When reading the definition of a well-pointed category on for example Wikipedia, only the following condition is given:

  1. The terminal object 1 is a generator.

However the general definition of a well-pointed topos also mentions:

  1. The terminal object is not a zero object (i.e. it is not initial at the same time)

I can see how being initial would ruin the odds of being a generator as there is only one unique morphism from the initial object to any other object. But if this was the only reason for including condition 2, then condition 1 would suffice.

Is there a more genuine reason for including condition 2?

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    $\begingroup$ Pretty sure it just rules out the trivial category, since any well-pointed categories with a zero object will be trivial. So no, I don't think there's any deep reason. $\endgroup$ – Malice Vidrine Jan 27 at 7:42
  • $\begingroup$ (Actually there's probably even less reason to mention it in the case of toposes: a topos with a zero object is already trivial, so the lack of a zero object is usually assumed well before one gets to talking about well-pointedness.) $\endgroup$ – Malice Vidrine Jan 27 at 7:46
  • $\begingroup$ And is there any reason to rule out the trivial category except for that it is not very useful? $\endgroup$ – NDewolf Jan 27 at 8:58
  • $\begingroup$ Not really. It's one of those things where you'd write a lot of proofs that use phrases like "except for the trivial category" or "this is obviously satisfied in the trivial case, so consider any other case," unless you just rule it out up front for convenience. $\endgroup$ – Malice Vidrine Jan 27 at 9:08

Malice's comment is spot on: There are important theorems which only hold for well-pointed toposes if well-pointed is defined as it is.

One example for such a theorem is that the internal logic of the topos coincides with the external one (where we only speak about global elements). For the trivial topos (whose category consists of just one object), this equivalence fails quite hard, because in the internal logic of the trivial topos any statement whatsoever holds, including "$\bot$" (falsity), which does not hold externally.

Your question is one of those where switching to a constructive metatheory yields some further insights. Namely, constructively, we need even more conditions (conditions which are classically always satisfied). Then it becomes obvious that "excluding the trivial topos" is actually a red herring.

More details on both points can be found over at the nLab.


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