nLab computational trilogy

Idea

A profound cross-disciplinary insight has emerged – starting in the late 1970s, with core refinements in recent years – observing that three superficially different-looking fields of mathematics,

• computation/programming languages

• formal logic/type theory

• ∞-category theory/∞-topos theory (algebraic topology)

are but three different perspectives on a single underlying phenomenon at the foundations of mathematics:

Classical

Plain

Under the identifications

1. propositions as types,

2. programs as proofs,

3. relation between type theory and category theory

the following notions are equivalent:

In intuitionistic logic and type theory:	In programming languages and computation:	In category theory and topos theory:
A proof of a proposition, or a term of some type.	A program/λ-term with output of some data type.	A generalized element of an object.

whence these three subjects are but three perspectives on a single underlying phenomenon.

This insight dates from the late 1970s; an early record is Lambek & Scott 86; it is explicitly highlighted as a trilogy (Wikipedia: “three works of art that are connected and can be seen either as a single work or as three individual works”) in Melliès 06, Sec. 1:

From Melliès 06

(Notice that Melliès 06 on p.2 does mean to regard λ-calculus as programming language.)

In Harper 11 the profoundness of the trilogy inspires the following emphatic prose, alluding to the doctrinal position of ‘trinitarianism’:

The central dogma of computational trinitarianism holds that Logic, Languages, and Categories are but three manifestations of one divine notion of computation. There is no preferred route to enlightenment: each aspect provides insights that comprise the experience of computation in our lives.

Computational trinitarianism entails that any concept arising in one aspect should have meaning from the perspective of the other two. If you arrive at an insight that has importance for logic, languages, and categories, then you may feel sure that you have elucidated an essential concept of computation–you have made an enduring scientific discovery.

For more detailed review see Eades 12, Sec. 3.

Parametrized

More is true: Since

1. computation happens in contexts and is proof relevant;

2. categories give rise to their systems of slice categories and are in general (∞,1)-categories;

3. types may depend on other types and are in general homotopy types

the traditional computational trilogy above enhances to read as follows:

In dependent homotopy type theory:	In programming languages and computation:	In locally cartesian closed (∞,1)-categories/(∞,1)-toposes:
A term of some type in context.	A program outputting some data type in context.	A generalized element of an object in a slice.

See also Shulman 18.

In this deeper form yet another equivalence – to algebraic topology (Sati Schreiber 20, p. 5) – opens up, as generalized elements in an (∞,1)-topos may equivalently be regarded as cocycles in (non-abelian) cohomology, and in twisted cohomology if in a slice (∞,1)-category (Sati Schreiber 20 p. 6, FSS 20), whence we have a computational tetralogy:

In dependent homotopy type theory:	In programming languages and computation:	In locally cartesian closed ∞-categories/∞-toposes:	In non-abelian cohomology param. homotopy theory:
A term of some type in context.	A program of some data type in context.	An element of an object in a slice.	A cocycle in twisted cohomology.

(graphics from SS22)

Quantum

Plain

An analogous trilogy is seen under passage:

1. from logic/type theory to linear logic/linear type theory;

2. from computation to quantum computation;

3. from cartesian closed categories to closed monoidal categories

This is the main point of Melliès 06, Sec. 1, only that where Melliès shows “proof nets” (p. 4) we refer to them as “quantum computation” for better emphasis, following Abramsky-Coecke 04, Abramsky & Duncan 05, Duncan 06; going back to Pratt 92:

From Melliès 06, p. 4

See also Baez & Stay 09.

Parametrized

Combining the classical parametrized trilogy with the plain quantum trilogy, as one passes

• from classical computation to classically controlled quantum computation on linear spaces of quantum states parametrized over classical data types;

• from dependent intuitionistic homotopy type theory to dependent linear type theory of dependent stable homotopy types;

• from locally cartesian closed categories/(∞,1)-categories to indexed monoidal categories/(∞,1)-categories of parametrized spectra; which in the language of algebraic topology is the context of twisted generalized cohomology theory.

there appears the “classically controlled quantum computational tetralogy”:

(graphics from SS22)

In dependent linear homotopy type theory:	In classically controlled quantum programming languages:	In indexed monoidal ∞-cats of par. spectra:	In Whitehead-generalized twisted cohomology theory:
A term of some type in context.	A quantum circuit controlled by classical data.	An element of an object in a slice.	A cocycle in twisted cohomology.

(along the lines of Schreiber 14, Nuiten 13,

• with parametrized stable homotopy theory understood as twisted cohomology theory as in Ando, Blumberg & Gepner 10, Ando, Blumberg, Gepner & Hopkins 14, Fiorenza, Sati, Schreiber 20;

• with dependent linear homotopy type theory understood as, e.g., in Riley, Finster & Licata 21 following Schreiber 13 Prop. 4.1.9;

• with classically controlled quantum computation seen as dependent linear type theory, as stated fully explicitly in Fu, Kishida & Selinger 20, Fu, Kishida, Ross & Selinger 20 and more tentatively before in Vakar 14, Vakar 15, Vakar 17, following Schreiber 14)

$\,$

(from SS22)

Rosetta stone

The following shows a rosetta stone dictionary with more details:

(NB. This table shows the computational aspect mostly under “type theory”…)

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

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

Related concepts

• syntax-semantics duality

• relation between category theory and type theory

• programs as proofs, propositions as types

• initiality conjecture

References

In the introduction of

• Paul-André Melliès, Functorial boxes in string diagrams, Procceding of Computer Science Logic 2006 in Szeged, Hungary. 2006 (hal:00154243, pdf, pdf)

the insight is recalled to have surfaced in the 1970s, with an early appearance in print being the monograph

• Joachim Lambek, Phil Scott, Introduction to Higher Order Categorical Logic, Cambridge Studies in Advanced Mathematics Vol. 7. Cambridge University Press, 1986 (ISBN:978-0-521-24665-1)

See also at History of categorical semantics of linear type theory for more on this.

A exposition of the relation between the three concepts is in

• Robert Harper, The Holy Trinity (2011) (web, wayback machine snapshot)

• Harley Eades, Section 3 of: Type Theory and Applications, 2012 (pdf, pdf)

• Dan Frumin, Computational trinitarianism, Feb 2014 (prezi slides)

An exposition with emphasis on linear logic/quantum logic and the relation to physics is in

• John Baez, Mike Stay: Physics, Topology, Logic and Computation: A Rosetta Stone, in New Structures for Physics, Lecture Notes in Physics 813 Springer (2011) 95-174 [arXiv:0903.0340, doi:10.1007/978-3-642-12821-9_2]

Discussion in the context of homotopy type theory:

• Mike Shulman, Homotopical trinitarianism: A perspective on homotopy type theory, 2018 (pdf slides, pdf)

For further references see at programs as proofs, propositions as types, and relation between category theory and type theory.

Textbooks on the foundations of mathematics and foundations of programming language which connect via the common theme of type theory/categorical logic include the following:

• Paul Taylor, Practical Foundations of Mathematics (web)

• William Lawvere, Robert Rosebrugh, Sets for Mathematics, Cambridge UP 2003 (book homepage, GoogleBooks, pdf)

• Robert Harper, Practical Foundations for Programming Languages, Cambridge University Press (2016) (ISBN:9781107150300)

See also

• Joseph A. Goguen, A Categorical Manifesto. In Mathematical Structures in Computer Science 1 1 (1991) 49-67 (doi:10.1017/S0960129500000050, CiteSeerX.