Objection 1. It seems that to be everywhere does not belong to God alone. For the universal, according to the Philosopher (Poster. i), is everywhere, and always; primary matter also, since it is in all bodies, is everywhere. But neither of these is God, as appears from what is said above (Article 3). Therefore to be everywhere does not belong to God alone.
Objection 2. Further, number is in things numbered. But the whole universe is constituted in number, as appears from the Book of Wisdom (Wisdom 11:21). Therefore there is some number which is in the whole universe, and is thus everywhere.
Objection 3. Further, the universe is a kind of "whole perfect body" (Coel. et Mund. i). But the whole universe is everywhere, because there is no place outside it. Therefore to be everywhere does not belong to God alone.
Objection 4. Further, if any body were infinite, no place would exist outside of it, and so it would be everywhere. Therefore to be everywhere does not appear to belong to God alone.
Objection 5. Further, the soul, as Augustine says (De Trin. vi, 6), is "whole in the whole body, and whole in every one of its parts." Therefore if there was only one animal in the world, its soul would be everywhere; and thus to be everywhere does not belong to God alone.
Objection 6. Further, as Augustine says (Ep. 137), "The soul feels where it sees, and lives where it feels, and is where it lives." But the soul sees as it were everywhere: for in a succession of glances it comprehends the entire space of the heavens in its sight. Therefore the soul is everywhere.
On the contrary, Ambrose says (De Spir. Sanct. i, 7): "Who dares to call the Holy Ghost a creature, Who in all things, and everywhere, and always is, which assuredly belongs to the divinity alone?"
I answer that, To be everywhere primarily and absolutely, is proper to God. Now to be everywhere primarily is said of that which in its whole self is everywhere; for if a thing were everywhere according to its parts in different places, it would not be primarily everywhere, forasmuch as what belongs to anything according to part does not belong to it primarily; thus if a man has white teeth, whiteness belongs primarily not to the man but to his teeth. But a thing is everywhere absolutely when it does not belong to it to be everywhere accidentally, that is, merely on some supposition; as a grain of millet would be everywhere, supposing that no other body existed. It belongs therefore to a thing to be everywhere absolutely when, on any supposition, it must be everywhere; and this properly belongs to God alone. For whatever number of places be supposed, even if an infinite number be supposed besides what already exist, it would be necessary that God should be in all of them; for nothing can exist except by Him. Therefore to be everywhere primarily and absolutely belongs to God and is proper to Him: because whatever number of places be supposed to exist, God must be in all of them, not as to a part of Him, but as to His very self.
Reply to Objection 1. The universal, and also primary matter are indeed everywhere; but not according to the same mode of existence.
Reply to Objection 2. Number, since it is an accident, does not, of itself, exist in place, but accidentally; neither is the whole but only part of it in each of the things numbered; hence it does not follow that it is primarily and absolutely everywhere.
Reply to Objection 3. The whole body of the universe is everywhere, but not primarily; forasmuch as it is not wholly in each place, but according to its parts; nor again is it everywhere absolutely, because, supposing that other places existed besides itself, it would not be in them.
Reply to Objection 4. If an infinite body existed, it would be everywhere; but according to its parts.
Reply to Objection 5. Were there one animal only, its soul would be everywhere primarily indeed, but only accidentally.
Reply to Objection 6. When it is said that the soul sees anywhere, this can be taken in two senses. In one sense the adverb "anywhere" determines the act of seeing on the part of the object; and in this sense it is true that while it sees the heavens, it sees in the heavens; and in the same way it feels in the heavens; but it does not follow that it lives or exists in the heavens, because to live and to exist do not import an act passing to an exterior object. In another sense it can be understood according as the adverb determines the act of the seer, as proceeding from the seer; and thus it is true that where the soul feels and sees, there it is, and there it lives according to this mode of speaking; and thus it does not follow that it is everywhere.
Source: Summa Theologica (New Advent)