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A conical enriched limit is stronger than a limit in the underlying category
Statement
Whenever a conical enriched limit exists, its image in the underlying ordinary category is an ordinary limit.
Facts & Assumptions
Given: A diagram in a -category.
A conical enriched limit is an enriched weighted limit of constant-unit shape (Conical enriched limit).
Passing to the underlying category can lose enriched hom-object information (The underlying category can lose genuinely enriched information).
Proof
If is a conical enriched limit, then the enriched universal morphism gives, after applying the underlying-hom functor to each hom-object, exactly the ordinary cone bijection in the underlying category. So every conical enriched limit is an ordinary limit after forgetting enrichment.
So the conical enriched universal property is stronger than the underlying ordinary one: once the former exists, the latter follows by forgetting to global elements of the enriched hom-objects.
Depends on
Used by
Dependency tree · two levels
5 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
- Emily Riehl, Categorical Homotopy Theory, Example 7.5.2 (standard reference, not scraped)