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solifine Retweeted
For years, the MMDDYY and DDMMYY crowds have pointed out patterns in “the date” that are meaningless to me, but finally today’s date is a palindrome in the one true format, YYYYMMDD!
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I talked about an example of this before, namely when X is the nerve of a category. More can be found in §3.6 of Dyckerhoff–Kapranov's "Higher Segal Spaces". ⌊16⌋https://twitter.com/solifine/status/1213137923870208000 …
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If X happens to be "2-Segal" then this cooperad is "invertible". This means we can reverse the structure maps (which are bijections), and so our simplicial set is also an operad. ⌊15⌋
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For each simplicial set X, we have produced an X(1)-colored cooperad [in (Set,×,∗)]. ⌊14⌋
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For 1≤j≤n, the jth "input color" of x∊X(n) is given by the interval inclusion [1] → [n] in 𝚫 that sends 0 to j-1 and 1 to j. ⌊13⌋
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More specifically, this is an X(1)-colored cooperad. The "output color" of an element x ∊ X(n) is found using the endpoint preserving function [1] → [n] in 𝚫 (that is, 0↦0, 1↦n). ⌊12⌋
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Because the maps γ were not injective, the cocomposition map X(Σmᵢ) → X(r)×(X(m₁)×⋯×X(mᵣ)) lands in a smaller subset. ⌊11⌋
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It follows that the maps γ from above give functions X(Σmᵢ) → X(r)×(X(m₁)×⋯×X(mᵣ)) and the unique map ∅ → [1] gives X(1) → hom(∅,X) = ∗. ⌊10⌋
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Are you satisfied that the standard simplices form an operad? If so, let X be a simplicial set. We know that hom(A⨿B,X) = hom(A,X)×hom(B,X) and X(n) = hom(Δⁿ,X). (here, hom means "simplicial set morphisms", while earlier it meant morphisms in 𝚫) ⌊09⌋
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To see that this is an operad, one must check that the "two ways" of getting from [r] ⨿ (∐_{i=1}^r [mᵢ] ⨿ (∐_{j=1}^{mᵢ} [nᵢⱼ])) to [Σnᵢⱼ] are the same. ⌊08⌋pic.twitter.com/3XKmMFPros
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Define γ(i,r)=m₁+⋯+mᵢ and γ(k,mᵢ)=γ(i-1,r)+k. Here's a picture of where the images of the first few elements land inside of [Σmᵢ]. ⌊07⌋pic.twitter.com/bew6poaFAF
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(ofc you should be careful, bc in real life you could have, like, m₁=m₂=r=5 and then you'll get confused) ⌊06⌋
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To write the operadic multiplications γ: [r] ⨿ ([m₁] ⨿ ⋯ ⨿ [mᵣ]) → [Σmᵢ], maybe it's good to write elements like (k,r) and (k,mᵢ) to specify which component we're starting in. ⌊05⌋
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What are the operad structure maps? The tensor unit for ⨿ is ∅, so we don't have any choice about the "unit" ∅ → Δ¹ = [1] in our operad. ⌊04⌋
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The objects of the simplicial category 𝚫 are [n] = {0 < 1 < ⋯ < n}, for n≥0, and instead of writing Δⁿ for the representable simplicial set hom(-,[n]), let's just write [n]. (Mostly so we can actually write [mᵢ] on twitter!) ⌊03⌋
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First off, the standard simplices Δⁿ (as n≥0 varies) form an operad in simplicial sets. Wait wait, simplicial sets with COPRODUCT as the monoidal product! How does this work? ⌊02⌋
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Every simplicial set X is a (nonsymmetric) cooperad, whose n-ary cooperations are precisely the n-simplices of X. Why? Where do the cooperadic cocompositions come from? ⌊01⌋
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For the pentagon axiom to hold, we need ∂ω=1 as a 4-cochain. As an example when n=2, we can take ω(i,j,k) = -1 when i=j=k=1, and ω(i,j,k)=1 otherwise.
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The fundamental example for this talk is Vec_{ℤ/n}^ω. The objects are elements i,j etc of ℤ/n, and i⊗j ≔ i+j. The interesting part is the associators, which are specified by invertible elements ω(i,j,k) in the ground field.
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Next up is Victor Ostrik with an introduction to fusion categories. In the passage from classical symmetries and quantum symmetries, finite groups are to groups as fusion categories are to (ℂ-)linear tensor categories.
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