1.Introduction
1.1 Summary

2 Preliminaries
2.1 Bicategories
2.2 Monads and adjunctions
2.2.1 Comparison functors
2.2.2 Distributive laws
2.3 String diagrams
2.4 Monoidal and module categories
2.4.1 Braidings
2.4.2 Closedness
2.4.3 Rigidity and pivotality 
2.4.4 The Drinfeld centre 
2.5 Linear and abelian categories 
2.5.1 Finite and locally finite abelian categories
2.5.2 Tensor and ring categories 
2.6 Algebra and module objects
2.7 Monoidal bicategories 
2.7.1 String diagrams in monoidal bicategories
2.8 Coends
2.9 (Co)completions 

3 Duality theory for monoidal categories
3.1 Tensor representability 
3.1.1 Grothendieck–Verdier duality 
3.1.2 The free tensor representable category is not rigid 
3.2 Functor categories
3.2.1 Cauchy completions
3.3 Applications 
3.3.1 Boolean algebras
3.3.2 Mackey functors 
3.3.3 Crossed modules

4 Twisted centres
4.1 Heaps 
4.2 Pivotal structures and twisted centres 
4.2.1 Twisted centres and their Picard heaps 
4.2.2 Quasi-pivotality 
4.2.3 Pivotality of the Drinfeld centre 

5 Monadic Tannaka–Krein reconstruction
5.1 Bimonads 
5.2 Hopf monads 
5.3 (Co)module monads 
5.3.1 Reconstruction for comodule monads 

6 Monadic twisted centres
6.1 Cross products and distributive laws 
6.2 Centralisable functors and the central bimonad 
6.3 Centralisers and comodule monads 
6.4 The Drinfeld and anti-Drinfeld double of a Hopf monad 
6.5 Pairs in involution for Hopf monads 

7 Duoidal R-matrices
7.1 Duoidal categories 
7.1.1 Double opmonoidal monads 
7.2 R-matrices 
7.2.1 From R-matrices to duoidal structures and back 
7.3 Linearly distributive monads 

8 Infinite and non-rigid reconstruction
8.1 Extending module structures 
8.1.1 Extendable monads 
8.1.2 Semisimple monads 
8.1.3 Linton coequalisers via multiactegories 
8.2 Internal projective and injective objects 
8.3 Reconstruction for lax module endofunctors 
8.3.1 Rigid monoidal and (finite) tensor categories 
8.4 An Eilenberg–Watts theorem for lax module monads 

9 Hopf trimodules
9.1 Bicomodules and their graphical calculus 
9.2 From Hopf trimodules to lax module functors 
9.3 Contramodule reconstruction over Hopf trimodule algebras 
9.3.1 Morita equivalence for contramodules 
9.4 A semisimple example of non-rigid reconstruction 
9.5 A Hopf trimodule algebra constructing the fibre functor 
9.6 The fundamental theorem of Hopf modules 
9.6.1 Hopf trimodules and twisted antipodes 
9.6.2 Fusion operators for Hopf monads 

Bibliography