Existence of the homotopy category functor h from simplicial sets to Cat is equivalent to specific weighted colimits in Cat, making the nerve embedding reflective and establishing that Cat is complete and cocomplete.
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Revisiting colimits in $\mathbf{Cat}$ and homotopy category
Existence of the homotopy category functor h from simplicial sets to Cat is equivalent to specific weighted colimits in Cat, making the nerve embedding reflective and establishing that Cat is complete and cocomplete.