Gurski N. Coherence in Three-Dimensional Category Theory

Cambridge: Cambridge University Press, 2013. — 288 p.Higher category theory is an increasingly important discipline with applications in topology, geometry, logic and theoretical computer science. This comprehensive treatment covers essential material for any student of coherence, or for any researcher wishing to apply higher categories or coherence results in fields such as algebraic topology.Tricategories
Gray-monads
An outline
Acknowledgements
Background
Bicategorical background
Bicategorical conventions
Mates in bicategories
Coherence for bicategories
The Yoneda embedding
Coherence for bicategories
Coherence for functors
Gray-categories
The Gray tensor product
Cubical functors
The monoidal category Gray
A factorization
Tricategories
The algebraic definition of tricategory
Basic definition
Adjoint equivalences and tricategory axioms
Trihomomorphisms and other higher cells
Unpacked versions
Calculations in tricategories
Comparing definitions
Examples
Primary example: Bicat
Fundamental -groupoids
Free constructions
Graphs
The category of tricategories
Free Gray-categories
Basic structure
Structure of functors
Structure of transformations
Pseudo-icons
Change of structure
Triequivalences
Gray-categories and tricategories
Cubical tricategories
Gray-categories
Coherence via Yoneda
Local structure
Global results
The cubical Yoneda lemma
Coherence for tricategories
Coherence via free constructions
Coherence for tricategories
Coherence and diagrams of constraints
A non-commuting diagram
Strictifying tricategories
Coherence for functors
Strictifying functors
Gray-monads
Codescent in Gray-categories
Lax codescent diagrams
Codescent diagrams
Codescent objects
Codescent as a weighted colimit
Weighted colimits in Gray-categories
Examples: coinserters and coequifiers
Codescent
Gray-monads and their algebras
Enriched monads and algebras
Lax algebras and their higher cells
Total structures
The reflection of lax algebras into strict algebras
The canonical codescent diagram of a lax algebra
The left adjoint, lax case
The left adjoint, pseudo case
A general coherence result
Weak codescent objects
Coherence for pseudo-algebras