What conditions must be set for defining a key comparison in std::map to ensure correct code compilation?

For the correct operation of std::map, keys must be comparable using a comparator that defines a strict weak ordering. By default, std::less<Key> is used, which requires that for keys a and b, the following conditions hold:

• Antisymmetry: if a < b is true, then b < a is false.
• Transitivity: if a < b and b < c, then a < c.
• No cycles: a < a is always false.

If you use your own comparator, it must implement the operator bool operator()(const Key&, const Key&) const with the same properties. This is necessary for correct key sorting and proper data structure operation.