Can I API the category leadsource?

We are able to pull the individual leadsource.
Can we do the same with the Lead Category?

You can do so with the Data Service in the Legacy XMLRPC API.

I meant to ask, how do I pull the Leadsource category for the contact?
It seems we can only do the leadsource or id, but not the category? please verify.

The LeadSourceId relates to the LeadSourceCategory via the LeadSource table, accessible via the Data Service. You will have to retrieve it via a second call, although I would cache the LeadSource and LeadSourceCategory tables locally, as it is unlikely to change frequently.