🔢Category Theory Unit 8 – Colimits and Kan Extensions

Key Concepts and Definitions

• Category theory studies objects and morphisms between them in an abstract setting
• Colimits generalize the concept of gluing objects together in a category
• Kan extensions provide a way to extend functors along natural transformations
• Diagrams in a category consist of objects and morphisms arranged in a specific shape
• Cocones are special objects and morphisms that sit above a diagram and satisfy certain properties
• Universal properties characterize objects and morphisms uniquely up to isomorphism
• Adjunctions are relationships between functors that form a natural bijection between certain sets of morphisms

Colimits: The Basics

• Colimits are dual to limits and represent the gluing together of objects in a category
• Colimits are defined by a universal property involving cocones over a diagram
• The colimit object is the apex of the universal cocone and receives a unique morphism from any other cocone
• Colimits can be thought of as a generalization of various constructions (coproducts, coequalizers, pushouts)
• The existence and uniqueness of colimits depend on the specific category being considered
  • In $Set$, colimits always exist and can be constructed explicitly
  • In other categories, additional conditions may be required for colimits to exist

Types of Colimits

• Initial objects are colimits of the empty diagram and represent objects with a unique morphism to every other object
• Coproducts are colimits of discrete diagrams and generalize the disjoint union of sets
• Coequalizers are colimits of parallel pair diagrams and identify elements based on a given relation
• Pushouts are colimits of span diagrams and capture the gluing of objects along a common subobject
• Directed colimits are colimits of diagrams indexed by directed categories and allow for the construction of infinite objects
  • Examples include colimits of sequences (chains) and colimits of directed systems
• Filtered colimits are a special case of directed colimits that preserve certain finite limits

Constructing Colimits

• Colimits can be constructed explicitly in certain categories using specific techniques
• In $Set$, coproducts are constructed as disjoint unions and coequalizers by taking quotients
• Pushouts in $Set$ are constructed by taking the disjoint union and then identifying elements based on the span morphisms
• In categories with all small colimits, colimits can be constructed using coproducts and coequalizers
  • This is known as the co-Yoneda lemma and provides a general recipe for constructing colimits
• In $Ab$ (abelian groups), colimits can be constructed using direct sums and cokernels
• In $Top$ (topological spaces), colimits are constructed using disjoint unions and quotient spaces

Introduction to Kan Extensions

• Kan extensions provide a way to extend functors along natural transformations
• Given functors $F: C \to D$ and $G: C \to E$, a Kan extension of $F$ along $G$ is a functor $K: E \to D$ with a natural transformation $\alpha: F \Rightarrow K \circ G$
• Kan extensions can be thought of as the best approximation of a functor along a given natural transformation
• Kan extensions are characterized by a universal property involving certain natural transformations
• The existence and uniqueness of Kan extensions depend on the specific categories and functors involved
  • In categories with all small limits and colimits, Kan extensions always exist

Left and Right Kan Extensions

• There are two types of Kan extensions: left and right
• Left Kan extensions are characterized by an initial property and can be constructed as colimits in the target category
  • Given $F: C \to D$ and $G: C \to E$, the left Kan extension $Lan_G(F): E \to D$ satisfies $Lan_G(F)(e) = colim_{(c,f:Gc \to e)} F(c)$
• Right Kan extensions are characterized by a terminal property and can be constructed as limits in the target category
  • Given $F: C \to D$ and $G: C \to E$, the right Kan extension $Ran_G(F): E \to D$ satisfies $Ran_G(F)(e) = lim_{(c,f:e \to Gc)} F(c)$
• The universal property of Kan extensions provides a bijection between certain sets of natural transformations
  • For left Kan extensions: $Nat(Lan_G(F),H) \cong Nat(F,H \circ G)$
  • For right Kan extensions: $Nat(H,Ran_G(F)) \cong Nat(H \circ G,F)$

Applications and Examples

• Kan extensions have numerous applications in various areas of mathematics and computer science
• In representation theory, Kan extensions are used to construct induced representations and extend representations along group homomorphisms
• In algebraic topology, Kan extensions are used to define cohomology theories and extend them along continuous maps
• In category theory itself, Kan extensions are used to construct adjoint functors and establish equivalences between categories
  • The Yoneda lemma can be seen as a special case of Kan extensions
• In functional programming, Kan extensions are used to define free monads and extend computations along natural transformations
• Specific examples of Kan extensions include:
  • The nerve functor $N: Cat \to sSet$ is a right Kan extension of the Yoneda embedding along the inclusion $\Delta \to Cat$
  • The geometric realization functor $|-|: sSet \to Top$ is a left Kan extension of the standard simplex functor along the Yoneda embedding

Common Challenges and Solutions

• Computing Kan extensions explicitly can be challenging, especially in categories without nice colimit or limit constructions
  • In such cases, one may need to resort to more abstract characterizations or use adjunctions to simplify the computation
• Verifying the universal property of Kan extensions can be tedious and require careful diagram chasing
  • Using the Yoneda lemma or other categorical tools can help simplify the verification process
• Determining the existence of Kan extensions in a given situation may require additional assumptions on the categories and functors involved
  • Knowing common sufficient conditions (e.g., the existence of all small limits and colimits) can help in establishing the existence of Kan extensions
• Understanding the relationship between left and right Kan extensions and their dual nature can be confusing at first
  • Focusing on the universal properties and the colimit/limit constructions can help clarify the distinction and the duality between them
• Applying Kan extensions to specific problems may require some creativity and insight
  • Looking at examples from various fields and understanding the general principles behind Kan extensions can provide guidance and inspiration for problem-solving