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

1.1

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

givenL1

1.2

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

givenL1

2.1

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

step 1.1L2

2.2

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

step 1.2L2

3.1

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

step 2.1step 2.2∎

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)