Alphabeta Math

Limits and Colimits: Examples and Counterexamples

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✓ 17 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Category Theory
Combinatorics
Real Analysis
Set Theory
Topology
2 · Logical flowchart

3 · Items

Example

Products in Set are Cartesian products and coproducts are tagged disjoint unions

Example

Equalizers in Set are agreement subsets and coequalizers are quotients by the generated equivalence relation

Example

Pullbacks in Set are fibre products and pushouts are quotients of tagged disjoint unions

Example

A pullback in Top is the fibre product with the subspace topology inherited from the product

Example

The equalizer of two group homomorphisms is their agreement subgroup

Example

The colimit of an increasing chain of sets is its union

Example

In a poset regarded as a category, products are infima, coproducts are suprema, and equalizers are automatic

Example

The singleton set and trivial group are terminal, while the empty set and trivial group are initial

Counterexample

Assuming Choice, nonempty sets have all small products but a parallel pair with no equalizer and hence a diagram with no limit

Counterexample

A monotone functor between poset categories preserves every monomorphism but need not preserve pullbacks

Counterexample

A limit in a full subcategory need not be the ambient limit

Counterexample

Filtered colimits in Set need not commute with countably infinite products

False statement

FALSE: every category has all small limits

False statement

FALSE: a functor preserving binary products and equalizers must preserve all finite limits

False statement

FALSE: the underlying-set functor Top→Set fails to preserve some small limit

False statement

FALSE: colimits in Grp are computed by taking the Set-colimit of the underlying diagram

Example

Under the definable-class diagram convention, the empty set is the product of the large family of all sets

Sources