What’s the big deal with HoTT?

June 22, 2013

Now that the Homotopy Type Theory book is out, a lot of people are asking “What’s the big deal?”.  The full answer lies within the book itself (or, at any rate, the fullest answer to date), but I am sure that many of us who were involved in its creation will be fielding this question in our own ways to help explain why we are so excited by it.  In fact what I think is really fascinating about HoTT is precisely that there are so many different ways to think about it, according to one’s interests and backgrounds.  For example, one might say it’s a nice way to phrase arguments in homotopy theory that avoids some of the technicalities in the classical proofs by treating spaces and paths synthetically, rather than analytically.  Or one might say that it’s a good language for mechanization of mathematics that provides for the concise formulation of proofs in a form that can be verified by a computer.  Or one might say that it points the way towards a vast extension of the concept of computation that enables us to compute with abstract geometric objects such as spheres or toruses.  Or one might say that it’s a new foundation for mathematics that subsumes set theory by generalizing types from mere sets to arbitrary infinity groupoids,  sets being but particularly simple types (those with no non-trivial higher-dimensional structure).

But what is it about HoTT that makes all these interpretations and applications possible?  What is the key idea that separates HoTT from other approaches that seek to achieve similar ends?  What makes HoTT so special?

In a word the answer is constructivity.  The distinctive feature of HoTT is that it is based on Per Martin-Löf’s Intuitionistic Theory of Types, which was formulated as a foundation for intuitionistic mathematics as originally put forth by Brouwer in the 1930’s, and further developed by Bishop, Gentzen, Heyting, Kolmogorov, Kleene, Lawvere, and Scott, among many others.  Briefly put, the idea of type theory is to codify and systematize the concept of a mathematical construction by characterizing the abstract properties, rather than the concrete realizations, of the objects used in everyday mathematics.  Brouwer’s key insight, which lies at the heart of HoTT, is that proofs are a form of construction no different in kind or character from numbers, geometric figures, spaces, mappings, groups, algebras, or any other mathematical structure.  Brouwer’s dictum, which distinguished his approach from competing alternatives, is that logic is a part of mathematics, rather than mathematics is an application of logic.  Because for him the concept of a construction, including the concept of a proof, is prior to any other form of mathematical activity, including the study of proofs themselves (i.e., logic).

So under Martin-Löf’s influence HoTT starts with the notion of type as a classification of the notion of construction, and builds upwards from that foundation.  Unlike competing approaches to foundations, proofs are mathematical objects that play a central role in the theory.  This conception is central to the homotopy-theoretic interpretation of type theory, which enriches types to encompass spaces with higher-dimensional structure.  Specifically, the type $\textsf{Id}_A(M,N)$ is the type of identifications of $M$ and $N$ within the space $A$.  Identifications may be thought of as proofs that $M$ and $N$ are equal as elements of $A$, or, equivalently, as paths in the space $A$ between points $M$ and $N$.  The fundamental principles of abstraction at the heart of type theory ensure that all constructs of the theory respect these identifications, so that we may treat them as proofs of equality of two elements.  There are three main sources of identifications in HoTT:

1. Reflexivity, stating that everything is equal to itself.
2. Higher inductive types, defining a type by giving its points, paths, paths between paths, and so on to any dimension.
3. Univalence, which states that an equivalence between types determines a path between them.

I will not attempt here to explain each of these in any detail; everything you need to know is in the HoTT book.  But I will say a few things about their consequences, just to give a flavor of what these new principles give us.

Perhaps the most important conceptual point is that mathematics in HoTT emphasizes the structure of proofs rather than their mere existence.  Rather than settle for a mere logical equivalence between two types (mappings back and forth stating that each implies the other), one instead tends to examine the entire space of proofs of a proposition and how it relates to others.  For example, the univalence axiom itself does not merely state that every equivalence between types gives rise to a path between them, but rather that there is an equivalence between the type of equivalences between two types and the type of paths between them.  Familiar patterns such as “$A$ iff $B$” tend to become “$A\simeq B$“, stating that the proofs of $A$ and the proofs of $B$ are equivalent.  Of course one may choose neglect this additional information, stating only weaker forms of it using, say, truncation to suppress higher-dimensional information in a type, but the tendency is to embrace the structure and characterize the space of proofs as fully as possible.

A close second in importance is the axiomatic freedom afforded by constructive foundations.  This point has been made many times by many authors in many different settings, but has particular bite in HoTT.   The theory does not commit to (nor does it refute) the infamous Law of the Excluded Middle for arbitrary types: the type $A+(A\to \textbf{0})$ need not always be inhabited.  This property of HoTT is absolutely essential to its expressive power.  Not only does it admit a wider range of interpretations than are possible with the Law included, but it also allows for the selective imposition of the Law where it is needed to recover a classical argument, or where it is important to distinguish the implications of decidability in a given situation.  (Here again I defer to the book itself for full details.)  Similar considerations arise in connection with the many forms of Choice that can be expressed in HoTT, some of which are outright provable, others of which are independent as they are in axiomatic set theory.

Thus, what makes HoTT so special is that it is a constructive theory of mathematics.  Historically, this has meant that it has a computational interpretation, expressed most vividly by the propositions as types principle.  And yet, for all of its promise, what HoTT currently lacks is a computational interpretation!  What, exactly, does it mean to compute with higher-dimensional objects?  At the moment it is difficult to say for sure, though there seem to be clear intuitions in at least some cases of how to “implement” such a rich type theory.  Alternatively, one may ask whether the term “constructive”, when construed in such a general setting, must inevitably involve a notion of computation.  While it seems obvious on computational grounds that the Law of the Excluded Middle should not be considered universally valid, it becomes less clear why it is so important to omit this Law (and, essentially, no other) in order to obtain the richness of HoTT when no computational interpretation is extant.  From my point of view understanding the computational meaning of higher-dimensional type theory is of paramount importance, because, for me, type theory is and always has been a theory of computation on which the entire edifice of mathematics ought to be built.

The Homotopy Type Theory Book is out!

June 20, 2013

By now many of you have heard of the development of Homotopy Type Theory (HoTT), an extension of intuitionistic type theory that provides a natural foundation for doing synthetic homotopy theory.  Last year the Institute for Advanced Study at Princeton sponsored a program on the Univalent Foundations of Mathematics, which was concerned with developing these ideas.  One important outcome of the year-long program is a full-scale book presenting the main ideas of Homotopy Type Theory itself and showing how to apply them to various branches of mathematics, including homotopy theory, category theory, set theory, and constructive analysis.  The book is the product of a joint effort by dozens of participants in the program, and is intended to document the state of the art as it is known today, and to encourage its further development by the participation of others interested in the topic (i.e., you!).  Among the many directions in which one may take these ideas, the most important (to me) is to develop a constructive (computational) interpretation of HoTT.  Some partial results in this direction have already been obtained, including fascinating work by Thierry Coquand on developing a constructive version of Kan complexes in ITT, by Mike Shulman on proving homotopy canonicity for the natural numbers in a two-dimensional version of HoTT, and by Dan Licata and me on a weak definitional canonicity theorem for a similar two-dimensional theory.  Much work remains to be done to arrive at a fully satisfactory constructive interpretation, which is essential for application of these ideas to computer science.  Meanwhile, though, great progress has been made on using HoTT to formulate and formalize significant pieces of mathematics in a new, and strikingly beautiful, style, that are well-documented in the book.

The book is freely available on the web in various formats, including a PDF version with active references, an ebook version suitable for your reading device, and may be purchased in hard- or soft-cover from Lulu.  The book itself is open source, and is available at the Hott Book Git Hub.  The book is under the Creative Commons  CC BY-SA license, and will be freely available in perpetuity.

Readers may also be interested in the posts on Homotopy Type Theory, the n-Category Cafe, and Mathematics and Computation which describe more about the book and the process of its creation.

Univalent Foundations at IAS

December 3, 2012

As many of you may know, the Institute for Advanced Study is sponsoring a year-long program, called “Univalent Foundations for Mathematics” (UF), which is developing the theory and applications of Homotopy Type Theory (HTT).  The UF program is organized by Steve Awodey (CMU), Thierry Coquand (Chalmers), and Vladimir Voevodsky (IAS).  About two dozen people are in residence at the Institute to participate in the program, including Peter Aczel, Andrej Bauer, Peter Dybjer, Dan Licata, Per Martin-Löf, Peter Lumsdaine, Mike Shulman, and many others.  I have been shuttling back and forth between the Institute and Carnegie Mellon, and will continue to do so next semester.

The excitement surrounding the program is palpable.  We all have the sense that we are doing something important that will change the world.  A typical day consists of one or two lectures of one or two hours, with the rest of the day typically spent in smaller groups or individuals working at the blackboard.  There are many strands of work going on simultaneously, including fundamental type theory, developing proof assistants, and formulating a body of informal type theory.  As visitors come and go we have lectures on many topics related to HTT and UF, and there is constant discussion going on over lunch, tea, and dinner each day.  While there I work each day to the point of exhaustion, eager to pursue the many ideas that are floating around.

So, why is homotopy type theory so exciting?  For me, and I think for many of us, it is the most exciting development in type theory since its inception.  It brings together two seemingly disparate topics, algebraic topology and type theory, and provides a gorgeous framework in which to develop both mathematics and computer science.  Many people have asked me why it’s so important.  My best answer is that it’s too beautiful to be ignored, and such a beautiful concept bmust be good for something!  We’ll be at this for years, but it’s too soon to say yet where the best applications of HTT will arise.  But I am sure in my bones that it’s as important as type theory itself.

Homotopy type theory is based on two closely related concepts:

1. Constructivity.  Proofs of propositions are mathematical objects classified by their types.
2. Homotopy.  Paths between objects of a type are proofs of their interchangeability in all contexts.  Paths in a type form a type whose paths are homotopies (deformations of paths).

Homotopy type theory is organized so that maps and families respect homotopy, which, under the identification of paths with equality proofs, means that they respect equality.  The force of this organization arises from axioms that specify what are the paths within a type.   There are two major sources of non-trivial paths within a type, the univalence axiom, and higher inductive types.

The univalence axiom specifies that there is an equivalence between equivalences and equalities of the objects of a universe.  Unravelling a bit, this means that for any two types inhabiting a universe, evidence for their equivalence (a pair of maps that are inverse up to higher homotopy, called weak equivalence) is evidence for their equality.  Put another way, weak equivalences are paths in the universe.  So, for example, a bijection between two elements of the universe $\textsf{Set}$ of sets constitutes a proof of the equality (universal interchangeability) of the two sets.

Higher inductive types allow one to define types by specifying their elements, any paths between their elements, any paths between those paths, and so on to any level, or dimension.  For example, the interval, $I$, has as elements the endpoints $0, 1 : I$, and a path $\textsf{seg}$ between $0$ and $1$ within $I$.  The circle, $S^1$ has an element $\textsf{base}$ and a path $\textsf{loop}$ from $\textsf{base}$ to itself within $S^1$.

Respect for homotopy means that, for example, a family $F$ of types indexed by the type $\textsf{Set}$ must be such that if $A$ and $B$ are isomorphic sets, then there must be an equivalence between the types $F(A)$ and $F(B)$ allowing us to transport objects from one “fiber” to the other.  And any function with domain $\textsf{Set}$ must respect bijection—it could be the cardinality function, for example, but it cannot be a function that would distinguish $\{\,0,1\,\}$ from $\{\,\textsf{true},\textsf{false}\,\}$.

Univalence allows us to formalize the informal convention of identifying things “up to isomorphism”.  In the presence of univalence equivalence types (spaces) are, in fact, equal.  So rather than rely on convention, we have a formal account of such identifications.

Higher inductives generalize ordinary inductive definitions to higher dimensions.  This means that we can now define maps (computable functions!) between, say, the 4-dimensional sphere and the 3-dimensional sphere, or between the interval and the torus.  HTT makes absolutely clear what this even means, thanks to higher inductive types.  For example, a map out of $S^1$ is given by two pieces of data:

1. What to do with the base point.  It must be mapped to a point in the target space.
2. What to do with the loop.  It must be mapped to a loop in the target space based at the target point.

A map out of $I$ is given similarly by specifying

1. What to do with the endpoints.  These must be specified points in the target space.
2. What to do with the segment.  It must be a path between the specified points in the target space.

It’s all just good old functional programming!  Or, rather, it would be, if we were to have a good computational semantics for HTT, a topic of intense interest at the IAS this year.  It’s all sort-of-obvious, yet also sort-of-non-obvious, for reasons that are difficult to explain briefly.  (If I could, they would probably be considered obvious, and not in need of much explanation!)

A game-changing aspect of all of this is that HTT provides a very nice foundation for mathematics in which types ($\infty$-groupoids) play a fundamental role as classifying all mathematical objects, including proofs of propositions, which are just types.  Types may be classified according to the structure of their paths—and hence propositions may be classified according to the structure of their proofs.  For example, any two proofs of equivalence of two natural numbers are themselves equivalent; there’s only one way to say that $2+2=4$, for example.  Of course there is no path between $2+2$ and $5$.  And these two situations exhaust the possibilities: any two paths between natural numbers are equal (but there may not even be one).  This unicity of paths property lifts to function spaces by extensionality, paths between functions being just paths between the range elements for each choice of domain element.  But the universe of Sets is not like this: there are many paths between sets (one for each bijection), and these are by no means equivalent.  However, there is at most one way to show that two bijections between sets are equivalent, so the structure “peters out” after dimension 2.

The idea to apply this kind of analysis to proofs of propositions is a distinctive feature of HTT, arising from the combination of constructivity, which gives proofs status as mathematical objects, and homotopy, which provides a powerful theory of the equivalence of proofs.  Conventional mathematics ignores proofs as objects of study, and is thus able to express certain ideas only indirectly.  HTT brings out the latent structure of proofs, and provides an elegant framework for expressing these concepts directly.

Update: edited clumsy prose and added concluding paragraph.

Transformations as strict groupoids

May 30, 2011

The distinguishing feature of higher-dimensional type theory is the concept of equivalence of the members of a type that must be respected by all families of types.  To be sufficiently general it is essential to regard equivalence as a structure, rather than a property.  This is expressed by the judgement

$\displaystyle \Gamma\vdash \alpha::M\simeq N:A$

which states that $M$ and $N$ are equivalent members of type $A$, as evidenced by the transformation $\alpha$.  Respect for equivalence is ensured by the rule

$\displaystyle{{\Gamma,x:A\vdash B\,\textsf{type}\quad \Gamma\vdash \alpha :: M\simeq N:A \quad \Gamma\vdash P:B[M/x]}\over {\Gamma\vdash \textit{map}\{x:A.B\}[\alpha](P):B[N/x]}},$

which states that equivalent members determine equivalent instances of a family of types.  The equivalence between instances is mediated by the operation $\textit{map}\{x:A.B\}[\alpha](-)$, which sends members of $B[M/x]$ to members of $B[N/x]$.  We call this mapping the action of the family $x:A.B$ on the transformation $\alpha$.

For reasons that will only become apparent as we go along, it is important that “equivalence” really be an equivalence: it must be, in an appropriate sense, reflexive, symmetric, and transitive.  The “appropriate sense” is precisely that we require the existence of transformations

$\displaystyle{\Gamma\vdash \textit{id}::M\simeq M:A}$

$\displaystyle{{\Gamma\vdash\alpha::M\simeq N:A}\over{\Gamma\vdash\alpha^{-1}::N\simeq M:A}}$

$\displaystyle{{\Gamma\vdash \beta:N\simeq P:A\quad \Gamma\vdash \alpha:M\simeq N:A}\over{\Gamma\vdash\beta\circ\alpha:M\simeq P:A}}$

Moreover, these transformations must be respected by the action of any family, in a sense that we shall make clear momentarily.  Before doing so, let us observe that these transformations constitute the operations of a groupoid, which we may think of either as an equivalence relation equipped with evidence or a category in which every map is invertible (a generalized group).  While the former interpretation may not suggest it, the latter formulation implies that we should impose some requirements on how these transformations interact, namely the axioms of a groupoid:

1. Composition (multiplication) is associative: $\gamma\circ(\beta\circ\alpha)\equiv (\gamma\circ\beta)\circ\alpha::M\simeq N:A$.
2. Identity is the unit of composition: $\textit{id}\circ\alpha\equiv\alpha::M\simeq N:A$ and $\alpha\circ\textit{id}\equiv\alpha::M\simeq N:A$.
3. Inverses cancel: $\alpha^{-1}\circ\alpha\equiv\textit{id}::M\simeq M:A$ and $\alpha\circ\alpha^{-1}\equiv\textit{id}::N\simeq N:A$.
These conditions, which impose equalities on transformations, demand that the second-dimensional structure of a type form a strict groupoid.  I will come back to an important weakening of these requirements later.

We further require that the action of a type family preserve the groupoid structure.  For this it is enough to require that it preserve identities and composition:

$\displaystyle{\textit{map}\{x:A.B\}[\textit{id}](-) \equiv \textit{id}(-):B[M/x]}$

and

$\displaystyle{\begin{array}{c}\textit{map}\{x:A.B\}[\beta\circ\alpha](-)\\\equiv\\\textit{map}\{x:A.B\}[\beta](\textit{map}\{x:A.B\}[\alpha](-))\end{array}}$.

Thinking of a groupoid as a category, these conditions state that the action of a type family be (strictly) functorial.  (Here again we are imposing strong requirements in order to facilitate the exposition; eventually we will consider a relaxation of these conditions that will admit a richer range of applications.)

(The alert reader will note that I have not formally introduced the concept of a transformation between types, nor the equality of these, into the theory.  There are different ways to skin this cat; for now, I will be a bit loose about the axiomatics in order to focus attention on the main ideas.  Rest assured that everything can be made precise!)

By demanding that the groupoid axioms hold strictly (as equalities) and that the action of families be strictly functorial, we have simplified the theory considerably by restricting it to dimension 2.  To relax these restrictions requires higher dimensions.  For example, we may demand only that the groupoid conditions hold up to a transformation of transformations, but hold strictly from then on; this is the 3-dimensional case.  Or we can relax all such conditions to hold only up to a higher transformation, resulting in finite dimensional type theory.  Similar considerations will apply to other conditions that we shall impose on the action of families, in particular to specify the action of type constructors on transformations, which I will discuss next time.  The presentation of finite-dimensional type theory will be aided by the introduction of identity types (also called path types).  Identity types avoid the need for an ever-expanding nesting of transformations between transformations between ….  More on that later!

Update (August 2012): Egbert Rijke has written lucidly on the topic of Yoneda’s Lemma and it’s relation to homotopy type theory in his Master’s Thesis, which I encourage readers to consult for a nice summary of higher-dimensional type theory.

Higher-Dimensional Type Theory

May 30, 2011

Ideas have their time, and it’s not for us to choose when they arrive.  But when they do, they almost always occur to many people at more or less the same time, often in a slightly disguised form whose underlying unity becomes apparent only later.  This is perhaps not too surprising, the same seeds taking root in many a fertile mind.  A bit harder to explain, though, is the moment in time when an idea comes to fruition.  Often all of the ingredients are available, and yet no one thinks to put two-and-two together and draw what seems, in retrospect, to be the obvious inference.  Until, suddenly, everyone does.  Why didn’t we think of that ages ago?  Nothing was stopping us, we just didn’t notice the opportunity!

The recent development of higher-dimensional structure in type theory seems to be a good example.  All of the ingredients have been present since the 1970’s, yet as far as I know no one, until quite recently, no one quite put together all the pieces to expose the beautiful structure that has been sitting there all along.  Like many good ideas, one can see clearly that the ideas were foreshadowed by many earlier developments whose implications are only now becoming understood.  My plan is to explain higher type theory (HTT) to the well-informed non-expert, building on ideas developed by various researchers, including Thorsten Altenkirch, Steve Awodey, Richard Garner, Martin Hofmann, Dan Licata, Peter Lumsdaine, Per Martin-Löf, Mike Shulman, Thomas Streicher, Vladimir Voevodsky, and Michael Warren.  It will be useful in the sequel to be familiar with The Holy Trinity, at least superficially, and preferably well enough to be able to move back and forth between the three manifestations that I’ve previously outlined.

One-dimensional dependent type theory is defined by derivation rules for these four fundamental forms of judgement (and, usually, some others that we suppress here for the sake of concision):

$\displaystyle \Gamma\vdash A\,\mathsf{type}$

$\displaystyle \Gamma\vdash M : A$

$\displaystyle \Gamma\vdash M \equiv N : A$

$\displaystyle \Gamma\vdash A\equiv B$

A context, $\Gamma$, consists of a sequence of declarations of variables of the form $x_1:A_1,\dots,x_n:A_n$, where it is presupposed, for each $1\leq i\leq n$, that $x_1:A_1,\dots,x_{i-1}:A_{i-1}\vdash A_i\,\mathsf{type}$ is derivable.

The key notion of dependent type theory is that of a family of types indexed by (zero or more) variables ranging over a type.  The judgement $\Gamma\vdash A\,\mathsf{type}$ states that $A$ is a family of types indexed by the variables given by $\Gamma$.  For example, we may have $\vdash\textit{Nat}\,\textsf{type}$, specifying that $\textit{Nat}$ is a closed type (a degenerate family of types), and $x{:}\textit{Nat}\vdash\textit{Seq}(x)\,\textsf{type}$, specifying that $\textit{Seq}(n)$ is a type (say, of sequences of naturals of length $n$) for each $\vdash n:\textit{Nat}$.  The rules of type theory ensure, either directly or indirectly, that the structural properties of the hypothetical/general judgement are valid.  In particular families of types respect equality of indices:

$\displaystyle{{\Gamma,x:A\vdash B\,\textsf{type}\quad \Gamma\vdash M\equiv N:A \quad \Gamma\vdash P:B[M/x]}\over {\Gamma\vdash P:B[N/x]}}.$

In words, if $B$ is a family of types indexed by $A$, and if $M$ and $N$ are equal members of type $A$, then every member of $B[M/x]$ is also a member of $B[N/x]$.

The generalization to two- (and higher-) dimensional type theory can be motivated in several ways.  One natural source of higher-dimensional structure is a universe, a type whose elements correspond to types.  For example, we may have a universe of sets given as follows:

$\displaystyle \vdash \textit{Set}\,\textsf{type}$

$\displaystyle x:\textit{Set}\vdash \textit{Elt}(x)\,\textsf{type}$

$\displaystyle \vdash \textit{nat}:\textit{Set}$

$\displaystyle \vdash \textit{Elt}(\textit{nat})\equiv\textit{Nat}$

$\displaystyle a:\textit{Set},b:\textit{Set}\vdash a\times b : \textit{Set}$

$\displaystyle a:\textit{Set},b:\textit{Set}\vdash \textit{Elt}(a\times b)\equiv \textit{Elt}(a)\times\textit{Elt}(b)$

and so forth, ensuring that $\textit{Set}$ is closed under typical set-forming operations whose interpretations are given by $\textit{Elt}$ in terms of standard type-theoretic concepts.

In many situations, including much of informal (yet entirely rigorous) mathematics, it is convenient to identify sets that are isomorphic, so that, for example, the sets $\textit{nat}\times\textit{nat}$ and $\textit{2}\to\textit{nat}$ would be interchangeable.  In particular, these sets should have the “same” (type of) elements.  But obviously these two sets do not have the same elements (one consists of pairs, the other of functions, under the natural interpretation of the sets as types), so we cannot hope to treat $\textit{Elt}(\textit{nat}\times\textit{nat})$ and $\textit{Elt}(\textit{2}\to\textit{nat})$ as equal, though we may wish to regard them as equivalent in some sense.  Moreover, since two sets can be isomorphic in different ways, isomorphism must be considered a structure on sets, rather than a property of sets.  For example, $\textit{2}$ is isomorphic to itself in two different ways, by the identity and by negation (swapping).  Thus, equivalence of the elements of two isomorphic sets must take account of the isomorphism itself, and hence must have computational significance.

It is precisely the desire to accommodate equivalences such as this that gives rise to higher dimensions in type theory.  Specifically, we introduce two-dimensional structure by adding a new judgement to type theory stating that two members of a type are related by a specified transformation:

$\displaystyle \Gamma\vdash \alpha :: M\simeq N : A$

Crucially, families of types must respect transformation:

$\displaystyle{{\Gamma,x:A\vdash B\,\textsf{type}\quad \Gamma\vdash \alpha :: M\simeq N:A \quad \Gamma\vdash P:B[M/x]}\over {\Gamma\vdash \textit{map}\{x:A.B\}[\alpha](P):B[N/x]}}.$

A transformation should be thought of as evidence of interchangeability of the members of a type; the map operation puts the evidence to work.

Returning to our example of the universe of sets, let us specify that a transformation from one set to another is an pair of functions constituting a bijection between the elements of the two sets:

$\displaystyle{ {\begin{array}{c} \Gamma,x:\textit{Elt}(a)\vdash f(x):\textit{Elt}(b) \\ \Gamma,x:\textit{Elt}(b)\vdash g(x):\textit{Elt}(a) \\ \Gamma,x:\textit{Elt}(a)\vdash g(f(x))\equiv x:\textit{Elt}(a) \\ \Gamma,x:\textit{Elt}(b)\vdash f(g(x))\equiv x:\textit{Elt}(b) \end{array}} \over {\Gamma\vdash\textit{iso}(f,g)::a\simeq b:\textit{Set}}}$

(The equational conditions here are rather strong; I will return to this point in a future post.  For now, let us just take this as the defining criterion of isomorphism between two sets.)

Evidence for the isomorphism of two sets induces a transformation on types given by the following equation:

$\displaystyle{ {\Gamma\vdash M:\textit{Elt}(a)}\over {\Gamma\vdash \textit{map}\{\textit{Elt}\}[\textit{iso}(f,g)](M)\equiv f(M) : \textit{Elt}(b)}}$

(suppressing the obvious typing premises for $f$ and $g$).  In words an isomorphism between sets $a$ and $b$ induces a transformation between their elements given by the isomorphism itself.

This, then, is the basic structure of two-dimensional type theory, but there is much more to say!  In future posts I intend to develop the ideas further, including a discussion of these topics:

1. The definition of $\textit{map}\{x:A.B\}$ is given by induction over the structure of $x:A.B$.  The above equation covers only one case; there are more, corresponding to each way of forming a family of types $x:A.B$.  The extension to function types will expose the role of the inverse of the isomorphism between sets.
2. The judgement $\alpha::M\simeq N:A$ may be internalized as a type, which will turn out to correspond to the identity type in Martin-Löf’s type theory, albeit with a different interpretation given by Altenkirch.  The identity type plays an important role in the extension to all higher dimensions.
3. To ensure coherence and to allow for greater expressiveness we must also discuss equality and equivalence of transformations and how these influence the induced transformation of families of types.  In particular transformations admit a groupoid structure which expresses reflexivity, symmetry, and transitivity of transformation; these conditions can be considered to hold strongly or weakly, giving rise to different applications and interpretations.
4. Higher-dimensional type theory admits a fascinating interpretation in terms of homotopy theory which types are interpreted as spaces, members as points in those spaces, and transformations as paths, or homotopies.  This, together with a generalization of the treatment of universes outlined above, is the basis for Voevodsky’s work on univalent foundations of mathematics.
5. One may consider relaxing the groupoid structure on transformations to a “monoidoid” (that is, category) structure by not requiring symmetry (inverses).  The structure of type theory changes significantly in the absence of symmetry, posing significant open problems, but admitting a wider range of applications of higher-dimensional structure in both CS and mathematics.
To keep up to date with the latest developments in this area, please visit the Homotopy Type Theory blog!

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