Index: Representables

Representables

Routing Summary

This folder covers representable functors and the Yoneda lemma — the central theorem of the book. Representability unifies all universal properties.

• Need hom-functors $H_A$, representable functors, generalized elements? → Representable Functors
• Need the Yoneda bijection $[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,X)\cong X(A)$? → Yoneda Lemma
• Need the Yoneda embedding, full faithfulness, or uniqueness of representations? → Yoneda Embedding and Consequences

Concept Map

Concept	Note	Type	Depends On	Key Result
Covariant hom functor $H^A$	Representable Functors	definition	Functors	$\mathcal{A}(A,-):\mathcal{A}\to \mathbf{Set}$
Contravariant hom functor $H_A$	Representable Functors	definition	Functors	$\mathcal{A}(-,A):{\mathcal{A}}^{\mathrm{op}}\to \mathbf{Set}$
Representable functor	Representable Functors	definition	Functor Categories	$F\cong \mathcal{A}(A,-)$ for some $A$
Generalized element	Representable Functors	definition	Representable Functors	Map $S\to A$ = element of $A$ of shape $S$
Universal element	Representable Functors	theorem	Representable Functors	Rep. = universal element $u\in F(A)$
Yoneda lemma	Yoneda Lemma	theorem	Representable Functors	$[{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,X)\cong X(A)$ natural
Yoneda bijection	Yoneda Lemma	theorem	Yoneda Lemma	$\alpha \mapsto {\alpha }_A(1_A)$; $x\mapsto {\tilde{x}}_B(f)=(Xf)(x)$
Yoneda embedding $H^{\bullet }$	Yoneda Embedding and Consequences	definition	Yoneda Lemma	$A\mapsto H_A=\mathcal{A}(-,A)$, functor $\mathcal{A}\to [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}]$
Yoneda embedding fully faithful	Yoneda Embedding and Consequences	theorem	Yoneda Lemma	$\mathcal{A}(A,B)\cong [{\mathcal{A}}^{\mathrm{op}},\mathbf{Set}](H_A,H_B)$
Uniqueness of representations	Yoneda Embedding and Consequences	theorem	Yoneda Embedding and Consequences	$H_A\cong H_{A^{\prime }}$ iff $A\cong A^{\prime }$
Representables preserve limits	Yoneda Embedding and Consequences	theorem	Yoneda Embedding and Consequences	$\mathcal{A}(A,\lim D)\cong \lim \mathcal{A}(A,D-)$

Notes

• Representable Functors — CONTAINS: hom-functor definitions, representability definition, generalized elements, examples (forgetful functor on Grp, non-representable example), universal element theorem
• Yoneda Lemma — CONTAINS: full statement of Yoneda lemma with proof sketch, covariant version, Yoneda and representability corollary, product example, naturality
• Yoneda Embedding and Consequences — CONTAINS: Yoneda embedding definition, full faithfulness theorem, uniqueness of representations, adjoints unique, tensor product unique, Yoneda preserves limits

Sources

• 1612.09375v2.pdf — Basic Category Theory, Ch. 4.1–4.3

See Also

• Limits via Representables — Limits as representable functors (Ch. 6.1)
• Limits in Presheaf Categories — Density theorem