Why Lagrange point colonization
When most people think about space colonization, they think about colonies on the Moon or Mars. Some people conflate the concept of space colonization with faster-than-light travel (which is impossible), and think about conquering other stellar systems. This is NOT what Republic of Lagrangia is about. We are aimed at colonizing free space by using space habitats. A space habitat is simply a large space station able to house a large number of people, most designs rely on centrifugation for generating artificial gravity.
The great advantage of space habitats is that we can locate them anywhere we wish. They can be relocated if necessary, if for example we live in the close neighbourhood of a hostile space colony (or a hostile Earth), we can move away from them. Another possibility is avoiding collision with meteorites. This is impossible if you are living on the Moon or Mars, technically speaking it is possible to move these objects, but it take you an immense amount of energy to make even a slight displacement and due to their masses almost any significant displacement will have unintended consequences for other bodies in the Solar System (Earth in particular).
The next question is of course, where should we locate our space habitat? There are a lot of possible location, each of them has their benefits and disadvantages. Most people would think about placing space habitats in an orbit around the Earth. Main advantage is that they are close to Earth, therefore travel time is short (from a few hours to a few days). This location is historically defended by most space advocacy groups. The foremost problem with cislunar space colonies is that they suffer from regular eclipses. This problem can be reduced by placing space habitats farther away from Earth, but this also increase travel time. Another problem is the lack of resources in the Earth system.
An obvious, but misleading, objection is that there are resources on Earth and on the Moon. Why should you go to live in outer space if you need to import everything you need from Earth? True, during the first days of space colonization a lot of stuff need to be imported, but after some time space colonies will become more and more self-sufficient. And the Moon is rich in some resources, especially titanium, but is also poor in others (most importantly hydrogen, carbon and nitrogen are hard to find on the Moon). Secondly the Moon is massive enough to require a relative high escape velocity (compared with Near Earth Objects) and it cost more energy to reach the Moon than the Near Earth Objects.
An attractive location for positioning space colonies are the fifth and fourth Sun-Earth Lagrange points, which are in co-orbit with the Earth. Bodies placed in or around these points have a stable orbit. Further these points do not suffer from eclipses caused by the Earth, therefore we can rely on continuous operational Solar power. One problem is that it takes several months to reach them, but we have to realize that a few centuries ago it would take several months to cross the Atlantic. But this did not stop European countries from colonizing the Americas. Yes, the long travel time causes several challenges, but we believe that man can overcome this. Proper planning and design are a key to success.
Another advantage of the fourth and fifth Earth-Sun Lagrange points is the presence of so-called Trojan asteroids. Currently the existence one such an object is confirmed, it is reasonable to assume other Earth Trojans exists. As I will explain in the next section, asteroids are the treasures of the Solar System. Even if there no other Earth Trojans, or those which exists are of poor composition, then L4 and L5 can be used as a destination for relocated Near Earth Asteroids. Some space advocates argue in favor of capturing asteroids and to relocate them into an orbit around our planet. I do not believe this is a good idea, we can easily see the danger of this mission. Many people on this planet would consider this as an unacceptable risk.
The long travel time from Earth to the fourth and fifth Lagrange points has also benefits. If for some reason the communities of the Lagrange points got in conflict with terrestrial nations, they effectively protected from military aggression from the Earth. Any hostile missile has to transverse for months through space and can be detected remotely by proper equipment.