Higher Proper ads Philip Hackney University of Louisiana at Lafayette - PowerPoint PPT Presentation

Recent Developments in the Theory of Higher Proper ads Philip Hackney University of Louisiana at Lafayette 3rd Conference on ope - ad Theory and Related Topics Jilin University September 2020

What is... a PROPERAD? (Bruno Vallette 2003, Ross Duncan 2006) Ocntm - l ) Oln ) 0cm ) Operand Kien : OC ! ;D 0cm ;D Pln ;k) ⑦ P(m;j ) - P( ntm - l ; ktj - l ) l > O Proper ad : ① y x Ik rt . Ilk ⑤ Htt > mm O ' l . . ✓ fl → 111 ) - t - - - Il

What are properads for? not bi algebras co algebras , but Oper ads model algebras and can . Proper ads handle job this Mutt consult : can , ¥ to a proper ad with generators : There two is ^ relations : ¥9 = 1¥ I }q , ¥1 = , - Yo lo lo ¥ ? - , are bialgebras proper ad Algebras over this -

major goal Cisinski–Moerdijk–Weiss approach to ∞ -operads using “dendroidal objects” No 2- § a category of built on trees rooted hey What aspects of this approach to ∞ -operads can be adapted to study ∞ -properads?

(directed) graphs (with legs) Data : If E(G) edges O e. ezez I V(G) vertices * ④ in, out: V(G) → ℘ (E(G)) eke 's ihlvt-Eei.ez.es ? o a Conventions : ✓ \ - connected • acyclic ( in directed ) sense J / ) - all edges down point

bad subgraphs

bad subgraphs . . ^ I tie

good subgraphs " . - . * # ) run

good subgraphs I ÷ Re .

structure of Sb(G) the set of “good subgraphs” of G - : 2e le " E(G) → Sb(G) " subgraphs Elementary : a , V(G) → Sb(G) # e) = Ee } in in, out: Sb(G) → ℘ (E(G)) " out Che )= { e ? boundary functions " in (a) = in Lv ) out (G) = oath ) t ECH )uE( K ) C ECG ) Hoke Sb ( G) unions: Do & UH )uVCK ) CVCG ) good subgraph ? determine a E Sb ( G ) Huk this If so , is

Hongyif ) properadic Chu & graphical category me ✓ Γ

properadic graphical category Γ ) ( Chu , H . 2020 Objects: graphs f : G → H Morphisms: E- (G) → E( H ) fo of pair of functions consists : a : Sb ( G) → Sb ( H ) f that so p( ECG ) ) in P( ECG ) ) Sb ( G ) a , commutes / If @ Cfo ) ↳ Plfo ) , , PIEIHI ) SHH ) PIECH ) ) G , then b) If J , Ju k subgraphs of K , good are u f. ( K ) f. ( Jok ) f , SJ ) - - subgraph of H , First example : if a good then G is we have G -74

can be built of of smaller graphs iterated graphs , out unions so by ( b ) axiom f Sb (G) → Sb ( H ) , : on elementary graphs determined by its is value by the composite functions i. e . fo ) - > ECH ) ECG sbtcosfssbtiti , and VCG ) - s SBCG ) # Sb CH )

generators for Γ ÷ ÷ . .

Structure of the category Γ like the simplicial category D → Theorem (H., Robertson, Yau 2015) Γ is a generalized Reedy category the sense of Berger & Moerdijk Ob ( 5) → IN deg : - generated by degeneracy maps T 1- t ( inner & outer ) face maps generated by f Theorem (H., Robertson, Yau 2018) → * Tft With this structure, Γ is an Eilenberg–Zilber category 1- - split epi morphisms = 1- t mono morphisms = f active - bpdneF.gs/TnertEsubgrapLIchsio , → Theorem (Joachim Kock 2016) Γ has an active-inert orthogonal factorization system G → H f : G → H f : active inert - - f fo induces bijection 's isomorphic to is a in (G) E in ( H ) inclusion subgraph out (G) = out ( H ) Face generated by outer face maps generated by Tint degeneracy maps inner face maps and

= simplicial sets ) Γ presheaves = graphical sets ( like D- presheaves → See X : T functors the Segal is determined X XCG ) by the condition just when satisfies its elementary X of values subgraphs on or l y l r V a o l n I of o ¥ € 1 ¥ ' ' A " G subgraphs elementary of G

- x L is n - ¥ ' G subgraphs elementary of G - HYE ) XCG ) - XCX )xX( 97K¥ ) UI lim HH ) elem H

properads 1- " → Set Segal pre sheaf A X : when the map is line X ( H ) X ( G ) a bijection G for all is elem HEG ( or Theorem ( H . Robertson , Yau 2015 ) ) Definition : preheat ( in Set ) thing A is the Segal proper ad same as a . / xxx . Hey , inert E XCG ) - Xtc : ) • x ¢ ← / ' ' "÷÷¥ :S :* y ' Tactive v ← Xfc } ) [ coloured proprad ) If Al

X : TOP -7 Spaces ∞ -properads Segal is just when = line XCH ) XCG ) → WEAK HOMOTOPY EQUIVALENCE is a elem for all G HEG X : TOP -7 Spaces An presheaf - proper ad is x a Reedy fibre - t which is ' X( 2) hcimotopically discrete , has and • is Segal * . xn ,xsXkY ) = ) - Hc : ) XC x / v Xtc ;)

There models for possible other - proper ads : are x x - categories doP → set - orbach • tops > Speed POE f Natan equivalence model just written , ( Chu , H . 2020 ) ( from H . , Robertson 2016 ) down .

⑦ → enriched ∞ -properads FinSet* ✓ x - category ( based on ideas from Hougyichu thesis ) symmetric monoidal W - ( wop x - cat → , ⑧ ) 1- -1 / / ← Cartesian fibration V ✓ op T - Fisette VHT G - VEGH TW of whose decorated Objects graphs vertices are are W of by objects v 111 ⑤ → I HIT U IN

" ( TW ) ( a notion due to Chu - Haugseug ) has algebraic pattern structure an of the Segal condition make so we can sense pre sheaf for a X : ( Tw ) " - Spaces i÷¥ ⇒⇒ ÷÷÷÷÷¥¥*¥* !

Fibre wise represent ability : : W x ( Cam ( - ) ) spay • ) - X ( Cnn ( w ) ) Maplin , W -1 / / picks out colors " x xntm { a . . . . . .am ; b . . . . . ,bn } → X ( 7 ) ) = • E W Mapp ( ai . - → am ; b . . . . . ,bn " - - W ) W-enrichedx-properad-xn.fi Spaces which is • Segal wise representable Fibre •

Thank You !