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Are product morphisms for a categorical product in Top the same as for categorical product morphisms in Set?

More generally: How product morphisms for Top are characterized?

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What is a "product morphism"? This is not standard terminology. –  Zhen Lin Aug 17 '12 at 1:39
    
@ZhenLin: McLane –  porton Aug 17 '12 at 5:27
    
I don't find any mentioning of a product morphism in Mac Lane. Are you talking about projections? –  roman Aug 17 '12 at 7:31
    
@ZhenLin en.wikipedia.org/wiki/Product_%28category_theory%29 - "product of morphisms" –  porton Aug 17 '12 at 8:17
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That isn't standard terminology at all. See this MO question. –  Zhen Lin Aug 17 '12 at 8:32
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1 Answer

up vote 5 down vote accepted

Products in $\textbf{Top}$ are "the same" as in $\textbf{Set}$ in the following sense:

  • If $X_i$ ($i \in I$) is a family of topological spaces, then the underlying set of the product $\prod_i X_i$ is the product of the underlying sets, and the projections in $\textbf{Top}$ are the same as in $\textbf{Set}$.

  • Similarly for the relationship between a family of continuous maps $f_i : Y \to X_i$ and the combined map $f = \langle f_i \rangle : Y \to \prod_i X_i$.

But that is all we are able to say. In technical terms, the above amounts to claiming that the forgetful functor $U : \textbf{Top} \to \textbf{Set}$ preserves all small products. One can prove this by abstract nonsense by observing that $U$ has a left adjoint, namely the functor $\textbf{Set} \to \textbf{Top}$ that equips a set with the discrete topology. It is not true that $U$ reflects or creates products: this is because there are in general many possible topologies on the set $\prod_i X_i$ that make the projections $\pi_j : \prod_i X_i \to X_j$ continuous without necessarily having the universal property of a product.

If we restrict to the full subcategory $\textbf{KHaus}$ of compact Hausdorff spaces, it is true that the forgetful functor $U : \textbf{KHaus} \to \textbf{Set}$ preserves and reflects all small products. This is because $\textbf{Haus}$ is monadic over $\textbf{Set}$ – roughly speaking, this means $\textbf{KHaus}$ behaves a bit like a category of algebraic structures.

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