category theory

# Contents

## Idea

Type theory and certain kinds of category theory are closely related. By a syntax-semantics duality one may view type theory as a formal syntactic language or calculus for category theory, and conversely one may think of category theory as providing semantics for type theory. The flavor of category theory used depends on the flavor of type theory; this also extends to homotopy type theory and certain kinds of (∞,1)-category theory.

## Overview

flavor of type theory$\;$equivalent to$\;$flavor of category theory
intuitionistic propositional logic/simply-typed lambda calculuscartesian closed category
multiplicative intuitionistic linear logicsymmetric closed monoidal category(various authors since ~68)
first-order logichyperdoctrine(Seely 1984a)
classical linear logicstar-autonomous category(Seely 89)
extensional dependent type theorylocally cartesian closed category(Seely 1984b)
homotopy type theory without univalence (intensional M-L dependent type theory)locally cartesian closed (∞,1)-category(Cisinski 12-(Shulman 12)
homotopy type theory with higher inductive types and univalenceelementary (∞,1)-topossee here
dependent linear type theoryindexed monoidal category (with comprehension)(Vákár 14)

## Theorems

We discuss here formalizations and proofs of the relation/equivalence between various flavors of type theories and the corresponding flavors of categories.

### First-order logic and hyperdoctrines

###### Theorem

The functors

constitute an equivalence of categories

$FirstOrderTheories \stackrel{\overset{Lang}{\leftarrow}}{\underset{Cont}{\to}} Hyperdoctrines \,.$

### Dependent type theory and locally cartesian closed categories

We discuss here how dependent type theory is the syntax of which locally cartesian closed categories provide the semantics. For a dedicated discussion of this (and the subtle coherence issues involved) see also at categorical model of dependent types.

###### Theorem

There are 2-functors

that constitute an equivalence of 2-categories

$MLDependentTypeTheories \stackrel{\overset{Lang}{\leftarrow}}{\underset{Cont}{\to}} LocallyCartesianClosedCategories \,.$

This was originally claimed as an equivalence of categories (Seely, theorem 6.3). However, that argument did not properly treat a subtlety central to the whole subject: that substitution of terms for variables composes strictly, while its categorical semantics by pullback is by the very nature of pullbacks only defined up to isomorphism. This problem was pointed out and ways to fix it were given in (Curien) and (Hofmann); see categorical model of dependent types for the latter. However, the full equivalence of categories was not recovered until (Clairambault-Dybjer) solved both problems by promoting the statement to an equivalence of 2-categories, see also (Curien-Garner-Hofmann). Another approach to this which also works with intensional identity types and hence with homotopy type theory is in (Lumsdaine-Warren 13).

We now indicate some of the details.

#### Type theories

For definiteness, self-containedness and for references below, we say what a dependent type theory is, following (Seely, def. 1.1).

###### Definition

A Martin-Löf dependent type theory $T$ is a theory with some signature of dependent function symbols with values in types and in terms (…) subject to the following rules

1. type formation rules

1. $1$ is a type (the unit type);

2. if $a, b$ are terms of type $A$, then $(a = b)$ is a type (the equality type);

3. if $A$ and $B[x]$ are types, $B$ depending on a free variable of type $A$, then the following symbols are types

1. $\prod_{a : A} B[a]$ ([dependent product]), written also $(A \to B)$ if $B[x]$ in fact does not depend on $x$;

2. $\sum_{a : A} B[a]$ (dependent sum), written also $A \times B$ if $B[x]$ in fact does not depend on $x$;

2. term formation rules

1. $* \in 1$ is a term of the unit type;

2. (…)

3. equality rules

1. (…)

#### Category of contexts

###### Definition

Given a dependent type theory $T$, its category of contexts $Con(T)$ is the category whose

• objects are the types of $T$;

• morphisms $f : A \to B$ are the terms $f$ of function type $A \to B$.

Composition is given in the evident way.

###### Proposition

$Con(T)$ has finite limits and is a cartesian closed category.

###### Proof

Constructions are straightforward. We indicated some of them.

Notice that all finite limits (as discussed there) are induced as soon as there are all pullbacks and equalizers. A pullback in $Con(T)$

$\array{ P &\to& A \\ \downarrow && \downarrow^{\mathrlap{f}} \\ B &\stackrel{g}{\to}& C }$

is given by

$P \simeq \sum_{a : A} \sum_{b \in B} (f(a) = g(b)) \,.$

The equalizer

$P \to A \stackrel{\overset{f}{\to}}{\underset{g}{\to}} B$

is given by

$P = \sum_{a : A} (f(a) = g(a)) \,.$

Next, the internal hom/exponential object is given by function type

$[A,B] \simeq (A \to B) \,.$
###### Proposition

$Con(T)$ is a locally cartesian closed category.

###### Proof

Define the $Con(T)$-indexed hyperdoctrine $P(T)$ by taking for $A \in Con(T)$ the category $P(T)(A)$ to have as objects the $A$-dependent types and as morphisms $(a : A \vdash X(a) : type) \to (a : A \vdash Y(a) : type)$ the terms of dependent function type $(a : A \vdash t : (X(a) \to Y(a)))$.

This is cartesian closed by the same kind of argument as in the previous proof. It is now sufficient to exhibit a compatible equivalence of categories with the slice category $Con(T)_{/A}$.

$Con(T)_{/A} \simeq P(T)(A) \,.$

In one direction, send a morphism $f : X \to A$ to the dependent type

$a : A \vdash f^{-1}(a) \coloneqq \sum_{x : X} (a = f(x)) \,.$

Conversely, for $a : A \vdash X(a)$ a dependent type, send it to the projection $\sum_{a : A} X(a) \to A$.

One shows that this indeed gives an equivalence of categories which is compatible with base change (Seely, prop. 3.2.4).

###### Definition

For $T$ a dependent type theory and $C$ a locally cartesian closed category, an interpretation of $T$ in $C$ is a morphism of locally cartesian closed categories

$Con(T) \to C \,.$

An interpretation of $T$ in another dependent type theory $T'$ is a morphism of locally cartesian closed categories

$Con(T) \to Con(T') \,.$

#### Internal language

###### Proposition

Given a locally cartesian closed category $C$, define the corresponding dependent type theory $Lang(C)$ as follows

• the non-dependent types of $Lang(C)$ are the objects of $C$;

• the $A$-dependent types are the morphisms $B \to A$;

• a context $x_1 : X , x_2 : X, \cdots , x_n : X_n$ is a tower of morphisms

$\array{ X_n \\ \downarrow \\ \cdots \\ \downarrow \\ X_2 \\ \downarrow \\ X_1 }$
• the terms $t[x_A] : B[x_A]$ are the sections $A \to B$ in $C_{/A}$

• the equality type $(x_A = y_A)$ is the diagonal $A \to A \times A$

### Homotopy type theory and locally cartesian closed $(\infty,1)$-categories

All of the avive has an analog in (∞,1)-category theory and homotopy type theory.

###### Proposition

Every presentable and locally cartesian closed (∞,1)-category has a presentation by a type-theoretic model category. This provides the categorical semantics for homotopy type theory (without, possibly, the univalence axiom).

This includes in particular all (∞-stack-) (∞,1)-toposes (which should in addition satisfy univalence). See also at internal logic of an (∞,1)-topos.

Some form of this statement was originally formally conjectured in (Joyal 11), following (Awodey 10). For more details see at locally cartesian closed (∞,1)-category.

### Univalent homotopy type theory and elementary $(\infty,1)$-toposes

More precise information can be found on the homotopytypetheory wiki.

A (locally presentable) locally Cartesian closed (∞,1)-category (as above) which in addition has a system of object classifiers is an ((∞,1)-sheaf-)(∞,1)-topos.

It has been conjectured in (Awodey 10) that this object classifier is the categorical semantics of a univalent type universe (type of types), hence that homotopy type theory with univalence has categorical semantics in (∞,1)-toposes. This statement was proven for the canonical $(\infty,1)$-topos ∞Grpd in (Kapulkin-Lumsdaine-Voevodsky 12), and more generally for (∞,1)-presheaf $(\infty,1)$-toposes over elegant Reedy categories in (Shulman 13).

In these proofs the type-theoretic model categories which interpret the homotopy type theory syntax are required to provide type universes that behave strictly under pullback. This matches the usual syntactically convenient universes in type theory (either a la Russell or a la Tarski), but more difficult to implement in the categorical semantics. More flexibly, one may consider syntactic type universes weakly à la Tarski (Luo 12, Gallozzi 14). These are more complicated to work with syntactically, but should have interpretations in a (type-theoretic model categories presenting) any (∞,1)-topos. Discussion of univalence in this general flexible sense is in (Gepner-Kock 12). For the general syntactic issue see at

While (∞,1)-sheaf (∞,1)-toposes are those currently understood, the basic type theory with univalent universes does not see or care about their local presentability as such (although it is used in other places, such as the construction of higher inductive types). It is to be expected that there is a decent concept of elementary (∞,1)-topos such that homotopy type theory with univalent type universes and some supply of higher inductive types has categorical semantics precisely in elementary (∞,1)-toposes (as conjectured in Awodey 10). But the fine-tuning of this statement is currently still under investigation.

Notice that this statement, once realized, makes (or would make) Univalent HoTT+HITs a sort of homotopy theoretic refinement of foundations of mathematics in topos theory as proposed by William Lawvere. It could be compared to his elementary theory of the category of sets, although being a type theory rather than a theory in first-order logic, it is more analogous to the internal type theory of an elementary topos.

## References

An elementary exposition of in terms of the Haskell programming language is in

The equivalence of categories between first order theories and hyperdoctrines is discussed in

• R. A. G. Seely, Hyperdoctrines, natural deduction, and the Beck condition, Zeitschrift für Math. Logik und Grundlagen der Math. (1984) (pdf)

Surveys inclue

• Tom Hirschowitz, Introduction to categorical logic (2010) (pdf)

(see the discussion building up to the theorem on slide 96)

• Roy Crole, Deriving category theory from type theory, Theory and Formal Methods 1993 Workshops in Computing 1993, pp 15-26

• Maria Maietti, Modular correspondence between dependent type theories and categories including pretopoi and topoi, Mathematical Structures in Computer Science archive Volume 15 Issue 6, December 2005 Pages 1089 - 1149 (pdf)

The equivalence between linear logic and star-autonomous categories is due to

• R. A. G. Seely, Linear logic, $\ast$-autonomous categories and cofree coalgebras, Contemporary Mathematics 92, 1989. (pdf, ps.gz)

and reviews/further developments are in

• G. M. Bierman, What is a Categorical Model of Intuitionistic Linear Logic? (web)

• Andrew Graham Barber, Linear Type Theories, Semantics and Action Calculi, 1997 (web, pdf)

• Paul-André Melliès , Categorial Semantics of Linear Logic, in Interactive models of computation and program behaviour, Panoramas et synthèses 27, 2009 (pdf)

An adjunction between the category of type theories with product types and toposes is discussed in chapter II of

• Joachim Lambek, P. Scott, Introduction to higher order categorical logic, Cambridge University Press (1986) .

The equivalence of categories between locally cartesian closed categories and dependent type theories was originally claimed in

• R. A. G. Seely, Locally cartesian closed categories and type theory, Math. Proc. Camb. Phil. Soc. (1984) 95 (pdf)

following a statement earlier conjectured in

• Per Martin-Löf, An intuitionistic theory of types: predicative part, In Logic Colloquium (1973), ed. H. E. Rose and J. C. Shepherdson (North-Holland, 1974), 73-118. (web)

The problem with strict substitution compared to weak pullback in this argument was discussed and fixed in

• Pierre-Louis Curien, Substitution up to isomorphism, Fundamenta Informaticae, 19(1,2):51–86 (1993)

• Martin Hofmann, On the interpretation of type theory in locally cartesian closed categories, Proc. CSL ‘94, Kazimierz, Poland. Jerzy Tiuryn and Leszek Pacholski, eds. Springer LNCS, Vol. 933 (CiteSeer)

but in the process the equivalence of categories was lost. This was finally all rectified in

• Pierre Clairambault, Peter Dybjer, The Biequivalence of Locally Cartesian Closed Categories and Martin-Löf Type Theories, in Typed lambda calculi and applications, Lecture Notes in Comput. Sci. 6690, Springer 2011 (arXiv:1112.3456)

and

Another version of this which also applie to intensional identity types and hence to homotopy type theory is in

The analogous statement relating homotopy type theory and locally cartesian closed (infinity,1)-categories was fromally conjectured around

following earlier suggestions by Steve Awodey. Explicitly, the suggestion that with the univalence axiom added this is refined to (∞,1)-topos theory appears around

Details on this higher categorical semantics of homotopy type theory are in

with lecture notes in

• Chris Kapulkin, Type theory and locally cartesian closed quasicategories, Oxford 2014 (video)

Models specifically in (constructive) cubical sets are discussed in

• Marc Bezem, Thierry Coquand, Simon Huber, A model of type theory in cubical sets, 2013 (web, pdf)

• Ambrus Kaposi, Thorsten Altenkirch, A syntax for cubical type theory (pdf)

• Simon Docherty, A model of type theory in cubical sets with connection, 2014 (pdf)

A precise definition of elementary (infinity,1)-topos inspired by giving a natural equivalence to homotopy type theory with univalence was then proposed in

Categorical semantics of univalent type universes is discussed in

Discussion of weak Tarskian homotopy type universes is in

A discussion of the correspondence between type theories and categories of various sorts, from lex categories to toposes is in

• Maria Emilia Maietti, Modular correspondence between dependent type theories and categories including pretopoi and topoi, Math. Struct. in Comp. Science (2005), vol. 15, pp. 1089–1149 (gzipped ps) (doi)

Revised on May 20, 2015 17:47:20 by Urs Schreiber (195.113.30.252)