I$ defining equalizer diagrams, or product diagrams, or any small $I$ yields the notion of pointwise computation.
What should not be misunderstood is : $D^C$ might have limits of shape $I$ without $D$ having them. In particular, there may be two functors $F,G$ with a product $F\times G$ such that $(F\times G)(c)$ is not a product of $F(c), G(c)$: this can happen if $D$ doesn't have all products.
In the context of topos theory, we'll often have $D= Set$ which has all small limits (and colimits- of course the above applies to colimits as well), and so for any small $C$, limits are "computed pointwise" in $Set^C$. In particular this means that the equalizer of $f,g : F\to G$ will be defined for an object $c$ in $C$ as $\{x\in F(c)\mid f_c(x)=g_c(x)\}$.