How to describe this aggregation using the Nest .NET Library?

I have an aggregation that also has a filter and I am having trouble composing it with the .NET Nest Library:

{
  "aggs": {
    "positive_lat": {
      "filter": {
        "bool": { 
          "must_not": [
            {
              "term": {
                "latitude": 0
              }
            }
          ]
        }
      }, 
      "aggs" : {
        "min_lat": {
          "min": {
             "field": "latitude"
          }
        }
      }
    }
  }
}

How does one do something like that? Is it possible?

Yep, it's possible:

var client = new ElasticClient();

// use your document type instead of object
var searchResponse = client.Search<object>(s => s
    .Aggregations(a => a
        .Filter("positive_lat", f => f
            .Filter(q => q
                .Bool(b => b
                    .MustNot(mn => mn
                        .Term("latitude", 0)
                    )
                )
            )
            .Aggregations(aa => aa
                .Min("min_lat", m => m
                    .Field("latitude")
                )
            )
        )
    
    )
);

// do something with min latitude
var minLatitude = searchResponse.Aggregations.Filter("positive_lat").Min("min_lat").Value;

Take a look at the Writing Aggregations documentation, as well as the usage examples for aggregations.