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Oh, I see what you mean. You mean that for the definition of groups, topology, etc. I need to refer to a mathematical theory like set theory within first-order
by auggierose 2y ago
Oh, I see what you mean. You mean that for the definition of groups, topology, etc. I need to refer to a mathematical theory like set theory within first-order logic. Ok, let's do that. But then there is a definition of what "T is a topos" means, in terms of set theory based on first-order logic, right? Obviously, that definition cannot include the category of sets, because that category wouldn't be a set itself, so I am fine with considering only "small" toposes which are representable as sets, just as I am fine with just considering "small" groups, "small" topologies, and "small" categories.
- soist 2y agoSo what is your question? The standard definition of a topos spells out the structural constraints for subobject classifier, finite limits, and Cartesian closure. You can write out these definitions for the topos of sets and any "small" structures in your topos which are points of the specified logical/structural variety will be small toposes internal to the topos of sets.
- auggierose 2y agoOk. So I need to stay awake until I understand these category theoretic concepts, and then I know what a topos is, I guess. Thank you, that was a very helpful conversation.
- soist 2y agoNo problem.