So, can anyone explain to me when you are to use the infinitive as the complement of the verb "to be"?
I am confused; I do not understand what your problem actually is. So I will try to deconstruct your original question in order to help.
Aside from some archaic forms, infinitives in Polish language end in either -ć or -c, such as być or pomóc. Polish infinitives express aspect, such as robić (imperfective) or zrobić (perfective). This forum has several threads about aspects.
I am not sure I understand you. You started with the example "Dlaczego mu nie pomóc", followed by its English translation "Why not help him", which formally should be "Why not to help
him", and then wonder why is "pomóc" in the infinitive? You might as well ask the same question about the English expression, because this is one of these rare cases where Polish and English use the same structure and can be easily translated word for word. Let us see it again:
Why not to help
Dlaczego nie pomóc
mu. (Less elegant version of the original "Dlaczego mu nie pomóc", but perfectly acceptable)
Why=dlaczego, not=nie, to help=pomóc
[Going slightly off topic: Consider the following expression "Dlaczego mu nie pomagać
", where the imperfective "pomagać" is used rather than perfective "pomóc". ]
Where would any form of "to be" possibly fit in there (being omitted of course)?
How about this famous phrase for a start?To be
or not to be
- that is the question.Być
albo nie być
- oto jest pytanie.
I am going to be
I hope this helped.