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For a small category, the Yoneda functor preserves and reflects all existing small limits
Statement
For a small category , its Yoneda embedding preserves and reflects every small limit that is defined in .
Facts & Assumptions
Given: A small category and a small diagram in it.
The Yoneda embedding sends to (The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding).
Covariant representable functors preserve small limits (Every covariantly representable functor to Set preserves all existing small limits).
Limits in functor categories are computed pointwise (For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise).
Fully faithful functors reflect limits (Fully faithful functors reflect limits and colimits).
The Yoneda embedding is fully faithful (The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective).
Proof
At , evaluation of the Yoneda image of a limiting cone is its image under by [F1]. This is a limiting cone of sets by [L1].
Since every evaluation is limiting, [L2] says the Yoneda-image cone is a limit in the presheaf category. Thus Yoneda preserves the small limit.
By [L4], Yoneda is fully faithful, so [L3] makes it reflect every limit. The smallness of ensures that the stated presheaf category and embedding are formed under the library convention.
Depends on
- Every covariantly representable functor to Set preserves all existing small limits
- For small source and index categories, chosen target limits and colimits compute the corresponding functor-category limits and colimits pointwise
- Fully faithful functors reflect limits and colimits
- The Yoneda assignment and the small-source Yoneda functor, traditionally called the Yoneda embedding
- The Yoneda functor is fully faithful, and it is a full embedding when its object map is injective
Used by
- FALSE: the Yoneda embedding preserves colimits False statement
Dependency tree · two levels
22 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- E. Riehl, Category Theory in Context, Theorem 3.5.5 (standard reference, not scraped)