basic constructions:
strong axioms
natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory
A preset is a set without an equality relation. Conversely, a set may be defined as a preset $X$ equipped with an equality relation (technically an equivalence prerelation on $X$).
In his seminal work The Foundations of Constructive Analysis (1967), Errett Bishop explained what you must do to define a set (see also Bishop set) in three steps:
If you only do (1), then you don't have a set, according to Bishop; you only have a preset.
A given preset may define many different sets, depending on the equality relation. For example, the set $Q$ of positive rational numbers may be defined using the same preset as the set $Z \times Z$ of ordered pairs of positive integers, but the equality relation is different; two pairs $(a,b)$ and $(c,d)$ of integers are equal iff $a = c$ and $b = d$, while two rational numbers $a/b$ and $c/d$ are equal iff $a d = b c$. (Of course, these definitions require that one already have the set $Z$ of positive integers, including its equality relation, and the operation of multiplication on it.)
As functions go between sets, so prefunctions go between presets. Bishop used the term ‘operation’ instead of ‘prefunction’, but ‘operation’ has many other meanings.) Even if $X$ and $Y$ are sets, a prefunction from $X$ to $Y$ is not the same as a function from $X$ to $Y$, because a prefunction need not preserve equality; that is, we may have $a = b$ without $f(a) = f(b)$. Conversely, we may define a function as a prefunction (between sets) that preserves equality.
For example, consider the identity prefunction on the underlying preset of both $Q$ and $Z \times Z$, as defined above. From $Z \times Z$ to $Q$, this is a function, since $a/b = c/d$ if $(a,b) = (c,d)$. But from $Q$ to $Z \times Z$, it is not a function, since (for example) $2/4 = 3/6$ but $(2,4) \neq (3,6)$. A related example is the operation of taking the numerator of a (positive) rational number; from $Q$ to $Z$, we may view this as a prefunction but not as a function, although it is a function on $Z \times Z$.
In general, the prefunctions from $X$ to $Y$ form a preset, since there is no way to compare them for equality. (Of course, it is still impredicative, at least in the classical sense, to form this preset.) However, if $Y$ is a set, then these prefunctions do form a set, with $f = g$ defined to mean that $f(a) = g(a)$ for every $a$ in $X$. If $X$ is also a set, then the function set from $X$ to $Y$ is a subset of this set of prefunctions.
A (say binary) prerelation between $X$ and $Y$ may be thought of as a prefunction from $X \times Y$ to truth values. Even if one is too predicative to allow a (pre)set of truth values, still one may have a notion of prerelation, by fiat if nothing else. Note that one can compare prerelations for equality; $R = S$ means that $a \sim_R b$ if and only if $a \sim_S b$. (In other words, a preset of truth values becomes a set under the biconditional, so we can compare functions to it.) We define a relation between sets to be a prerelation that respects equality.
Many properties of relations can also be predicated of prerelations, but not all. In particular, prerelations may be reflexive, symmetric, and transitive, so we have a notion of equivalence prerelation, which completes the definition of sets in terms of presets. A prerelation may also be entire, but it makes no sense to ask if it is functional unless $Y$ is a set. In that case, there is a correspondence between prefunctions and functional entire prerelations as usual. In general, however, there is no way to define the prerelation corresponding to a given prefunction (which would be a sort of pre-graph). In other words, the idea that functions are certain relations (namely the functional entire ones) does not extended to prefunctions and prerelations unless $Y$ is a set.
Many foundations based on type theory, such as those of Per Martin-Löf and Thierry Coquand, use types (sometimes called ‘sets’, but they don't have quotients) which behave something like presets (and are sometimes even called ‘presets’). A set (sometimes called ‘setoid’) is defined as above, as a type with an equality relation. However, these usually come equipped with ‘identity’ relations, which are equality relations in all but name; this amounts to saying that every preset has a free set, a completely presented set. (Note that the cofree set on a preset always exists; it is a subsingleton.) They usually also adopt an axiom of choice for prefunctions that, together with the identity relations, proves the presentation axiom (a weak form of the full axiom of choice) for general sets.
It is possible to develop type-theoretic foundations in which presets are not equipped with identity relations (only metamathematical identity or interconvertibilty judgements); see preset for some discussion. The presentation axiom is not provable in the base theory, although it is provable in the impredicative version (where identity relations can be defined, following Leibniz's definition of equality). A similar result holds for SEAR+ε.
The sorts in Michael Makkai's FOLDS are presets. FOLDS is very different from the other foundations considered above, since it is based strictly on prerelations and has no notion of prefunction. As far as I can tell, it therefore does not prove the presentation axiom.
If you are willing to accept the presentation axiom, then you can define a notion of preset internal to a given theory of sets: as a projective set. (With the full axiom of choice, therefore, a preset is simply a set.) Alternatively, you might forgo presets as such but define a prefunction between sets to be an entire relation; although not everything translates, some of the properties are similar.
To make the principle of equivalence hold automatically, a category should have only a preset of objects and only its hom-sets as sets. Then a category whose set of objects is a set may be called a strict category, which is really a special case of a strict ∞-category. Alternatively, one may keep sets as sets but adopt preclasses; then a small category is strict but a large category is not.
In constructive mathematics, we want the real numbers to form a linearly ordered Heyting field $R$ with completeness for located Dedekind cuts. Using power sets and a set $N$ of natural numbers, one may form $R$ directly as a subset of $\mathcal{P}N$ (or something equivalent), but what if you wish to be (at least weakly) predicative? Using function sets, one may form the Cauchy reals as a subquotient of $N^N$, but these are complete in the desired sense only if a weak form of countable choice (which follows from either the presentation axiom or excluded middle) holds. (Essentially, there may not be enough sequences of natural numbers.) Alternatively, if you have them, you can use prefunction sets and form $R$ as a subquotient of the set of presequences of natural numbers.
The construction of $R$ above may also be done with entire relations if the axiom of fullness holds (see also real numbers object). Conversely, the axiom of fullness follows from the existence of presets of prefunctions; in addition to defining a functional entire prerelation, a prefunction between sets also defines an entire relation, and the set of these satisfies fullness. (This is related to the idea that prefunctions between sets may be formalised as entire relations.)
See also the discussion at net about how to force the domain of a net to be partial order, by using either entire relations or prefunctions as nets.
Sometimes one finds a foundation of predicative mathematics in which it appears to be impossible to construct quotient sets, leading to much hand-wringing. (For a simple example, simply remove the axiom of power sets from ZFC as normally presented.) However, if you reinterpret the nominal ‘sets’ as presets and define a set as a preset equipped with an equivalence prerelation, then quotient sets exist after all. (In impredicative mathematics, there is a more familiar construction of a quotient set available, as a subset of a power set.)
Assuming (as usual) that the original foundation had equality relations on its sets, there will be identity prerelations on the presets, leading to a special class of sets (those which arise from equipping a preset with its identity prerelation) which we may again call the completely presented sets.
When you do this, the new kind of set is usually called a ‘setoid’, and then there may be hand-wringing about the need to use setoids instead of sets as one would like. But if you didn't have quotient sets originally, then you shouldn't have been talking about ‘sets’ in the first place; theories of sets without quotient sets are really theories of presets. (You can also use the term ‘type’ if it seems appropriate.)
In general, the category of sets is the ex/lex completion of the original category of presets (at least assuming certain structure in the original theory).
computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory