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fixed functor domain error: lifted functor is from pTop, not ppTop (plus misc. edits to meet character minimum)
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edit approved Nov 29, 2019 at 11:29
wchargin
edit approved Nov 29, 2019 at 11:29
wchargin
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Let ppTop$\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{Gp}$ where GP is the category of groups.

Now we consider the category pTop$\text{pTop}$ consisting of path-connected topological spaces and we can naturally define the fundamental groupoids instead of fundamental groups on pTop$\text{pTop}$. If we want to define the fundamental group then we need to choose a base point. Notice that there is a forgetful functor $\text{For}:\text{ppTop}\to \text{pTop}$.

My question is: could we lift the functor $FG: \text{ppTop}\to \text{Gp}$ to a functor $\widetilde{FG}: \text{ppTop}\to \text{Gp}$$\widetilde{FG}: \text{pTop}\to \text{Gp}$ such that $\widetilde{FG}\circ \text{For}=FG$.? If not, how to construct a contradiction?

Let ppTop denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{Gp}$ where GP is the category of groups.

Now we consider the category pTop consisting of path-connected topological spaces and we can naturally define the fundamental groupoids instead of fundamental groups on pTop. If we want to define the fundamental group then we need to choose a base point. Notice that there is a forgetful functor $\text{For}:\text{ppTop}\to \text{pTop}$.

My question is: could we lift the functor $FG: \text{ppTop}\to \text{Gp}$ to a functor $\widetilde{FG}: \text{ppTop}\to \text{Gp}$ such that $\widetilde{FG}\circ \text{For}=FG$. If not, how to construct a contradiction?

Let $\text{ppTop}$ denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{Gp}$ where GP is the category of groups.

Now we consider the category $\text{pTop}$ consisting of path-connected topological spaces and we can naturally define the fundamental groupoids instead of fundamental groups on $\text{pTop}$. If we want to define the fundamental group then we need to choose a base point. Notice that there is a forgetful functor $\text{For}:\text{ppTop}\to \text{pTop}$.

My question is: could we lift the functor $FG: \text{ppTop}\to \text{Gp}$ to a functor $\widetilde{FG}: \text{pTop}\to \text{Gp}$ such that $\widetilde{FG}\circ \text{For}=FG$? If not, how to construct a contradiction?

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Zhaoting Wei
Zhaoting Wei
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Can we define fundamental groups functorially for non-pointed path connected topological spaces?

Let ppTop denote the category of pointed and path connected topological spaces with morphisms base-preserve continuous maps. The fundamental group gives a functor $FG: \text{ppTop}\to \text{Gp}$ where GP is the category of groups.

Now we consider the category pTop consisting of path-connected topological spaces and we can naturally define the fundamental groupoids instead of fundamental groups on pTop. If we want to define the fundamental group then we need to choose a base point. Notice that there is a forgetful functor $\text{For}:\text{ppTop}\to \text{pTop}$.

My question is: could we lift the functor $FG: \text{ppTop}\to \text{Gp}$ to a functor $\widetilde{FG}: \text{ppTop}\to \text{Gp}$ such that $\widetilde{FG}\circ \text{For}=FG$. If not, how to construct a contradiction?