Timeline for Large "internal" categories and "finite" products
Current License: CC BY-SA 4.0
6 events
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| Nov 5 at 14:51 | comment | added | Zhen Lin | Ah, yes, that's true. Maybe that's why I didn't pursue that line of thought. But I think, practically, we want a much nicer filtration than that, and we want to be able to construct indices of the filtration by recursion. Maybe first we need to understand the endofunctors of $\mathcal{E}$ that permit recursion... hmmm... | |
| Nov 5 at 14:08 | comment | added | Simon Henry | @ZhenLin Isn't that always true internally? If $E$ is locally small internal category and $Y \to E$ is a $Y$-indexed collection of objects of $E$, then I can build an internal (small) category which is a full subcategory of $E$ and whose object of objects is $Y$. So internally for any small familly of small category with maps to $E$ I can factor all of them trhough some common small category (so this is "small directed" internally). Or maybe you want to impose this as an external condition? | |
| Nov 4 at 22:41 | comment | added | Zhen Lin | I never worked out the details, but basically we want the large category to be a highly directed union of internal categories (possibly as a structure rather than a property), and then restrict to functors that can be similarly expressed as a functor between internal categories. The difficulty is in making sense of "highly directed". In ZFC we would want an ordinal-indexed increasing sequence of internal categories. | |
| Nov 4 at 15:07 | comment | added | Simon Henry | I agree with your answer, but I guess my question is, How do you find such suitable categories that contains the basic large categories we want to consider (like the category of sets). For example, I'm not convinced we can find them as full subcategories of the categories of presheaves. | |
| Nov 4 at 11:16 | comment | added | David Roberts♦ | It vaguely feels like comprehension is lurking somewhere, but I don't really understand it enough to be able to say anything sensible. | |
| Nov 4 at 9:21 | history | answered | Zhen Lin | CC BY-SA 4.0 |