How can Time Series Analysis be done with Categorical Variables
By definition time-series ARIMA models assume that, given a numerical observation at time $t-k$, the value of the numerical variable $X$ at time $t$ can be approximated as
$$
X_t = \sum_{j=1}^p a_j X_{t-j} + \varepsilon_t + c
$$
where $\varepsilon$ is a white noise error term and the $a_j$ are parameters to be determined. The idea is that the numerical variable $X$ at time $t$ only depends on some of its values at previous times; as you can see, by construction the above works for numerical variables only. Then one introduces some more conditions about moving averages and deviations to be verified and is able to prove that, under such conditions, the form of the coefficients $a_j$ can be determined.

The standard way to deal with categorical variables in these cases is to use one-hot encoding, namely you introduce dummy variables for each level of your category and fit against the dummy being 1 or 0, according to whether such category is present or not at time $t-k$. A similar question was asked here and you might want to have a look.

Another thing that you may want to ask yourself is whether time series is really what you are looking for, rather than just any classification model that provides a prediction given a set of categories and a seasonal variables.