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Copy file name to clipboardExpand all lines: static/pdf/comphaus_copresentable.tex
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\end{proof}
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\end{lemma}
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This shows that $\CompHaus^{\op}$ is equivalent to the category of algebras over the monad $S \mapsto\Hom_{\CompHaus}([0, 1]^S, [0, 1])$. We may view such algebras as being models of the algebraic theory of all continuous functions $[0,1]^S \to [0,1]$. In fact, we can show that any such function only depends on countably many coordinates in the domain, so this operations of this theory will be generated by the continuous functions $[0,1]^\omega\to [0,1]$.
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This shows that $\CompHaus^{\op}$ is equivalent to the category of algebras over the monad $S \mapsto\Hom_{\CompHaus}([0, 1]^S, [0, 1])$. We may view such algebras as being models of the algebraic theory of all continuous functions $[0,1]^S \to [0,1]$. In fact, we can show that any such function only depends on countably many coordinates in the domain, so that operations of this theory will be generated by the continuous functions $[0,1]^\omega\to [0,1]$.
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\begin{lemma}
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The object $[0,1]$ of $\CompHaus$ is $\aleph_1$-copresentable.
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