nLab
Bishop set

Idea

A Bishop set is a notion of set in constructive mathematics.

In (Bishop) the notion of set is specified by stating that a set has to be given by a description of how to build elements of this set and by giving a binary relation of equality, which has to be an equivalence relation.

A function from a set $A$ to a set $B$ is then given by an operation, which is compatible with the equality (i.e. two elements which are equal in A are mapped to two elements which are equal in B), and is described as “a finite routine $f$ which assigns an element $f(a)$ of $B$ to each given element $a$ of $A$”. This notion of routine is left informal but must “afford an explicit, finite, mechanical reduction of the procedure for constructing $f(a)$ to the procedure for constructing $a$.”

These ideas have been formalized in type theory, where they serve to set up set theory in an ambient type theoretic logical framework. See Formalization in type theory below.

Formalization in type theory

It is direct and natural to represent formally the notion of Bishop sets in type theory. See for instance (Palmgren05) for an exposition and (Coquand-Spiwack, section 3) for a brief list of the axioms.

In the context of type theory a Bishop set is sometimes called a setoid.

Properties

The predicative topos of Bishop sets

In much of set theory the category Set of all sets is a Topos. For Bishop sets formalized in type theory this is not quite the case. Instead:

Theorem

The category of Bishop sets in Martin-Löf dependent type theory is a strong predicative topos.

This is (van den Berg, theorem 6.2), based on (Moerdijk-Palmgren, section 7).

Related concepts

• ETCS

computational trinitarianism = propositions as types +programs as proofs +relation type theory/category theory

logic	category theory	type theory
true	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	initial object	empty type
proposition	(-1)-truncated object	h-level 1-type/h-prop
proof	generalized element	program
conjunction	product	product type
disjunction	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	internal hom	function type
negation	internal hom into initial object	function type into empty type
universal quantification	dependent product	dependent product type
existential quantification	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
equivalence	path space object	identity type
equivalence class	quotient	quotient type
induction	colimit	inductive type, W-type, M-type
higher induction	higher colimit	higher inductive type
completely presented set	discrete object/0-truncated object	h-level 2-type/preset/h-set
set	internal 0-groupoid	Bishop set/setoid
universe	object classifier	type of types
modality	closure operator monad	modal type theory, monad (in computer science)

References

The original publication is

• Errett Bishop, Foundations of Constructive Analysis. New York: McGraw-Hill (1967)

in the context of constructive analysis. A detailed discussion is in

• M. Hofmann, Extensional constructs in Intensional Type theory, Springer (1997)

Reviews include

• Erik Palmgren, Bishop’s set theory (2005) (pdf)

• Erik Palmgren, Constructivist and Structuralist Foundations: Bishop’s and Lawvere’s Theories of Sets (2009) (web)

The formalization of Bishop set is reviewed for instance in section 3 of

• Thierry Coquand, Arnaud Spiwack, Towards constructive homological algebra in type theory (pdf)

(there with an eye towards further formalization of homological algebra).

Some of the text above is taken from this section.

The predicative topos formed by Bishop sets in type theory is discussed in

• Ieke Moerdijk, Erik Palmgren, Wellfounded trees in categories. Ann. Pure Appl. Logic, 104(1-3):189–218, (2000)

• Benno van den Berg, Predicative toposes (arXiv:1207.0959)

A formalization of constructive set theory in terms of setoids in intensional type theory coded in Coq is discussed in

• Erik Palmgren, Olov Wilander, Constructing categories of setoids of setoids in type theory, (pdf, Coq)

See also

• UF-IAS-2012, On the setoid model of type theory (video)

• UF-IAS-2012, Setoid model of type theory formalized