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In usual presentations of categorical set theories such ascategorical set theories? such as ETCS?, the set theory is usually a theory of sets and functions? in an abstract category, with elements? being defined as the global elements?, the morphisms? out of the terminal object? . However, this approach poses a few conceptual and practical issues, namely that in ordinary mathematical practice, elements are not the same as functions out of the terminal set, although they are isomorphic to each other, and function evaluation of elements defined as function composition of functions out of the terminal set is an abuse of notation.$\mathbb{1}$, although $A$ is in bijection with the function set $A^\mathbb{1}$, and function evaluation of elements defined as function composition of functions out of the terminal set is an abuse of notation.

There are other structural set theories?, such as SEAR , which explicitly put in the elements of a set as a primitive of the theory: theory. this In is such equivalent a to theory saying involving that sets and functions, function evaluation would be a primitive of the category theory, is rather an than derived from function composition ofconcrete category rather than an abstract category. In such a presentation involving sets and functions, function evaluation? would be a primitive of the theory, rather than derived from function composition of global elements? , and the axiom of function extensionality is likewise defined directly on the elements. The Such presentation of such a concrete categorical set theory reads more like a traditional presentation of set theory in terms of sets and elements, rather than category theory. However, in contrast toSEAR, this theory is essentially the same structurally as ETCS?: as a theory of sets and functions.

In this presentation, we will be adapting Tom Leinster’s presentation of ETCS to the concrete setting. Some of these axioms might be redundant, but that was true of Tom Leinster’s original presentation.Tom Leinster?’s presentation of ETCS.

Basic primitives

Our theory has the following primitives:

• Some things called sets;

• For every set $A$, these things called elements in $A$, with elements $a$ in $A$ written as $a \in A$;

• For every set $A$ and $B$, these things called functions from $A$ to $B$, with functions $f$ from $A$ to $B$ written as $f:A \to B$;

• For every set $A$ and $B$, an operation called function evaluation assigning every element $a \in A$ and function $f:A \to B$ an element $f(a) \in B$;

• For every set $A$, a function $id_A:A \to A$ called the identity function of $A$;

• For every set $A$, $B$, and $C$, an operation called function composition assigning every function $f:A \to B$ and $g:B \to C$ a function $g \circ f:A \to C$;

Equality of elements and functions

For every set $A$ and elements $a \in A$ and $b \in A$, there is a relation $a = b$ called equality of elements, such that

• for every element $a \in A$, $a = a$
• for every element $a \in A$ and $b \in A$, $a = b$ implies that $b = a$
• for every element $a \in A$, $b \in A$, and $c \in A$, $a = b$ and $b = c$ implies that $a = c$.

For every set $A$ and $B$ and functions $f:A \to B$ and $g:A \to B$, there is a relation $f = g$ called equality of functions, such that

• for every function $f:A \to B$, $f = f$
• for every function $f:A \to B$ and $g:A \to B$, $f = g$ implies that $g = f$
• for every function $f:A \to B$, $g:A \to B$, and $h:A \to B$, $f = g$ and $g = h$ implies that $f = h$.

Basic axioms for functions

Axiom of identity functions. For every set $A$ and for every element $a \in A$, $id_A(a) = a$.

Axiom of composition/evaluation equivalence. For every set $A$, $B$, and $C$, and for every element $a \in A$, $(g \circ f)(a) = g(f(a))$.

Axiom of extensionality. For every set $A$ and $B$ and for every function $f:A \to B$ and $g:A \to B$, if $f(x) = g(x)$ for all elements $x \colon A$, then $f = g$.

The associativity and unit laws of function composition follow from the axioms:

• For every set $A$ and $B$, function $f:A \to B$, and element $a \in A$,

  $$(g \circ id_A)(a) = g(id_A(a)) = g(a)$$

  and extensionality implies that $g \circ id_A = g$.

• For every set $A$ and $B$, function $f:A \to B$, and element $a \in A$,

  $$(id_B \circ g)(a) = id_B(g(a)) = g(a)$$

  and extensionality implies that $id_B \circ g = g$.

• For every set $A$, $B$, $C$, and $D$, function $f:A \to B$, $g:B \to C$, and $h:C \to D$, and element $a \in A$,

  $$(h \circ (g \circ f))(a) = h((g \circ f)(a)) = h(g(f(a)) = (h \circ g)(f(a)) = ((h \circ g) \circ f)(a)$$

  and extensionality implies that $h \circ (g \circ f) = (h \circ g) \circ f$.

Thus, these axioms imply that the collection of sets, functions, and elements form a category.

Other axioms

Axiom of singletons. There is a set $\mathbb{1}$, called a singleton, with a unique element $* \in \mathbb{1}$, called a point.

Axiom of Cartesian products. For every set $A$ and $B$, there is a set $A \times B$, called a Cartesian product of $A$ and $B$, with a function $p_A:A \times B \to A$ called the projection onto $A$ and a function $p_B:A \times B \to B$ called the projection onto $B$, such that given two elements $a \in A$ and $b \in B$ there is a unique element $a, b \in A \times B$ such that $p_A(a, b) = a$ and $p_B(a, b) = b$.

Axiom of fibers. For every set $A$ and $B$, element $b \in B$, and function $f:A \to B$, there is a set $f^{-1}(b)$ called the fiber of $f$ over $b$ with a function $i:f^{-1}(b) \to A$, such that for every element $a \in f^{-1}(b)$, $f(i(a)) = b$. A fiber of $f$ over $b$ is also called the solution set of the equation $f(x) = b$.

An injection is a function $f:A \to B$ such that for every element $a \in A$ and $b \in A$, if $f(a) = f(b)$, then $a = b$.

Axiom of truth values. There is a set $\Omega$ called a set of truth values with an element $\bot \in \Omega$ called true such that for every set $A$ and $B$ and injection $f:A \to B$, there is a unique function $\chi_A:B \to \Omega$ called the classifying function of $A$ such that $A$ is the fiber of $\chi_A$ over $\bot$.

Axiom of function sets. For every set $A$ and $B$ there is a set $B^A$ called the function set with a function $(-)((-)):B^A \times A \to B$ such that for every set $C$ and function $f:C \times A \to B$ there is a function $g:C \to B^A$ such that for all elements $c \in C$ and $a \in A$, $g(c)(a) = f(c, a)$.

Axiom of natural numbers. There exists a set $\mathbb{N}$ called a set of natural numbers with an element $0 \in \mathbb{N}$ and a function $s:\mathbb{N} \to \mathbb{N}$ such that for every other set $A$ with an element $a \in A$ and a function $g:A \to A$, there is a unique function $f:\mathbb{N} \to A$ such that $f(0) = a$ and $f(s(a)) = g(f(a))$.

A surjection is a function $f:A \to B$ such that for every $b \in B$ the fiber of $f$ at $b$ is inhabited. A right inverse of $f$ is a function $g:B \to A$ such that for all elements $a \in A$, $f(g(a)) = a$. A choice set is a set $B$ such that all surjections into $B$ have right inverses.

Axiom of choice. Every set is a choice set.

These axioms imply that the collection of sets, functions, and elements form a model of ETCS?.

References

• Tom Leinster?, Rethinking set theory, (arxiv:1212.6543)