TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-01

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In a biclosed monoidal category tensor is cocontinuous in each variable

Statement

Let $C$ be a biclosed monoidal category. For every object $X$ , the functors $- \otimes X$ and $X \otimes -$ preserve every colimit that exists in $C$ .

Facts & Assumptions

Given: A biclosed monoidal category $C$ and an object $X$ .

[L1]

Biclosed means that $- \otimes X$ and $X \otimes -$ are left adjoints for every $X$ (Left-closed, right-closed, and biclosed monoidal categories).

[L2]

Every left adjoint preserves all colimits that exist in its domain (Left adjoints preserve every colimit that exists).

Proof

technique · direct

1.1givenL1

Since $C$ is biclosed, the functor $- \otimes X$ has a right adjoint and is therefore a left adjoint by [L1].

1.2givenL1

Likewise $X \otimes -$ has a right adjoint and is a left adjoint.

2.1step 1.1L2

Apply [L2] to step 1.1 to conclude that $- \otimes X$ preserves every colimit that exists in $C$ .

2.2step 1.2L2

Apply [L2] to step 1.2 to conclude that $X \otimes -$ preserves every colimit that exists in $C$ .

3.1step 2.1step 2.2∎

Hence tensoring with a fixed object is cocontinuous in each variable.

Depends on

Used by

The internal hom preserves limits in the covariant variable and sends colimits to limits in the contravariant variable Theorem

Dependency tree · two levels

7 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, Category Theory in Context, 2nd ed., Section 4.4 and Theorem 4.2.1 (standard reference, not scraped)