Our honorable sheikh! We talk about people who have excuses and are allowed to break their fast. What are the categories of inability that allow Muslim to break his fast?

All praise is due to Allah, the Lord of all the worlds. May Allah send peace and blessings upon our master Muhammad and upon his family and his companions.
The shari`ah intends ease and does not intend to make things difficult or cause ruin to people. This is one of the characteristics of this gracious shari`ah. Almighty Allah said “[These are] the ones who shall follow the Messenger, the unlettered Prophet, whom they find inscribed in the Torah [that is] with them and in the Evangel; who enjoins them with what is right and forbids them from what is wrong, making lawful for them wholesome things, and prohibiting for them impure things; and relieving them from the burden [of strict obligation] and the yokes that were [laid] upon them.” This is a complementary statement to your introduction.
Accordingly, we say that people of excuses include travelers and sick people as explicitly stated in the Quran “But one among you who is sick or is on a journey [shall fast] the same number of other days.” The sick person is excusable, and fasting is not obligatory upon him. He is not even permitted to fast if he is badly sick. But scholars differed in the degree of the disease that allows breaking fasting. All the scholars agree that severe sickness drops the obligation of fasting if it causes such harm that may lead- or almost lead- to perishing, which is the case of maximum necessity. There is also a slight sickness that does not remit the obligation of fasting except according to the view of a group of scholars such as al-Bukhary who held the view that the word “sick” covers every person who feels sick and so he dropped the obligation of fasting in the case of slight sickness.
This position finds support in reports attributed to some scholars from among the Successors. But scholars differed concerning mild sickness. The correct position is that if fasting will lead to prolonging or increasing the sickness, the obligation of fasting becomes remitted in the sense that one is permitted to give up fasting but he does not have to. This means that if he fasts, his fasting will be valid, according to the correct scholarly position, because sickness remits the immediate obligation not the reason of the obligation. This is the difference between sickness, on the one hand, and menstruation and postnatal bleeding, on the other hand. In case of menstruation and postnatal bleeding a woman is not permitted to fast. These are two other excuses considered by the lawgiver and fasting in these cases is not even valid or permissible, whereas if the sick person can endure fasting, his fasting will be valid according to the position of the majority. Traveling also remits the obligation of fasting.