How to filter a belongsToMany relationship? Laravel 5.4

I've created an application to filter data. Inside of index, I made a filter to filter products by description, model, status, stock and category. My category is organized through the 'erp_categoy' table and the relationship with the product is made through 'erp_product_category'(Receiving Product ID + Category ID). I created a BelongsToMany relationship in the Model: Product, Category and ProductCategory.

Product Model

public function category()
    {
        return $this->belongsToMany('App\Category', 'erp_product_category', 'erp_productid', 'erp_categoryid');
    }

Category Model

public function product()
    {
        return $this->belongsToMany('App\Product','erp_product_category', 'erp_categoryid', 'erp_productid');
    }

Product_category Model

public function product()
    {
        return $this->belongsTo('App\Models\Product', 'erp_categoryid', 'erp_categoryid');
    }

In my index I created a select one that lists all categories of the 'erp_category' table.

<select id="categoria" name="categoria" class="form-control" style="width: 150px;">
                                @foreach($cat as $categoria)
                                    <option value="" @if($categoria->erp_categoryid === session('categoria')) selected  @endif ></option>
                                @endforeach
                            </select>

I've extended my JavaScript and stored the values in a session 'category'.

<script src=""></script>
    <script>
        var postSearch = '';
        var searchRequest = {
            'categoria' :'',
        };

    </script>

And then I did the research through JS.

$(document).on('blur', '#categoria', function(){
        var categoria = $('#categoria').val();
        searchRequest['categoria'] = categoria;
        doSearch();
    });

function doSearch() {
    $.post(postSearch, {
            'search_data': JSON.stringify(searchRequest),
            '_token': $('meta[name=csrf-token]').attr('content'),
        }
        , function(data) {
            $('#product-table').html(data);
        });
}

I was expecting the product-table (table I created to return the values). I list my products through the category entered by select, but it returns the initial list of the index.

Product-table

<td>
    @foreach($prod->category as $categoria)
    
    @endforeach
</td>

Controller

public function search(Request $request)
{    
 $product = Product::query();
 $categoria = Category::query();
if ($request->isMethod('post'))
{
$data = json_decode($request->search_data);
$categoria;
$categoria = $data->categoria;
session(['categoria' => $categoria]);
if(strlen(session('categoria')) > 0)
{
 $product_ids = Product::whereHas('category', function ($query){
   $query->where('erp_category.erp_categoryid', '=', session('categoria'));
   })->get();
    $ids = [];

          foreach($product_ids as $product_data)
          {
              $ids[] = $product_data->erp_productid;
          }

          $product = $product->whereIn('erp_category.erp_categoryid', $ids);
      }
        $content = $product->paginate(10);
        $cat = Category::all();

        if ($request->isMethod('post'))
        {
            return view('admin.product-table')->with('product', $content)->with('cat',$cat);
        } else
        {
            return view('admin/product')->with('product', $content)->with('cat',$cat);
        }

Any suggestion?

via Chebli Mohamed