A survey of varying mandates.
The National Coalition for the Homeless released this fascinating map today illustrating the wide variation in policies across the country that mandate exactly how cold it has to be before cities are required to provide overnight shelter for every single homeless person (although plenty of cities have no such requirement):
In Washington, D.C., the law is called the Frigid Temperature Protection Act and requires the city to make available space in District buildings or facilities to anyone without shelter when the temperature falls below 26 degrees, although it's usually applied when the wind chill dips below 32.
The above map suggests that there's a point (albeit a varied one) beyond which it's simply not humane for cities to allow anyone to sleep outdoors. But it also raises the question of what it would take to extend the same resources the rest of the year, or at least throughout the winter.