The big universe, also called the macrocosm is divided into 4 worlds in the kabbalah : The Divine world, the spiritual world and the psychic world and the physical world. The 3 first worlds are made of 3 sphere each. Forming a trinity each. So we have 3 trinities. The Divine trinity, the spiritual trinity and the psychic trinity.

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In total we can count 10 spheres called the sepherot in kabbalah. But the esoteric tradition talks of 7 planes or spheres of existence ( The atmic plane, the buddic plane, the causal plane, the mental plane, the astral plane and the etheric plane and the physical world). Now it may sound that kabbalah count of spheres differs from the traditional 7 counts. No, there is no contradiction. While kabbalah is looking more at the macrocosm (the bigger universe), the traditional esoteric teaching is looking at the microcosm. So the esoteric only mentions the 7 planes it is because those are 7 ones we may possibly reach in our small inner universe . After the atmic plane (the summit of spirituality), we enter the Divine spheres which are out of our direct reach. The 3 Divine sphere is what kabbalah name receptively Ketter, Hockmah and binah.

The spiritual world
What we call the spiritual world is made of 3 planes (atmic, buddic and causal ). Going from the physical world towards the spiritual world, the causal plane is the first we reach. So the causal world is the entry point into the spiritual world. In other words, for us humans, the spiritual world starts from the causal plane. The first manifestation of the spirit in us comes from this sphere (causal plane) of the universe. And for the beings of the causal world (as it is also populated with beings) the mental world is the entry point to the psychic world (the world of souls). Many spiritual masters will refer to the causal plane as the spiritual world, this is just to avoid being too technical and too detailed.

In kabbalah, the casual world is named Tiphereth. Tiphereth is one of the most famous sphere in kabbalah for it is situated in the central line of the kabbalah tree of life. In that line, tiphereth faces Ketter (the Divine sphere) directly. This is why it represents the perfect image of Ketter (the first manifestation of God) in the spiritual world. Tiphereth is the sun of our inner universe. It is even called the solar sphere sometimes. It is the source of light (wisdom), the wisdom of our spirit.
From the image above, you can see the correspondences Between kabbalah and the traditional esoteric:

Malkuth = physical plane.
Yesod = Etheric/Astral plane.
Hod = Upper Astral plane
NetZach = Mental plane
Tiphereth = Causal plane
Geburah = Buddic plane
Chesed = Atmic Plane.

In terms of dimensions, the physical plane belongs to the 3D, the Etheric plane belongs to the 4D, The astral and Mental belongs to the 5D, and the causal plane belongs to the 6D.

Love and light