I recently started studying topoi and the book I am using defines them as categories that have all finite limits and co-limits, exponentiation and sub-object classifiers. The book briefly remarks that finite co-completeness is actually unnecessary as the other three conditions imply it. However, the book itself does not prove this fact. As such, could anyone suggest an easily accessible proof of this result? I am specifically looking for an online-accessible paper/book on the proof.
2026-03-30 00:00:19.1774828819
Finite Limits, Exponentiation and Sub-Object Classifiers imply Finite Co-Limits
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This is a non-trivial theorem of Paré [1974, Colimits in topoi]. You can also find a proof in some textbooks, e.g. Sheaves in geometry and logic.