CounterexampleConstruction: AI-generatedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 2026-08-13

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A monotone functor between poset categories preserves every monomorphism but need not preserve pullbacks

Statement refuted

Every functor that preserves monomorphisms preserves pullbacks.

Facts & Assumptions

Given: The diamond poset $P = {0, a, b, 1}$ with $0 < a < 1$ , $0 < b < 1$ , and $a, b$ incomparable; and the two-element chain $Q = {0 < 1}$ .

[F1]

Pullbacks have the compatible-pair universal property (Pullbacks and pushouts as limits and colimits of cospans and spans).

[L1]

A functor preserves a chosen limit exactly when its canonical comparison is an isomorphism (A functor preserves a chosen limit exactly when its canonical comparison to the chosen target limit is an isomorphism, and dually for colimits).

[F2]

Monomorphisms cancel on the left (Monomorphism and epimorphism by left and right cancellation).

[F3]

A poset is a category with at most one arrow between any two objects, and monotone maps are functors (A preorder is a category with at most one morphism between any two objects, and its functors are exactly monotone maps).

Counterexample

technique · finite posets

1.1

Define the monotone map $F : P \to Q$ by $F (0) = 0$ and $F (a) = F (b) = F (1) = 1$ . By [F3] it is a functor. Every arrow in either poset category is monic, since two parallel arrows are automatically equal; hence $F$ preserves every monomorphism.

F2F3

1.2

In $P$ , the pullback of $a \to 1 \leftarrow b$ is the meet $a \land b = 0$ , as follows directly from [F1] after translating arrows to inequalities. Its image is $0$ .

F1F3

2.1

The image cospan in $Q$ is $1 \to 1 \leftarrow 1$ , whose pullback is $1$ . The canonical comparison is the noninvertible arrow $0 \to 1$ , so [L1] says that $F$ does not preserve this pullback. This refutes the statement.

F1L1step 1.2∎

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