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It is known that in a locally $\lambda$-presentable category $\lambda$-filtered colimits commute with $\lambda$-small limits, and hence with pullbacks. I guess that this fails for $\mu$-filtered colimits for $\mu<\lambda$?

Thus, I am looking for

  • a regular cardinal $\lambda$
  • a locally $\lambda$-presentable category $\mathcal{C}$, which I would like to require to be additive,
  • an ordinal $\mu < \lambda$
  • a diagram $(X_{\alpha})_{\alpha<\mu}$ in $\mathcal{C}$,
  • a morphism $Y \to S$ in $\mathcal{C}$
  • compatible morphisms $X_{\alpha} \to S$ in $\mathcal{C}$,

such that the canonical morphism

$$\varinjlim_{\alpha<\mu} (X_{\alpha} \times_S Y) \to \bigl(\,\varinjlim_{\alpha<\mu} X_{\alpha}\,\bigr) \times_S Y$$

is not an isomorphism. Can you give such an example?

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  • 3
    $\begingroup$ Why the downvote? $\endgroup$ – HeinrichD May 23 '17 at 19:22
  • $\begingroup$ Why is $\cal C$ additive? $\endgroup$ – Fosco May 23 '17 at 20:16
  • $\begingroup$ @FoscoLoregian: I would like to require this, because currently I work with additive categories. Hence, a counterexample should not come from non-additivity. $\endgroup$ – HeinrichD May 24 '17 at 6:04
  • $\begingroup$ The first version of this question was about products, but they are just coproducts in the additive case, so that the question was trivial. I edited the question - it is now about pullbacks, which is the situation which I am actually interested in. $\endgroup$ – HeinrichD May 24 '17 at 6:23

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