The set of natural transformations between two functors F,G :C→DF,G:C→D can be expressed as the end Nat(F,G)≅∫AHomD(F(A),G(A)). Nat(F,G)≅∫A HomD (F(A),G(A)).
Define set of natural cotransformations from FF to GG to be the coend CoNat(F,G)≅∫AHomD(F(A),G(A)). CoNat(F,G)≅∫AHomD (F(A),G(A)).
Let: - F=B∙(Σ4)∗/F=B∙ (Σ4 )∗/ be the under ∞∞-category of the nerve of the delooping of the symmetric group Σ4Σ4 on 4 letters under the unique 00-simplex ∗∗ of B∙Σ4B∙ Σ4 . - G=B∙(Σ7)∗/G=B∙ (Σ7 )∗/ be the under ∞∞-category nerve of the delooping of the symmetric group Σ7Σ7 on 7 letters under the unique 00-simplex ∗∗ of B∙Σ7B∙ Σ7 .
How many natural cotransformations are there between FF and GG?
Also I'm curious as to what percentage of the questions in this benchmark are of this type / difficulty, vs the seemingly much easier example of "In Greek mythology, who was Jason's maternal great-grandfather?".
I'd imagine the latter is much easier for an LLM, and almost trivial for any LLM with access to external sources (such as deep research).
pass rate really only matters in context of the difficulty of the tasks
> In Greek mythology, who was Jason's maternal great-grandfather?
https://www.google.com/search?q=In+Greek+mythology%2C+who+wa...
ideally a model would be able to answer this accurately and completely.
For example, Jason's mother was Philonis, daughter of Mestra, daughter of Daedalion, son of Hesporos. So Jason's maternal great-grandfather was Hesporos.
And eyeballing the benchmarks, it'll probably reach a >50% rate per query by the end of the year. Seems to double every model or two.
I mean I too can complain that my iPhone doesn’t automatically screen out spammers and send my mom flowers on Mother’s Day.
Pixel phone launched in 2016.