False statementConstruction: AI-adaptedVerification: AI-generatedprecheck passaudited 2026-08-13

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FALSE: a functor preserving binary products and equalizers must preserve all finite limits

Statement refuted

Every functor that preserves binary products and equalizers preserves all finite limits.

Facts & Assumptions

Given: The terminal category $1$ and the functor $F : 1 \to S e t$ sending its sole object to $\emptyset$ .

[L1]

Finite-limit criteria require nullary product data, equivalently a terminal object, in addition to binary products and equalizers (Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals).

[F1]

Continuous means preserving all small limits (Preservation, reflection, and creation of limits and colimits; continuous and cocontinuous functors).

[F2]

Morphisms of $S e t$ are functions (Sets and functions form the large locally small category $S e t$ ).

Refutation

technique · constant-empty functor

1.1

The binary product of the sole object of $1$ with itself is that object, and $\emptyset \times \emptyset = \emptyset$ . The image product cone consists of identity functions on $\emptyset$ , so $F$ preserves the binary product.

1.2

Every parallel pair in $1$ is $(1, 1)$ and has identity equalizer. Its image is $(1_{\emptyset}, 1_{\emptyset})$ , whose identity is also an equalizer. Thus $F$ preserves equalizers.

1.3

The sole object of $1$ is terminal, but $\emptyset$ is not terminal in $S e t$ because no function ${*} \to \emptyset$ exists. So $F$ does not preserve the empty product, hence does not preserve all finite limits and is not continuous by [F1].

F1F2

2.1

This refutes the statement and exhibits exactly the missing nullary case in [L1].

L1step 1.1step 1.2step 1.3∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

18 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

E. Riehl, Category Theory in Context, Theorem 3.5.17 (standard reference, not scraped)