CorollaryStatement: AI-adaptedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-08-31

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.

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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

1.1L1L2

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 $S e t$ .

1.2L1L2

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 $C a t$ .

1.3L1L3

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

2.1step 1.1step 1.2step 1.3∎

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

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
The category of small categories is cartesian closed Theorem

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)