In a course I'm learning about Topos theory, there are a lot of exercises which require you to prove explicitly some category is an elementary topos: i.e. to construct exponentials and a subobject classifier, and to show that it has all finite limits. Despite having worked my way through a number of these questions, I still find them very difficult and have not yet found a consistent method of approaching them. Examples include:

  • The functor category $[\mathcal{C}^{\text{ op}}, \bf{Set}]$ for a small category $\mathcal{C}$.

  • The category $B(G)$ of continuous $G$-sets with $G$-actions (for some group $G$), $(X,\epsilon)$ and as morphisms $(X,\epsilon ) \to (Y, \epsilon ')$ the functions $X \to Y$ which respect the group actions.

  • The functor category $[\mathcal{C}^{\text{ op}}, \bf{Set}]$ for a category $\mathcal{C}$ with slice category $\mathcal{C}/c$ equivalent to a small category for each object $c$.

  • The category of graphs which permit multiple directed edges between two vertices.

This is a soft question, since I am aware of how to do the first of these exercises (using the Yoneda lemma) and the third appears to require a little extra work and the result of the first, but in general it's not very clear to me how to go about trying to construct an exponential, or find a subobject classifier, or even necessarily to show that a category has all finite limits - you can use results like having products and equalisers to deduce having all limits, but then you still have to show the existence of certain limits, exponentials and a subobject classifier.

Intuitively speaking, is there any sensible way to go about trying to show that these things exist for a specific category? I guess you have to construct them explicitly for these sorts of exercises, but the trick with the Yoneda lemma for the top one will only work for functor categories; in general, how would you approach construction of these objects? My question is vague so I'm happy with vague answers, just trying to get better at tackling these sorts of exercises and any thoughts you can throw in would be much appreciated, thanks.

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    $\begingroup$ This is by no means an answer to the complete question, but for limits in functor categories, see here: en.wikipedia.org/wiki/Functor_category#Facts $\endgroup$ Feb 6 '12 at 3:13
  • $\begingroup$ That's very useful: I'm expecting to collect lots of small helpful bits of information here rather than a single all-encompassing answer. As a matter of interest, the link explains how products of functors act on objects (pointwise), but it isn't clear to me how a product of functors would behave on morphisms: any new morphism would have to be between 2 products of categories, and it's not clear to me how exactly that would work. Do you know how? Thanks for the help! $\endgroup$
    – Jak
    Feb 6 '12 at 3:46
  • $\begingroup$ I'm very much a clueless observer when it comes to category theory, but just as a thought until someone more knowledgeable arrives, perhaps you can consider a morphism from the product AxB to CxD to be in some sense a set of 4 separate morphisms from A and B to C and D? How exactly you would know what the appropriate morphism $(F \times G) (f)$ from $F(A) \times G(A) \to F(B) \times G(B)$ would be, given some $f: A \to B$ is unclear to me however. Perhaps someone could clarify briefly in the comments before things become too off-topic; otherwise I'd say ask a new question. $\endgroup$
    – Spyam
    Feb 6 '12 at 4:46
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    $\begingroup$ For $f : A \to B$, $(F \times G)(f) = F(f) \times G(f) : F(A) \times F(B) \to G(A) \times G(B)$. It's the result of sticking two arrows side by side. There is no general method of showing properties of a specific category: if there were, we wouldn't need to do any other kind of mathematics, would we? :p $\endgroup$
    – Zhen Lin
    Feb 6 '12 at 7:30
  • $\begingroup$ Well, one could hope! I didn't expect there to be a single general method, but I thought I might be able to gather some general approaches which might be useful in certain situations; the answer below is a good example of the sort of thing I was looking for. Perhaps my question is a little too vague for SE but I've learned some useful things from it so far! $\endgroup$
    – Jak
    Feb 6 '12 at 15:16

Maybe this is not an answer but an example.

If $\mathcal E$ is an elementary topos, then the category $\mathcal E^G$ of (right) $G$-objects for a group object in $\mathcal E$ is again an el. topos, and the proof is pretty easy using those objects you already have in $\mathcal E$. You can find it (and more) in Mac Lane-Moerdijk.


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