What was forgotten can be read off by counting maps – forgetting comes in three rungs: property, structure, stuff.
The corpus has dropped things many times. The conformal class drops a density from the metric (Paper 150), and a character table drops the squaring map (Paper 152). This paper shows that dropping has three rungs and decides which rung applies by counting maps. NO NEW MATHEMATICAL THEOREM IS CLAIMED. Adjoint functors (Kan), free constructions, the distinction between property, structure and stuff, and the classification of groups of order eight are all standard.
First, an adjunction is a count. A homomorphism out of a free group is nothing but a tuple of images of the generators. Taking a two-element set, the number of homomorphisms out of the free group is therefore the square of the order of the target, which is exactly the number of maps of sets. For the cyclic group of order six, the symmetric group on three letters, the symmetry group of the square and the Klein four-group alike, the two sides agreed (checked here). The adjunction is the statement that this agreement holds for every group at once, and naturally.
Second, forgetting has three rungs. A functor forgets a property if it is fully faithful, structure if it is faithful but not full, and stuff if it is not faithful. The example of forgetting structure is the forgetful functor from groups to sets: there are three homomorphisms from the cyclic group of order three to itself and twenty-seven maps. Distinct homomorphisms remain distinct as maps, so the functor is faithful, and most maps are not homomorphisms, so it is not full (computed here). The example of forgetting stuff is the functor sending a pair of sets to its first component: from a pair of two-element sets to another there are sixteen morphisms but only four in the image, so morphisms collide and the functor is not even faithful. And the example of forgetting a property is the inclusion of abelian groups into groups, which is fully faithful, since a group homomorphism between abelian groups already is a homomorphism of abelian groups. What is lost is not morphisms but which objects occur.
And that loss can be counted. Separating the five standard groups of order eight by their multisets of element orders shows all five to be pairwise distinct (checked here): the cyclic group of order eight gives one, two, four, four, eight, eight, eight, eight; the product of the cyclic groups of orders four and two gives one, two, two, two, four, four, four, four; the triple product of cyclic groups of order two gives one together with seven twos; the symmetry group of the square gives one, two, two, two, two, two, four, four; and the quaternion group gives one, two, four, four, four, four, four, four. Three of the five are abelian, and the two out of reach are the symmetry group of the square and the quaternion group – the very pair that Paper 152 found indistinguishable by their character tables. Where one functor cannot reach and where one invariant cannot see point at the same two objects.
Third, and this is the core. A missing right adjoint can be seen by counting a coproduct. A functor with a right adjoint preserves colimits, coproducts among them. In the category of abelian groups, however, the coproduct of the cyclic group of order three with itself has nine elements while the disjoint union of the underlying sets has six. For order two the two numbers agree at four, but that is a coincidence and they split at three. In the category of groups the failure is more extreme: the coproduct of the cyclic group of order two with itself is the infinite dihedral group. Realising as isometries of the integers the map that reverses sign and the map that subtracts from one, both have order two while their product sends three to two, one, zero, minus one, minus two, minus three, minus four, minus five, and so has infinite order (checked here). The underlying set of the coproduct is infinite while the disjoint union has four elements. Coproducts are not preserved, so no right adjoint exists – and since the left adjoint, the free construction, does exist, adjunction is not symmetric between left and right.
Fourth, some forgetful functors have both adjoints. The forgetful functor from topological spaces to sets is one: the discrete topology is left adjoint and the indiscrete is right adjoint. On a three-point set the discrete topology has eight open sets and the indiscrete has two. Against the Sierpinski space, the continuous maps from the discrete three-point space number eight and the maps of sets also number eight, while the continuous maps from the Sierpinski space to the indiscrete three-point space number nine and the maps of sets also number nine (computed here). Forgetting a topology is therefore a gentler act than forgetting a group law; one phrase, forgetful functor, names things that differ down to which adjoints exist.
Fifth, what is freely added does not balance what was forgotten. The underlying set of the cyclic group of order two has two elements, so the free group built on it is the free group on two letters, whose reduced words of lengths zero through three number one, four, twelve and thirty-six, fifty-three in all (counted here). The free group on two letters is infinite while the original group has two elements; the counit is onto with infinite kernel, and forgetting and freely rebuilding does not return the object. An adjunction is not an equivalence, and the gap is the counit.
Applied to the corpus: the map taking a metric to its conformal class is faithful and not full, which is the structure rung. In four dimensions the metric has ten components, the conformal class nine, and what drops is one, which Paper 150 identified as a density. In the vocabulary here, light forgets the density is the structure rung, and ten minus nine is the size of what was forgotten. Drop, ignore and approximate must therefore not be called by one name: all are functors, but whether the thing can be recovered depends on whether what was forgotten is a property, structure, or stuff.
IMPORTANT (scope): a census of established standard material (Layer 2). What is not claimed: any new mathematical theorem (there is none). What is added: deciding which of the three rungs a given forgetting sits on by counting maps, exhibiting the absence of a right adjoint as one number, the size of a coproduct, and saying which rung Paper 150’s light forgets the density belongs to. What remains: staring at the objects will not tell you what was forgotten; counting the maps will. And if what was forgotten is a property one need only choose the objects again, if it is structure it must be carried in from outside, and if it is stuff the distinction itself is gone – one word, drop, hiding the recoverable case and the unrecoverable one.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). What Was Forgotten Can Be Read Off by Counting Maps [Paper 153]. Zenodo.
DOI: 10.5281/zenodo.22084355
ORCID: 0009-0006-8587-3450