The category CPO of cpos and continuous functions has a reflexive object, i.e. an object $A$ such that $A\times A\simeq A$ and $A\simeq A^A$. Since CPO has countable products, my question is whether or not there is also an $X$ such that $X\simeq X^X$ and
$$X\simeq \Pi_{n\in \mathbb{N}}X$$
where the RHS is the product of $X$ with itself countably many times.
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1$\begingroup$ Cpo=complete partial order, I guess? $\endgroup$– David Roberts ♦Commented Jul 10, 2019 at 20:58
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$\begingroup$ @DavidRoberts According to wikipedia en.wikipedia.org/wiki/Complete_partial_order, the meaning of unadorned "cpo" is ambiguous... $\endgroup$– Mike ShulmanCommented Jul 10, 2019 at 23:06
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$\begingroup$ @Mike hmm, good thing I asked. "The category CPO of cpos" is such a stereotypical category theorist thing to say! $\endgroup$– David Roberts ♦Commented Jul 11, 2019 at 1:07
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$\begingroup$ You are right, I should have written the definitions. So for cpo i mean a directed complete partially ordered set (probably dcpo is a better abbreviation), i.e. a poset in which every directed subset has a supremum. An arrow is a monotone function that preserves suprema of directed sets. $\endgroup$– Steve K.Commented Jul 11, 2019 at 8:44
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