CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31‡ rests on unproved material (inherited)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

Set, Cat, and every complete category are cartesian monoidal

Statement

The categories $S e t$ and $C a t$ are cartesian monoidal. More generally, every complete category is cartesian monoidal.

Facts & Assumptions

Given: The standard product structures on sets and on small categories.

[L1]

Any category with binary products and a terminal object is monoidal under that product (A category with finite products is monoidal).

[L2]

Sets and functions form the category $S e t$ (Sets and functions form the large locally small category $S e t$ ), and small categories, functors, and natural transformations form the strict 2-category $C a t$ (Small categories, functors, and natural transformations form the strict 2-category $C a t$ ).

[L3]

A complete category has all small limits and in particular a terminal object and binary products (Finite, small, and large limits and colimits; complete and cocomplete categories).

Proof

technique · direct

1.1

The cartesian product of sets and the singleton set give binary products and a terminal object in $S e t$ , so [L1] yields a monoidal structure on $S e t$ .

L1L2

1.2

In $C a t$ , the product category is the categorical binary product and the one-object category is terminal, so [L1] yields a monoidal structure on $C a t$ .

L1L2

1.3

If $C$ is complete, then [L3] gives the required terminal object and binary products, and [L1] makes $C$ cartesian monoidal.

L1L3

2.1

Therefore $S e t$ , $C a t$ , and every complete category are cartesian monoidal.

step 1.1step 1.2step 1.3∎

Depends on

Used by

The cartesian monoidal structure on sets computed Example
The power-set functor is lax monoidal but not strong Example
FALSE: every lax monoidal functor has invertible structure maps False statement

Dependency tree · two levels

21 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

P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories, Chapter 2.3 (standard reference, not scraped)
E. Riehl, Category Theory in Context, Chapter 1 (standard reference, not scraped)