First of all, pc and pl types are types char* and long*. If you want to print the full type name using typeid pipe your output to c++filt, something like ./prog | c++filt --types.
To compare these two dates, you should convert std::string to time_t. For that use tm structure. To convert string to time use strptime() function from time.h header. After that create time_point value using from_time_t() and mktime(). And at the end convert time_point_t type to time_t with to_time_t() function.
Your code should be something like this:
auto end = chrono::system_clock::now();
string date_to_do_something = "Mon Jul 27 17:13:17 2021";
time_t end_time = chrono::system_clock::to_time_t(end);
// gives some weird types:pc, pl
cout << typeid(ctime(&end_time)).name() << endl;
cout << typeid(&end_time).name() << endl;
// Now how to compare?
tm t = tm{};
strptime(date_to_do_something.c_str(), "%a %b %d %H:%M:%S %Y", &t);
chrono::system_clock::time_point tp =
chrono::system_clock::from_time_t(mktime(&t));
time_t time = chrono::system_clock::to_time_t(tp);
if (time == end_time) {
// do something
} else {
// do something else
}