Index: Adjunctions

Adjunctions

Routing Summary

This folder covers adjoint functors: definition via hom-set bijection, units and counits, and the universal-property formulation via initial objects in comma categories.

• Need the definition of adjunction or core examples (free/forgetful)? → Adjoint Functors
• Need units, counits, or triangle identities? → Units and Counits
• Need the universal-property / initial-object formulation? → Adjunctions via Initial Objects
• Need results about adjoints and limits? → Adjoints and Limits

Concept Map

Concept	Note	Type	Depends On	Key Result
Adjunction (hom-set)	Adjoint Functors	definition	Functors, Nat. Trans.	$\mathcal{B}(FA,B)\cong \mathcal{A}(A,GB)$ natural
Transpose	Adjoint Functors	definition	Adjoint Functors	$\bar{f}:A\to GB$ from $f:FA\to B$
Adjoint uniqueness	Adjoint Functors	theorem	Adjoint Functors	Adjoints unique up to natural iso (Yoneda)
Unit $\eta$	Units and Counits	definition	Adjoint Functors	$1_{\mathcal{A}}\Rightarrow GF$; universal map
Counit $\epsilon$	Units and Counits	definition	Adjoint Functors	$FG\Rightarrow 1_{\mathcal{B}}$; evaluation map
Triangle identities	Units and Counits	theorem	Units and Counits	$(G\epsilon )\circ (\eta G)=1_G$, etc.
Initial/terminal objects	Adjunctions via Initial Objects	definition	—	Unique maps from/to every object
Comma category	Adjunctions via Initial Objects	definition	Adjoint Functors	$(A\downarrow G)$ = maps $A\to G-$
Adjunction ↔ initial objects	Adjunctions via Initial Objects	theorem	Units and Counits	Unit = initial object of $(A\downarrow G)$
Limits as terminal cones	Adjunctions via Initial Objects	theorem	Adjunctions via Initial Objects	$\lim D$ = terminal in Cone$(D)$

Notes

• Adjoint Functors — CONTAINS: hom-set definition, adjunction table of examples, Galois connections, four equivalent definitions, uniqueness theorem
• Units and Counits — CONTAINS: unit/counit definitions, triangle identities, recovering bijection from unit/counit, free/forgetful and product/hom examples
• Adjunctions via Initial Objects — CONTAINS: initial/terminal object definition, comma category definition, adjunction ↔ initial objects theorem, limits as terminal cones

Sources

• 1612.09375v2.pdf — Basic Category Theory, Ch. 2.1–2.3

See Also

• Adjoints and Limits — Right adjoints preserve limits
• Adjoint Functor Theorems — GAFT and SAFT