TheoremStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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When a category is tensored, every limit in it is a conical enriched limit

Statement

Assume $V$ is symmetric monoidal right closed, locally small, complete, and cocomplete, and let $B$ be a tensored $V$ -category. Then the ordinary limit of every small diagram in the underlying category $B_{0}$ , when it exists, is a conical enriched limit.

Facts & Assumptions

Given: A base $V$ as in the statement, a tensored $V$ -category $B$ , a small ordinary diagram, and its limit cone in $B_{0}$ .

[L1]

Tensors represent enriched hom-objects against the base: $B (X \otimes C, B) ≅ [X, B (C, B)]$ (Tensor and cotensor in a V-category).

[L2]

Conical enriched limits are the constant-unit weighted enriched limits (Conical enriched limit).

Proof

technique · direct

1.1

Because $B$ is tensored, [L1] identifies the enriched hom-object out of $X \otimes B$ with the hom-object $B (B, -)$ tested against $X$ . Thus each represented enriched hom-functor is a right adjoint in the underlying category and therefore preserves ordinary limits.

L1given

2.1

Apply step 1.1 to an ordinary limit object $L$ of the underlying diagram. For every test object $B$ , the ordinary limit bijection for $B_{0} (B, L)$ upgrades, through the tensor representation of [L1], to the enriched constant-weight bijection required by [L2].

L1L2step 1.1

3.1

Hence the ordinary limit is already a conical enriched limit.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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, Theorem 7.5.3 (standard reference, not scraped)