Do not the cat edition.
OP's mum is a universal construction
>>16897786all paths correspond to the same morphism induced by the universal property
There are only 9-10 category theorists on /sci/, a CT-exclusive splinter of /mg/ wouldn't last methinks.
Working on dessins / motives and ran into a basic open seam that I can’t quite close.Suppose [math]D[/math] is a stable symmetric monoidal triangulated category of geometric origin (e.g. a motivic category with six functors). One would like a canonical way to extract an abelian tensor category of “mixed objects” from [math]D[/math].Concretely:does there exist a [math]t[/math]-structure [math](D^{\le 0}, D^{\ge 0})[/math] on [math]D[/math] that is natural, compatible with [math]\otimes[/math] and the standard functorial operations, and whose heart [math]\mathcal{A} = D^{\le 0} \cap D^{\ge 0}[/math] recovers the expected abelian category (mixed Tate / 1-motives / etc. in known cases)?Equivalently, can one characterize a heart by a universal exactness property, e.g. an object lies in [math]\mathcal{A}[/math] iff all reasonable realization functors are concentrated in degree 0 and exact there?Locally (Artin, mixed Tate) this works, but globally the existence/uniqueness of such a heart seems obstructed. Is this fundamentally a failure of canonicity, or is there a known categorical obstruction theorem ruling it out?Not trying to solve motives here—just trying to understand whether a canonical abelianization of a motivic triangulated category is even a well-posed question.
>>16897929Wh- ....
>>16897958grothendieck was french so lots of weird terms in CT
>>16897929Dude I work in fucking algebraic combinatorics. I've read this post >20 times over and I still can't understand it. How do catfags dewit?
>>16897929isn't that beilinson-soule?
>>16897992Related but not the same.Beilinson–Soulé is a vanishing condition that would support a motivic t-structure (degree bounds), whereas I’m asking about the existence/canonicity of such a t-structure/heart itself.
>>16897786>do not the cat
>>16897998well that's the point, isn't it? there does exist a canonical t-structure iff beilison-soule holds; if you can construct one without assuming beilison-soule, you've proven it for your case
>>16898067In the Mixed Tate lane, I agree: a canonical motivic heart with the expected properties typically forces the same vanishing statements one would call ‘Beilinson–Soulé-type.’ So if one can construct the heart unconditionally in that lane, one is effectively proving those vanishings for that lane. I’m not claiming a global iff for all motives—only that the heart construction and vanishing are two sides of the same constraint in the controlled subcategory.
>>16897786Hello homosexuals and transgenders. I have recently been convinced that programming is very mathematical when I was trying to do dynamic programming problems and realised it was so much more fruitful to try to prove it properly instead of touching myself aimlessly. Actually I touch myself aimlessly normally quite a bit, these two might be related.Anyway please recommend me a category theory book ideally closely related to FP/discrete mathematical structures like trees/graphs/etc. and less so to advanced physics/real anal type structures.i know some toplogy/real anal/calculus. i am actually a bit weak in algebra i think so i might have to cover it in some way. the reason i am bad at algebra is that i wasn't interested in the applications when i was younger. i was interested in real anal type stuff because i wanted to learn complex anal because i was interested in physics. but i am more interested in algebra now because i see how it can be interesting and am a bit interested in cryptogrpahy.thank you very much!!!!
>>16898764
>>16898799thanks ill check it out and tell you how it made me feel
