Alphabeta MathUniversal Properties, Representables and the Yoneda Lemma — Examples
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✓ 14 results · all verified · 0 also independently AI-judged
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1 · Prerequisites
Abstract Algebra
Category Theory
Real Analysis
Set Theory
2 · Logical flowchart
3 · Items
Example
A Cartesian product represents X mapstoSet(X,A)timesSet(X,B)
Example
A tagged disjoint union represents X mapstoSet(A,X)timesSet(B,X)
Example
The function set B^A represents X mapstoSet(X× A,B)
Example
The free word monoid on X represents M mapstoSet(X,U(M))
Example
The free group on X represents G mapstoSet(X,U(G))
Example
ℤ[x] represents the underlying-set functor on unital rings
Example
The one-point space represents the underlying-set functor on Top
Example
A representable presheaf on a poset is the indicator of a principal down-set
Counterexample
The functor D(X)=X⊔ X on Set is not covariantly representable
Example
Two singleton sets give canonically isomorphic representations of the identity functor on Set
Example
The Yoneda embedding of the walking-arrow category computed objectwise
Example
For a monoid action, Yoneda says that an equivariant map from the regular action is determined by the identity element
False statement
The Yoneda lemma requires its category to be small
False statement
Non-isomorphic objects can have naturally isomorphic representable presheaves
Sources
- Justin Campbell, Harvard Math 55b tutorial notes, Example 1.1 and Definition 2.1
- Tom Leinster, Basic Category Theory, Section 3.1 (sums of sets)
- Emily Riehl, Category Theory in Context, Example 2.1.6(iv)
- David I. Spivak, Category Theory for Scientists, Definition 3.1.1.15 and Proposition 3.1.4.9
- Tom Leinster, Basic Category Theory, Examples 1.2.4(a) and 2.1.3(b)
- Emily Riehl, Category Theory in Context, Example 2.4.12(vi)
- Emily Riehl, Category Theory in Context, Example 2.1.5(ii)
- Tom Leinster, Basic Category Theory, Example 1.1.8(e) and Definitions 4.1.16--4.1.17
- Justin Campbell, Harvard Math 55b tutorial notes, Example 2.3
- Emily Riehl, Category Theory in Context, Example 2.1.5(i) and Corollary 2.3.2
- Tom Leinster, Basic Category Theory, Definition 4.1.21 and Corollary 4.3.7
- Emily Riehl, Category Theory in Context, Proposition 2.2.3 and Theorem 2.2.4
- Emily Riehl, Category Theory in Context, Theorem 2.2.4 and Remark 2.2.7
- Emily Riehl, Category Theory in Context, Proposition 2.3.1
- Tom Leinster, Basic Category Theory, Corollary 4.3.10