Is there a left adjoint to the inclusion of the full subcategory of 1-connected spaces into the category of all spaces?
Left adjoint to inclusion of 1-connected spaces
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algebraic-topology
category-theory
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0@MartinBrandenburg : Is the empty set not vacuously simply connected? – 2012-07-14
1 Answers
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Martin points out in the comments that one should include basepoints to make the question interesting. However, the inclusion functor $\text{1-connected based spaces} \to \text{based spaces}$ does not have a left adjoint because it does not preserve pullbacks. Consider the pullback in 1-connected based spaces of $f: (\mathbb R, 1) \rightarrow (\mathbb R, 0) \leftarrow (\mathbb R, 0) : g$ given by $f(x) = 1-x^2$ and $g(y) = y^2$. This pullback is some 1-connected space. But the pullback of the diagram in based spaces is $S^1$.
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0Fair point. The real problem here, though, is non-locally path connected, 1-connected spaces. – 2012-07-15