Limits and colimits in the category of presheaves

Question

We know that the category of presheaves (i.e. $Fct(C^{Op}, Set)$ ) is an elementary topos and in particular it is finitely complete and finitely cocomplete and so it has all finite limits and colimits.

For instance I know that the terminal object is a functor $\Bbb{1} :C^{Op}\rightarrow Set$ which sends all objects to a fixed singleton and all arrows to the identity from that sigleton to itself. I know also the binary product, the equalizer and the initial object. Can you help me to find the other remarkable limits and colimits (pullbacks, coproduct, coequalizer and pushout?)

We know also that an elementary topos is always a regular category, who are the internal equivalence relations and the kernel pairs?

Last question: I know that a mono in presheaves is a natural transformation between two functors $\eta :F\rightarrow G$ if and only if $\eta_A$ is injective for all $A\in Obj(C^{Op})$. Is it true that an arrow $\mu$ in the category of presheaves is an epi iff $\mu_A$ is surjective for all $A\in Obj(C^{Op})$ ?

Answer 1

The short mantra is that limits of diagrams of presheaves are computed "pointwise") (or "objectwise"). This effectively means that if you know how to compute limits in the category of sets, then you can easily compute limits of presheaves.

Let's take pullbacks for example. Suppose you have a diagram of presheaves

$$F \stackrel{\phi}{\to} H \stackrel{\psi}{\leftarrow} G$$

where $\phi, \psi$ are natural transformations, and you want to compute its limit which is the pullback, call it $P$. In particular, you have to be able to say what is the value of the set $P(c)$, for any object $c$ of $C$. Well, this set is nothing other than the pullback of

$$Fc \stackrel{\phi c}{\to} Hc \stackrel{\psi c}{\leftarrow} Gc$$

in the category of sets, where of course $\phi c$ denotes the component of the transformation $\phi$ at the object $c$. You just compute the limits, object $c$ by object $c$.

For each $c$, you'll get a commutative pullback square in $Set$, which have, in addition to the maps $\phi c, \psi c$, maps of the form $\pi_1 c: Pc \to Fc$ and $\pi_2 c: Pc \to Gc$.

I haven't yet said what is the function $P(f): Pc \to Pd$, where $f: c \to d$ is a morphism in $C^{op}$. But you can get this by exploiting the universal property of pullbacks. I should let you draw this yourself. Draw the pullback square for the object $d$, and then add to this diagram arrows of the form

$$Pc \to Fc \stackrel{Ff}{\to} Fd, \qquad Pc \to Gc \stackrel{Gf}{\to} Gd.$$

Then there are two paths going from $Pc$ to $Hd$, and the composites of these paths can be proved to be equal. Therefore, by the universal property of pullbacks, there is a unique fill-in $Pc \to Pd$, and this is your desired $Pf$. But in order to see that the composites are equal, you will need to add in more arrows in such a way to get a cubical shape. Two of the faces of the cube will be naturality squares, another face will be a commuting pullback at the object $c$. You should try it.

All limits and colimits of presheaves are computed in this pointwise way.