Higher representations and cornered Heegaard Floer homologyThanks: The first and second author thank the NSF for its support (grant DMS-1702305).
The first author gratefully acknowledges support from the NSF (grant DMS-1502686).
The second author gratefully acknowledges support from the NSF
(grant DMS-1161999) and from the Simons Foundation (grant #376202).
This material is based upon work supported by the NSF (Grant No. 1440140),
while the second author was in residence at the Mathematical Sciences Research
Institute in Berkeley, California, during the Spring 2018.
Andrew Manion
Address: A.M.: Department of Mathematics, North Carolina State University, 2108 SAS Hall, Raleigh, NC 27695, USA.
Email address: ajmanion@ncsu.edu
and
Raphaël Rouquier
Address: R.R.: UCLA Mathematics Department, Los Angeles, CA 90095-1555, USA.
Email address: rouquier@math.ucla.edu
Original source: arXiv:2009.09627v2
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Contents 1. Introduction1.1. Higher representations1.2. Higher tensor products1.3. Fukaya categories1.4. Heegaard Floer homology1.5. Singular curves1.6. Extended TQFT and further remarks1.7. Structure of the article1.8. Acknowledgments2. Differential and pointed structures2.1. Differential algebras and categories2.1.1. Categories2.1.2. Differential categories2.1.3. Objects2.1.4. Algebras2.1.5. G G 2.2. Bimodules2.2.1. Algebras2.2.2. Categories2.2.3. Bimodules and functors2.3. Pointed sets and categories2.3.1. Pointed sets2.3.2. Gradings and filtrations2.3.3. Pointed categories2.3.4. Differential pointed categories2.3.5. Pointed structures as 𝐅 2 {\mathbf{F}}_{2} 2.4. Symmetric powers3. Hecke algebras3.1. Differential graded nil Hecke algebras3.1.1. Coxeter groups3.1.2. Hecke algebras3.1.3. Traces3.1.4. Nil Hecke algebras3.1.5. Differential3.1.6. Differential graded pointed Hecke monoid3.2. Extended affine symmetric groups3.2.1. Finite case3.2.2. Definition3.2.3. Diagrammatic representation3.2.4. Length3.2.5. Extended affine Hecke algebra3.2.6. Positive versions3.2.7. Pointed versions4. 2 2 4.1. Monoidal category4.1.1. Definition4.1.2. 2 2 4.1.3. Pointed case4.2. Lax cocenter4.2.1. Lax bi-2 2 4.2.2. Category4.3. Diagonal action4.3.1. Category4.3.2. 1 1 4.3.3. 2 2 4.3.4. Functoriality4.3.5. Associativity4.4. Dual diagonal action4.4.1. Category4.4.2. Adjoint4.4.3. Relations4.4.4. 1 1 4.4.5. 2 2 4.5. Tensor product and internal Hom \operatorname{Hom}\nolimits 5. Bimodule 2 2 5.1. Differential algebras5.1.1. 2 2 5.2. Lax cocenter5.3. Diagonal action5.3.1. Algebra5.3.2. Left dual5.3.3. Action5.3.4. Tensor product case5.4. Dual diagonal action5.4.1. Algebra5.4.2. Left dual5.5. Differential categories5.5.1. Bimodule 2 2 5.5.2. Diagonal action5.6. Pointed categories5.7. Douglas-Manolescu’s algebra-modules6. Hecke 2 2 6.1. Regular 2 2 6.1.1. Bimodules6.1.2. Twisted description6.1.3. Actions6.1.4. Gluing6.2. Nil Hecke category6.2.1. Definition6.2.2. Length6.2.3. Filtration6.2.4. Non-commutative degree6.2.5. Differential6.2.6. Change of n n 6.3. Positive and finite variants6.3.1. Constructions6.3.2. Lipshitz-Ozsváth-Thurston’s strands algebras7. Strand algebras7.1. 1 1 7.1.1. Definitions7.1.2. Morphisms7.1.3. Quotients7.1.4. Paths7.1.5. Tangential multiplicity7.2. Curves7.2.1. Definitions7.2.2. Morphisms and subcurves7.2.3. Quotients7.2.4. Chord diagrams as singular curves7.2.5. Sutured surfaces and topological field theories7.3. Paths7.3.1. Admissible paths7.3.2. Pointed category of admissible paths7.3.3. Central extension7.3.4. Functoriality7.3.5. Pullback7.3.6. One strand bordered algebras7.3.7. Intersection multiplicity7.4. Strands7.4.1. Braids7.4.2. Degree7.4.3. Strands on S 1 S^{1} 7.4.4. Strand category7.4.5. Generation7.4.6. Decomposition at a point7.4.7. Differential7.4.8. Strands on non-singular curves7.4.9. Products and divisibility7.4.10. Subcurves7.4.11. Bordered Heegaard Floer algebras7.4.12. Fukaya categories from strand algebras8. 2 2 8.1. Action on ends of curves8.1.1. Definition8.1.2. Approximation8.1.3. 2 2 8.1.4. Twisted object description8.1.5. Right action8.1.6. Duality8.1.7. Actions for the line8.1.8. Action as functors8.1.9. Action on Fukaya categories8.2. Gluing8.2.1. Construction8.2.2. Bimodules8.2.3. Gluing map8.2.4. Equivalence relation8.2.5. Complement8.2.6. Large enough M M 8.2.7. Functoriality8.3. Diagonal action8.3.1. Isomorphism Theorem8.3.2. Setting8.3.3. Diagonal bimodule8.3.4. Matching of extended action8.3.5. Action of τ \tau References Read the whole chapter