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solifine
@solifine
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2. velj |
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i wrote an abstract about a segal operad yesterday
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solifine
@solifine
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2. velj |
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davidad 🎇
@davidad
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2. velj |
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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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solifine
@solifine
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2. velj |
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you should do one explaining how a simplicial set determines a cooperad
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solifine
@solifine
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1. velj |
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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⌋
twitter.com/solifine/statu…
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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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For each simplicial set X, we have produced an X(1)-colored cooperad [in (Set,×,∗)].
⌊14⌋
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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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solifine
@solifine
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1. velj |
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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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