Index: Limits and Colimits

Limits and Colimits

Routing Summary

This folder covers the theory of limits (products, equalizers, pullbacks, general limits) and their duals (colimits). Also covers how functors interact with limits.

• Need products, equalizers, terminal objects? → Products and Equalizers
• Need pullbacks, pushouts, or the mono-pullback characterisation? → Pullbacks
• Need the general limit definition, diagrams, cones? → General Limits
• Need colimits, coproducts, coequalizers, pushouts in detail? → Colimits
• Need preservation/reflection/creation of limits by functors? → Functors and Limits

Concept Map

Concept	Note	Type	Depends On	Key Result
Binary product $A\times B$	Products and Equalizers	definition	Categories	Universal property with projections
Terminal object	Products and Equalizers	definition	—	Empty product; unique map into it
Equalizer	Products and Equalizers	definition	—	Kernel-like construction $E\to X\rightrightarrows Y$
Monomorphism	Products and Equalizers	definition	—	Categorical injection
Products + equalizers → all limits	Products and Equalizers	theorem	Products and Equalizers	Prop. 5.1.26
Pullback $X{\times }_{Z}Y$	Pullbacks	definition	Products and Equalizers	Fibred product; universal square
Mono ↔ pullback square	Pullbacks	theorem	Pullbacks	$f$ mono iff diagonal iso
Pushout	Pullbacks	definition	General Limits	Dual of pullback; span colimit
Diagram $D:\mathcal{I}\to \mathcal{A}$	General Limits	definition	Functors	Shape category $\mathcal{I}$, diagram is a functor
Cone	General Limits	definition	General Limits	Family of maps into diagram, compatible
Limit = terminal cone	General Limits	definition	General Limits	Unique factorisation of all cones
Limit formula in Set	General Limits	example	General Limits	Compatible families in $\prod DI$
Complete category	General Limits	definition	General Limits	Has all small limits
Colimit = initial cocone	Colimits	definition	General Limits	Dual of limit
Coproduct $A\bigsqcup B$	Colimits	definition	—	Dual of product; injections
Coequalizer	Colimits	definition	—	Quotient-like construction
Colimit formula in Set	Colimits	example	Colimits	Quotient of coproduct
Epimorphism	Colimits	definition	—	Categorical surjection
Preserves limits	Functors and Limits	definition	General Limits	$F(\lim D)\cong \lim (FD)$
Reflects limits	Functors and Limits	definition	Functors and Limits	Limit in codomain → limit in domain
Creates limits	Functors and Limits	definition	Functors and Limits	Forgetful functors typically create
Right adjoints continuous	Functors and Limits	theorem	Functors and Limits	Proved in Adjoints and Limits

Notes

• Products and Equalizers — CONTAINS: binary products, arbitrary products, terminal objects, equalizers, products+equalizers generate all limits (Prop 5.1.26), monomorphisms
• Pullbacks — CONTAINS: pullback definition with commutative square, examples in Set/Top/Grp, mono ↔ pullback square lemma, pushout definition, pasting lemma
• General Limits — CONTAINS: diagram and cone definitions, limit = terminal cone, limit formula in Set, complete categories, products+equalizers generate limits
• Colimits — CONTAINS: cocone and colimit definitions, coproducts, coequalizers, colimit formula in Set, epimorphisms, cocomplete categories
• Functors and Limits — CONTAINS: preservation/reflection/creation definitions, forgetful functor creates limits example, right adjoints are continuous

Sources

• 1612.09375v2.pdf — Basic Category Theory, Ch. 5.1–5.3

See Also

• Synthesis — Limits via representables, adjoints and limits, functor theorem