U tweetove putem weba ili aplikacija drugih proizvođača možete dodati podatke o lokaciji, kao što su grad ili točna lokacija. Povijest lokacija tweetova uvijek možete izbrisati. Saznajte više
Yesterday we saw that there is a unique monochrome invertible operad. Today let's show that every category determines an invertible operad (that is, there is a fully-faithful functor Cat → iOp). [1/7]https://twitter.com/solifine/status/1212892580452732928 …
Note that this cannot be the usual inclusion of categories into operads (as operads concentrated in arity one), since we saw yesterday that the underlying category of an invertible operad is discrete. [2/7]https://twitter.com/solifine/status/1212894189593223168 …
Where does this come from? It looks like this: Cat ≃ 1-Seg ⊂ 2-Seg ≃ iOp where the second equivalence is the Dyckerhoff–Kapranov theorem from yesterday, and the first comes from taking the nerve of a category. [3/7]https://twitter.com/solifine/status/1212894498419888128 …
Let's just give the answer and check that it works. Given a category C, and morphisms f₁,f₂,…,fᵣ,g, define O(f₁,f₂,…,fᵣ;g) to be either empty or a point. It will be a point just when g=fᵣ∘fᵣ₋₁∘⋯∘f₂∘f₁. [4/7]
This should be suitably interpreted when r=0, that is, we should set O( ;idₓ) = ∗ and O( ;g) = ∅ whenever g is not an identity. [5/7]
Now we see what the operadic composition should be - there is only one choice! Both sides of this isomorphism are either a point or empty. [6/7]pic.twitter.com/nYSsR2Wije
In summary, every category C determines an operad O where the set of colors of O is the set of arrows of C and where elements of O(r) are strings of r composable arrows of C. This operad is invertible. [7/7]
Twitter je možda preopterećen ili ima kratkotrajnih poteškoća u radu. Pokušajte ponovno ili potražite dodatne informacije u odjeljku Status Twittera.
