Index: Synthesis

Synthesis

Routing Summary

This folder covers Chapter 6 of Leinster, where adjoints, representables, and limits are unified. Each major result connects all three themes.

• Need limits as representable functors, $\Delta ⊣\lim$? → Limits via Representables
• Need pointwise limits in presheaf categories or the density theorem? → Limits in Presheaf Categories
• Need the theorem “right adjoints preserve limits”? → Adjoints and Limits
• Need GAFT or SAFT? → Adjoint Functor Theorems
• Need cartesian closed categories or exponential objects? → Cartesian Closed Categories

Concept Map

Concept	Note	Type	Depends On	Key Result
Cone functor representable	Limits via Representables	theorem	Representables, Limits	$\mathrm{Cone}(A,D)\cong [\mathcal{I},\mathcal{A}](\Delta A,D)$
Limits unique up to iso	Limits via Representables	theorem	Limits via Representables	From Yoneda uniqueness
$\Delta ⊣\lim$	Limits via Representables	theorem	Limits via Representables	Limit functor is right adjoint
Limits commute with limits	Limits via Representables	theorem	Limits via Representables	Prop. 6.2.8
Pointwise limits in $[\mathcal{A},\mathcal{B}]$	Limits in Presheaf Categories	theorem	Functor Categories, Limits	$(\lim X_j)(A)=\lim X_j(A)$
Yoneda preserves limits	Limits in Presheaf Categories	theorem	Limits in Presheaf Categories	Cor. 6.2.12
Category of elements $\mathcal{E}(X)$	Limits in Presheaf Categories	definition	Functor Categories	Objects = $(A,x)$ with $x\in X(A)$
Density theorem	Limits in Presheaf Categories	theorem	Limits in Presheaf Categories	$X\cong {\mathrm{colim}}_{\mathcal{E}(X)}{H}_A$
Right adjoints preserve limits	Adjoints and Limits	theorem	Adjunctions, Limits	Thm. 6.3.1; proof via Yoneda
Left adjoints preserve colimits	Adjoints and Limits	theorem	Adjoints and Limits	Dual of above
Solution-set condition	Adjoint Functor Theorems	definition	Adjunctions	Small set of factorising maps
GAFT	Adjoint Functor Theorems	theorem	Adjoints and Limits	Continuous + solution-set ↔ left adjoint exists
SAFT	Adjoint Functor Theorems	theorem	Adjoint Functor Theorems	Well-powered + cogenerating ↔ continuous has left adjoint
Cartesian closed category	Cartesian Closed Categories	definition	Products, Adjunctions	$-\times B⊣(-{)}^B$; exponential $C^B$
Evaluation map	Cartesian Closed Categories	definition	Cartesian Closed Categories	Counit $C^B\times B\to C$
Set is CCC	Cartesian Closed Categories	example	Cartesian Closed Categories	$C^B$ = function set
CAT is CCC	Cartesian Closed Categories	example	Cartesian Closed Categories	${\mathcal{C}}^{\mathcal{B}}$ = functor category
Presheaf categories are CCC	Cartesian Closed Categories	theorem	Cartesian Closed Categories	$Z^Y(A)=\hat{\mathcal{A}}(H_A\times Y,Z)$
Vect_k is NOT CCC	Cartesian Closed Categories	example	Cartesian Closed Categories	Monoidal closed but not cartesian

Notes

• Limits via Representables — CONTAINS: cones are representable (Prop 6.1.1), limits unique (Cor 6.1.2), $\Delta ⊣\lim$ (Prop 6.1.4), limits commute with limits (Prop 6.2.8)
• Limits in Presheaf Categories — CONTAINS: pointwise limits theorem (Thm 6.2.5), limits commute, Yoneda preserves limits (Cor 6.2.12), category of elements, density theorem (Thm 6.2.17)
• Adjoints and Limits — CONTAINS: right adjoints preserve limits (Thm 6.3.1), proof sketch via Yoneda, examples (forgetful, hom, tensor, free), what is not preserved
• Adjoint Functor Theorems — CONTAINS: AFT for preorders (Prop 6.3.7), solution-set condition, GAFT (Thm 6.3.10), SAFT (Thm 6.3.13), why “special”
• Cartesian Closed Categories — CONTAINS: CCC definition, evaluation map, Set/CAT/presheaf CCC, Vect_k not CCC, monoidal closed, connection to lambda calculus

Sources

• 1612.09375v2.pdf — Basic Category Theory, Ch. 6.1–6.3 and Appendix

See Also

• Representables — Yoneda lemma used throughout synthesis
• Adjunctions — Adjunctions used in all synthesis results
• Limits and Colimits — Limits theory underlying synthesis