Factor analysis is used to reduce and summarize the data being analyzed , eliminating dependencies or independence from the data. It is also used to "identify factors that explain the correlations between the variables" , being that in this type of analysis all the variables are equal, that is to say independent, which means that no variable is more or less than another .

Having the required information, it is necessary to calculate a set of latent dimensions called factors, with which these relationships are explained. Therefore, factor analysis is a data reduction technique with which the hypotheses put forward can be tested.

Factorial Analysis can be of two types. The exploratory type, in which the number of factors is not known from the beginning and it is in the empirical application where this number is determined. In the confirmatory analysis, the factors are known from the outset, using the hypotheses for corroboration.

I'm not sure if you mean categorical observed or latent variables. Either way, the answer is yes. See https://www.wiley.com/en-us/Latent+Variable+Models+and+Factor+Analysis%3A+A+Unified+Approach%2C+3rd+Edition-p-9780470971925 and the general linear latent variable model described there. If the observed values are categorical (and latent metrical), then it common in education and called IRT (item response theory) or latent trait analysis. If both are categorical it is called latent class analysis, and used in some sociology and market research. If the observed are metric but you want categorical latent variables there are a few ways, but latent profile analysis is the most aligned to these others. See

Some papers on the applications of factor analysis follow

The Application and Misapplication of Factor Analysis in Marketing Research

David W. Stewart

https://booksc.xyz/book/18532500/1af071

Application of parallel factor analysis and x-ray photoelectron spectroscopy to the initial stages in oxidation of aluminium. I. The Al 2p photoelectron line

Do, T., McIntyre, N. S., Harshman, R. A., Lundy, M. E., Splinter, S. J.

https://booksc.xyz/book/15733/80b891

Thankyou all for yours valuable answers.It is clear that factor analysis is used in data reduction so therefore when I all variables were I had already reduced the data and all variables under the study are important say for example after pilot I am going for a final study I want know factors or variables influencing most with respect particular phenomenon in that case also I used factor analysis that's what I want to know and that was my actual doubt if anyone can put light on this issue I will thankful to you.

Factor analysis of the general self-efficacy scale and its relationship with individualism/collectivism among twenty-five countries: Application of multilevel confirmatory factor analysis

Chia-Huei WuJournal:Personality and Individual DifferencesY