202512272134

Tags : Discrete Homotopy Theory

Path Component Functor on Graphs

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The Path component functor ${\pi }_0$ takes a graph and sends it to its set of path-component. This functor is the left-adjoint to the discrete functor making the following diagram.

We also have that ${\pi }_0(X\otimes Y)={\pi }_{0}X\times {\pi }_{0}Y$

This makes this a Monoidal Functor.

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References

• Adjunctions
• Monoidal Functor