Timeline for Interpreting Yoneda as reducing categorical dimension
Current License: CC BY-SA 4.0
23 events
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| Apr 29 at 0:51 | comment | added | David Roberts♦ | @AlecRhea yeah, sorry, I don't really know! | |
| Apr 26 at 8:47 | comment | added | Alec Rhea | @DavidRoberts That sounds on point, but I’m looking for an exposition of two dimensional category theory that works with elements of the values of $2$-presheaves instead of abstractly in the $2$-presheaf $2$-topos. (also this reasoning should be ‘iterable’ in a sense; the ‘elements’ of $2$-presheaves are objects/arrows in a category, which admits a presheaf category which we can reason inside elemetwise). | |
| Apr 26 at 6:23 | comment | added | David Roberts♦ | @AlecRhea this sounds vaguely like "pointwise" reasoning, whereby you work with the elements of the values of a presheaf, rather than abstractly with objects of the presheaf topos. | |
| Apr 25 at 14:04 | comment | added | Alec Rhea | @PeterLeFanuLumsdaine What I mean by 'reducing algebraic dimension' is that this construction allows me to take stuff I would usually have to prove one-dimensionally (i.e. by working with arrows in a category) and prove it $0$-dimensionally by manipulating elements of sets of generalized global elements, for example as in the linked example at the end of the question (proving stuff about internal rings of matrices over internal ring objects a category with finite products and a terminal object). | |
| Apr 25 at 14:01 | comment | added | Alec Rhea | @PeterLeFanuLumsdaine Yes, the choice of terminology is still imperfect; I thought about 'reducing algebraic dimension', but there are already many standard notions of dimension in algebra distinct from what I mean. To fix terminology, by $0$-dimensional algebra I mean all the algebra we do by manipulating elements ($0$-dimensional objects, conceptually, since they're 'points') i.e. basic group/ring/field/set theory etc.. By $1$-dimensional algebra I mean all the algebra we do by manipulating arrows in a category ($1$-dimensional objects, conceptually, since we're manipulating lines). (cont.) | |
| Apr 25 at 9:43 | comment | added | Peter LeFanu Lumsdaine | More precisely, your construction describes a faithful functor $\mathrm{Psh}(C) \to \mathrm{Set}$ — i.e. a concretisation of $\mathrm{Psh}(C)$. This is a useful construction for many purposes, but it’s not shifting categorical dimension in any standard sense. | |
| Apr 25 at 9:42 | comment | added | Peter LeFanu Lumsdaine | Your description of this approach as collapsing things a dimension seems slightly inaccurate. For instance, you say “instead of […] diagrams in a category (1-dimensional algebra) […] I can work with functions acting on elements (0-dimensional algebra)”. But under your correspondence, presheaves are replaced by sets (still objects of a 1-category), and elements of presheaves are replaced by elements of sets (0-dimensional algebra in both cases). You haven’t reduced the dimension — you’ve gone from the 1-category of presheaves to the slightly simpler 1-category of sets. [cont’d] | |
| Apr 25 at 5:37 | history | edited | YCor | CC BY-SA 4.0 |
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| Apr 25 at 1:57 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Apr 24 at 20:46 | comment | added | Alec Rhea | @DavidRoberts I've tried to clarify what I meant; hopefully this resolves the issue? | |
| Apr 24 at 20:46 | comment | added | Alec Rhea | @ToddTrimble I did see it for future reference, and thank you. | |
| Apr 24 at 20:39 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Apr 24 at 20:24 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Apr 24 at 20:21 | history | undeleted | Alec Rhea | ||
| Apr 7 at 7:48 | comment | added | David Roberts♦ | I agree that "interpretation of the language of ZF" seems slightly off here, certainly if you are dealing with an arbitrary category. Without at least finite products you can't say much at all. This is a matter of the expressiveness of the logic, as well. The internal language of a regular category allows use of the regular fragment of FOL, less than the background full FOL assumed for ZF. | |
| Apr 7 at 0:46 | comment | added | Todd Trimble | I was going to say that maybe "completeness theorems" or "embedding theorems" (for statements in 2-category theory) might be useful buzzphrases. But now I don't know whether you'll see this comment -- you deleted too quickly for me to insert this comment in time. | |
| Apr 7 at 0:44 | history | deleted | Alec Rhea | via Vote | |
| Apr 7 at 0:43 | comment | added | Alec Rhea | @ToddTrimble I don’t mean to appropriate any language incorrectly; I’ll remove the question until I can find the correct way to phrase it. | |
| Apr 6 at 22:05 | comment | added | Todd Trimble | Like varkor, I'm also feeling internal resistance to how you're using "internal language", and maybe even more resistance to "interpretation of the language of ZF". which is single-sorted with a single binary predicate $\in$. But it sounds like maybe you want embedding/completeness theorems in the 2-categorical context. So for example, in the 1-categorical context, to prove a general statement about finite limits, it suffices to prove it for Set, since we have the Yoneda embedding which is fully faithful and left exact, to do the remaining work. | |
| Apr 5 at 23:20 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Apr 5 at 22:39 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Apr 5 at 22:23 | history | edited | Alec Rhea | CC BY-SA 4.0 |
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| Apr 5 at 15:03 | history | asked | Alec Rhea | CC BY-SA 4.0 |