laravel livewire refresh when updating variable

As you can see I am trying to make a list of categories, using ul/li on laravel livewire, the problem is that when I click on the "+" it transform to "-" as expected but retransform to "+", because I think DOM reloads.

controller:

<?php

namespace App\Http\Livewire\Product;

use App\Classes\Product\ShopifyCatList;
use Livewire\Component;

class ShopifyNewCategory extends Component
{
    public $categories, $html;

    private function getFirstCat()
    {
        $shopifyCatList = new ShopifyCatList();
        $this->categories = $shopifyCatList->getFirstCat();
    }

    public function addChildren($rank)
    {
        $shopifyCatList = new ShopifyCatList();
        $this->categories = $shopifyCatList->updateCategories($rank);
        $this->html = $shopifyCatList->updateHtml($this->html, $this->categories, $rank);
    }

    public function mount()
    {
        $this->getFirstCat();
        $this->html = '<li data-rank="' . $this->categories[0]['rank'] . '"><span class="sign" x-on:click="openList">+</span><span class="content">' . $this->categories[0]['name'] . '</span></li>';
    }

    public function render()
    {
        return view('livewire.product.shopify-new-category')->extends('layouts.app-livewire')->section('content');
    }
}

view:

@section('header')
    <div id="innerheader" style="display: flex;flex-direction: row;">
        <div style="flex:auto">
            <h1>Création nouvelle catégorie</h1>
        </div>
    </div>
@endsection
{{-- section content est mis dans le controller sinon form ne fonctionne pas correctement https://laracasts.com/discuss/channels/livewire/laravel-livewire-submitprevent-not-working-refreshing-page --}}
<form wire:submit.prevent="newCategory">
    @csrf
    <div class="form-group">
        <label for="name">Nom de la catégorie</label>
        <input type="text" class="form-control" id="name" name="name" placeholder="Nom de la catégorie"
            wire:model="name">
        @error('name')
            <span class="text-danger">{{ $message }}</span>
        @enderror
    </div>
    <div class="form-group">
        <label for="parent" id="parent">Catégorie parent: <b></b></label>
        <div x-data="categories" id="parent" class="border rounded bg-white overflow-auto" style="height:40vh;">
            <ul style="padding-left:2%">
                {!! $html !!}
            </ul>
        </div>
        @error('parent')
            <span class="text-danger">{{ $message }}</span>
        @enderror
    </div>
    <div class="d-flex justify-content-end">
        <button type="submit" class="btn btn-success mr-1" wire:click="updateValue('save')">Enregistrer</button>
        <button type="submit" class="btn btn-primary" wire:click="updateValue('saveRecreate')">Enregistrer et
            créer un autre</button>
    </div>
</form>

@push('styles')
    <style>
        .content ul {
            list-style: none;
        }

        span.sign {
            margin-right: 6px;
        }

        .content li {
            cursor: pointer;
        }

        .content li:hover {
            background-color: #39F856;
        }
    </style>
@endpush

<script>
    categories = () => {
        return {
            selected: '',
            labelNoticeSelected: $('label#parent b'),
            openList: (e) => {
                if (e.target.innerText === '+') {
                    e.target.innerText = '-';
                    @this.addChildren(e.target.parentNode.getAttribute('data-rank'));
                } else if (e.taget.innerText === '-') {
                    e.target.innerText = '+';
                    e.target.parentNode.parentNode.querySelector(
                            `li[data-rank='${e.target.parentNode.getAttribute('data-rank')}']+ul`).style
                        .display = 'none';
                }
            }
        }
    }
</script>

Here is all the code, I am also using alpinejs, when I click on the "+" I want to request db for children of the category I click on (that is ok) then I want it to transform "+" to "-" (that is not ok) so when I click on "-" it hides the children.

thanks for reading all of this and thanks to everyone trying to help.