4.2 Unit and counit of adjunction

Adjunctions are powerful tools in category theory, linking functors between categories. The unit and counit of an adjunction provide a way to measure how close these categories are to being retracts of each other.

Understanding units and counits is crucial for grasping adjunctions. These natural transformations, along with triangle identities, form the foundation for constructing and analyzing adjunctions, helping us explore relationships between different mathematical structures.

Unit and Counit of Adjunction

Unit and counit of adjunctions

Top images from around the web for Unit and counit of adjunctions

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  ∞-topos - Wikipedia View original

  Is this image relevant?

• 

  Adjoint functors - Wikipedia View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  ∞-topos - Wikipedia View original

  Is this image relevant?

1 of 3

Top images from around the web for Unit and counit of adjunctions

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  ∞-topos - Wikipedia View original

  Is this image relevant?

• 

  Adjoint functors - Wikipedia View original

  Is this image relevant?

• 

  L∞-algebras and their cohomology | Emergent Scientist View original

  Is this image relevant?

• 

  ∞-topos - Wikipedia View original

  Is this image relevant?

1 of 3

• Unit of adjunction
  • Natural transformation $\eta: 1_C \to GF$ maps identity functor on C to composition of F and G
  • Maps objects in category C to objects in C via GF creates a "round trip" through D
  • For each object $X$ in C, $\eta_X: X \to GF(X)$ represents the unit's action on X
• Counit of adjunction
  • Natural transformation $\epsilon: FG \to 1_D$ maps composition of F and G to identity functor on D
  • Maps objects in category D to objects in D via FG creates a "round trip" through C
  • For each object $Y$ in D, $\epsilon_Y: FG(Y) \to Y$ represents the counit's action on Y
• Naturality of unit
  • For any morphism $f: X \to X'$ in C, the following diagram commutes: $X \xrightarrow{\eta_X} GF(X)$ $\downarrow f \quad \quad \downarrow GF(f)$ $X' \xrightarrow{\eta_{X'}} GF(X')$
  • Ensures consistency of unit with morphisms in C (preserves structure)
• Naturality of counit
  • For any morphism $g: Y \to Y'$ in D, the following diagram commutes: $FG(Y) \xrightarrow{FG(g)} FG(Y')$ $\downarrow \epsilon_Y \quad \quad \downarrow \epsilon_{Y'}$ $Y \xrightarrow{g} Y'$
  • Ensures consistency of counit with morphisms in D (preserves structure)

Triangle identities for adjunctions

• Triangle identity for F
  • Composition $F \xrightarrow{F\eta} FGF \xrightarrow{\epsilon F} F$ equals identity on F
  • Ensures F "cancels out" the round trip through G (Set, Group)
• Triangle identity for G
  • Composition $G \xrightarrow{\eta G} GFG \xrightarrow{G\epsilon} G$ equals identity on G
  • Ensures G "cancels out" the round trip through F (Vector spaces, Modules)
• Proof strategy
  1. Use naturality of unit and counit to manipulate diagrams
  2. Apply definitions of functor composition to simplify expressions
  3. Utilize properties of identity morphisms to complete the proof
  4. Consider specific examples (Free group functor, Forgetful functor)

Correspondence of adjunctions vs natural transformations

• Adjunction as a quadruple $(F, G, \eta, \epsilon)$ defines relationship between categories
  • F: C → D (left adjoint) maps objects and morphisms from C to D
  • G: D → C (right adjoint) maps objects and morphisms from D to C
  • $\eta: 1_C \to GF$ (unit) measures how far C is from being a retract of D
  • $\epsilon: FG \to 1_D$ (counit) measures how far D is from being a retract of C
• Bijective correspondence establishes equivalence between adjunctions and natural transformations
  • Given an adjunction, unit and counit automatically satisfy triangle identities
  • Given unit and counit satisfying triangle identities, construct unique adjunction (Free-forgetful, Product-diagonal)
• Hom-set isomorphism $\text{Hom}_D(F(X), Y) \cong \text{Hom}_C(X, G(Y))$ relates morphisms in C and D
  • Bijection between morphisms F(X) → Y in D and morphisms X → G(Y) in C
  • Relationship to unit and counit through natural transformations
• Uniqueness of adjunction demonstrates fundamental nature of the concept
  • Prove that any two adjunctions with the same unit (or counit) are isomorphic
  • Implies adjunctions are determined by their unit or counit (up to isomorphism)

Construction of adjunctions from units

• Construction from unit

  1. Given $\eta: 1_C \to GF$, define left adjoint F using universal property
  2. Construct counit $\epsilon$ using properties of G and $\eta$
  3. Verify triangle identities to ensure adjunction properties

• Construction from counit

  1. Given $\epsilon: FG \to 1_D$, define right adjoint G using universal property
  2. Construct unit $\eta$ using properties of F and $\epsilon$
  3. Verify triangle identities to ensure adjunction properties

• Universal property approach leverages categorical definitions

  • Use unit to define universal arrows from X to G (initial objects)
  • Use counit to define universal arrows from F to Y (terminal objects)

• Examples of adjunction construction demonstrate practical applications

  • Free-forgetful adjunction (groups and sets)
  • Product-diagonal adjunction (cartesian product and diagonal functor)
  • Exponential-product adjunction (function spaces and cartesian product)