module 1Lab.intro where

# Introduction🔗

This page aims to present a first introduction to cubical type
theory, from the perspective of a mathematician who has heard about type
theory but has no previous familiarity with it. Specifically, the kind
of mathematician that we are appealing to is one who is familiar with
some of the ideas in category theory and homotopy theory — however, the
text also presents the concepts *syntactically*, in a way that
can be read without any prior mathematical knowledge.

Note that whenever code appears, all of the identifiers are hyperlinked to their definitions, which are embedded in articles that describe the concepts involved. For instance, in the code block below, you can click the symbol ≡ to be taken to the page on path types.

_ : 2 + 2 ≡ 4 _ = refl

For some operations, we use the built-in Agda definitions, which do not have explanations attached. This is the case for the + operation above. However, in any case, those definitions are either built-ins (in which case only the type matters), or defined as functions, in which case the implementation is visible. The most important built-ins for Cubical Agda, and those most likely to lead you to a built-in Agda page, have their own explanations, linked below:

- The
`1Lab.Path`

page explains the primitives`I`

,`i0`

,`i1`

,`_∧_`

,`_∨_`

,`~`

,`PathP`

,`Partial`

,`_[_↦_]`

, and`hcomp`

.

- The
`1Lab.Univalence`

page explains the primitives`Glue`

,`glue`

,`unglue`

.

With that out of the way, the first thing to do is the aforementioned
refresher on type theory. **If you’re already familiar with type
theory, feel free to skip to What’s
next?**. If you’re not familiar with some of the connections
that HoTT has to other areas of mathematics, like higher category theory
and topology, I recommend that you read the type theory section
anyway!

##
**Aside**: A note on terminology.

The text below refers to the foundational system that it is
presenting as “homotopy type theory”, while it would be technically more
correct to refer to it as **a** homotopy type theory. HoTT
is not a specific type theory, but a family of type theories that put
emphasis on the homotopy-theoretic interpretation of type theory. In
particular, a homotopy type theory is expected to validate univalence
and have some class of higher inductive types.

Below, we describe *cubical type theory*, which is one
particular homotopy type theory. In cubical type theory, we interpret
types as the Kan complices in the category of De Morgan cubical sets,
$\mathrm{PSh}(\square)$.
This interpretation gives rise to a model structure on the category of
cubical sets, referred to as a “cubical-type model structure”.

# Type theory🔗

**Warning**: The text below is subject to change! It’s
still very much a document in flux. In particular, the type theory
section doesn’t talk about induction, which is something that will be
expanded on in the future.

Type theory is a foundational system for mathematics which, in
contrast to a set-theoretic foundation like ZFC, formalises mathematical
*constructions* rather than mathematical *proofs*. That
is, instead of specifying which logical deductions are valid, and then
giving a set of axioms which characterise the behaviour of mathematical
objects, a type-theoretic foundation specifies directly which
mathematical constructions are valid.

Formally speaking, it is impossible to *construct* objects in
set theoretic foundations; Rather, one can, by applying the deduction
rules of the logical system and appealing to the axioms of the set
theory, prove that an object with the desired properties exists. As an
example, if we have a way of referring to two objects
$x$
and
$y$,
we can *prove that there exists* a set representing the ordered
pair
$(x, y)$,
using the axiom of pairing. Writing quasi-formally, we could even say
that this set is constructed using the expression
$\{x, \{x, y\} \}$,
but this is only a notational convenience.

In contrast, a type-theoretic foundational system, such as HoTT (what we use
here), specifies directly how to *construct* mathematical
objects, and proving a theorem becomes a special case of constructing an
object. To specify what constructions are well-behaved, we sort the
mathematical objects into boxes called *types*. Just like in set
theory, the word “set” is a primitive notion, in type theory, the word
“type” is a primitive notion. However, we can give an intuitive
explanation: “A type is a thing that things can be”. For instance,
things can be natural numbers, but things can not be “two” — hence, the
natural numbers are a type, but two is not. Formally, a type is
specified by giving:

A

**formation rule**, which tells us which expressions denote valid types at all. For instance, the formation rule for product types tells us that $\mathbb{N} \times \mathbb{N}$ denotes a well-formed type, but “$\mathbb{N} \times \to$” does not. The formation rules also have the duty of preventing us from running into*size issues*like Russell’s paradox, by constraining the use of universes.A handful of

**introduction rules**, which tell us how to piece previously-constructed objects together to form an object of a different type. Sticking with products as an example, if we have previously constructed objects called $a$ and $b$, we can construct an ordered pair $(a, b)$.Since the introduction rules are inseparable from their types, it is impossible to construct an object without knowing in which type it lives. If we used the rule which introduces ordered pairs, we know for sure that we built an object of the product type. To specify that an object lives in a certain type, we use the notation $a : A$ — which can be read as “$a$

**inhabits**$A$”, or “$a$**is a point of**$A$”, or “$a$**has type**$A$”.The notation for stating inhabitation agrees with mathematical parlance in when both make sense. For example, a function from the reals to the reals is commonly introduced with the notation $f : \mathbb{R} \to \mathbb{R}$. In type theory, this is made formal as a typing declaration: $f$ is a point of the type of functions $\mathbb{R} \to \mathbb{R}$, hence we know that the rules of function types can be used with $f$. However, when introducing a more complex object, we’re forced to break into informality and conventional shorthand: “Let G be a group”. In HoTT, we can directly write $G : \mathrm{Group}$

^{1}, which tells us that the rules for the type of groups apply to $G$.

A handful of

**elimination rules**, which say how inhabitants of that type can be manipulated. Keeping with the running example, the product type has elimination rules corresponding to projecting either the first or second component of a pair.Furthermore, each elimination rule comes with specified behaviour for when it “interacts” with the introduction rules, a process which we generally call

*reduction*. Repeatedly applying these “interactions” tells us how a given construction can have*computational behaviour*— by performing a delicate dance of elimination and introduction, we can go from $2 + 2$ to $4$.

Now that we have an intuitive motivation for the notions of type
theory, we can more precisely discuss the structure of *formal*
type theory and how it differs from formal (material) set theory. The
first (and perhaps most important) difference is that formal set theory
is a “two layered” system, but type theory only has “one layer”. In (for
example) ZFC, we have the
deductive framework of first-order logic (the “first layer”), and, using
the language of that logical system, we assert the axioms of set theory.
This remains valid if we switch set theories: For instance, IZF would substitute first-order logic with first-order
*intuitionistic* logic, and remove the assertion that the axiom
of choice holds.

In contrast, type theory only has one layer: The layer of types. What
in formal set theory would be a deduction in FOL, in type theory becomes
the construction of an inhabitant of a certain type, where propositions
are identified with certain types as in the table below. Note that up to
a first approximation, we can read each type former as denoting a
specific *set*, or as denoting a specific *space*.

This segues nicely into another difference between type theory and
set theory, which concerns the setup of their deductive systems. A
deductive system is a specification for how to derive
**judgements** using **rules**. We can think
of the deductive system as a game of sorts, where the judgements are to
be thought of as board positions, and the rules specify the valid
moves.

In the deductive system of first-order logic, there is only one kind
of judgement, stating that a proposition has a proof. If
$A$
is a proposition, then the judgement
“$A$
has a proof” is written
“$\vdash A$”.
Note that the judgement
$\vdash A$
is a part of the *meta*theory, and the proposition
$A$
is a part of the theory itself. That is,
$\vdash A$
is an **external** notion, and
$A$
is an **internal** notion.

In type theory, the basic judgement is inhabitation, written
$\vdash a : A$.
If
$A$
is a type that denotes a proposition, then this is analogous to the
judgement
$\vdash A$
of first-order logic, and we refer to
$a$
as a *witness* of
$A$,
or simply as a proof of
$A$.
Even better, if
$A$
is a purely logical proposition such that
$\vdash A$
holds without any appeal to the principle of excluded middle, then there
is a term
$a$
for which
$\vdash a : A$
holds in type theory.

When
$A$
is being thought of as a collection of things, we might imagine that
$a : A$
is analogous to
$a \in A$.
However, this is not the case: While
$a \in A$
is a *proposition* (an internal statement),
$a : A$
is a *judgement*, an external statement. That is, we can not use
$a : A$
with the propositional connectives mentioned in the table above — we can
not make statements of the form “if it is not the case that
$a : A$,
then
$b : B$”.
This is the second difference we will mention between set theory and
type theory: The *proposition*
$a \in A$
is an internal relation between previously-constructed terms
$a$
and
$A$,
but the external judgement
$\vdash a : A$
is indivisible, and it does not make sense to talk about
$a$
without also bringing up that it’s in
$A$.

The third and final difference is the treatment of equality, which is
arguably also the most important part of homotopy type theory. In set
theory, given sets
$a$
and
$b$,
we have a proposition
$a = b$,
which is given meaning by the *axiom of extensionality*:
$a = b$
whenever
$a$
and
$b$
behave the same with respect to
$\in$.
Correspondingly, in type theory, we have an equality *type*,
written
$a \equiv_A b$
(where
$a, b : A$),
giving us the internal notion of equality. But we also have an external
notion of equality, which encodes the aforementioned interactions
between introduction and elimination rules. This is written
$\vdash a = b : A$,
and it’s meant to indicate that
$a$
and
$b$
are the same *by definition* — hence we call it
**definitional equality**.

Since definitional equality is a judgement, it’s also not subject to the internal propositional connectives — we can not prove, disprove, assume or negate it when working inside type theory, for it does not make sense to say “if $a$ and $b$ are equal by definition, then […]”. Correspondingly, in the Agda formalisation, definitional equality is invisible, since it’s represented by the computation rules of the type theory.

In the rest of this section, we will recall the presentation of the most ubiquitous types. The idea is that if you’ve encountered type theory before, then skimming the explanation here should help snap your mind back into focus, while if this is your first time interacting with it, then the content here should be enough for you to understand the rest of the 1lab development.

# Functions🔗

If we have two previously-constructed types
$A$
and
$B$,
we can form the type of **functions from
$A$
to
$B$**,
written
$A \to B$.
Often, functions will also be referred to as **maps**. A
function is, intuitively, a rule prescribing how to obtain an inhabitant
of
$B$
given an inhabitant of
$A$.
In type theory, this is not only an intuition, but rather a
*definition*. This is in contrast with set theory, where
functions are defined to be relations satisfying a certain property.

Given a function
$f : A \to B$,
we can **apply** it to an inhabitant
$a : A$,
resulting in an inhabitant
$f\ a$
of the codomain
$B$,
called the **value of
$f$
at
$a$**.
For clarity, we sometimes write
$f(a)$
for the value of
$f$
at
$a$,
but in Agda, the parentheses are not necessary.

The most general form for constructing an inhabitant of
$A \to B$
is called **lambda abstraction**, and it looks like
$(\lambda (x : A). e)$,
though shorthands exist. In this construction, the subexpression
$e$
denotes the *body* of a function definition, and
$x$
is the parameter. You can, and should, think of the expression
$\lambda x. e$
as being the same as the notation
$x \mapsto e$
often used for specifying a map. Indeed, Agda supports definition of
functions using a more traditional, “rule-like” notation, as shorthand
for using
$\lambda$
abstraction. For instance, the following definitions of inhabitants
`f`

and `g`

of
$\mathrm{Nat} \to \mathrm{Nat}$
are definitionally the same:

In general, we refer to a way of constructing an inhabitant of a type
as an *introduction rule*, and to a way of consuming an
inhabitant of a type as an *elimination rule*. Concisely, the
introduction rule for
$A \to B$
is
$\lambda$
abstraction, and the elimination rule is function application. A general
principle of type theory is that whenever an elimination rule “meets” an
introduction rule, we must describe how they interact.

In the case of
$\lambda$-abstraction
meeting function application, this interaction is called
*$\beta$-reduction*^{2}, and it tells us how to compute a
function application. This is the usual rule for applying functions
defined by rules: The value of
$(\lambda x.\ e) e'$
is
$e$,
but with
$e'$
replacing
$x$
whenever
$x$
appears in
$e$.
Using the notation that you’d find in a typical^{3}
type theory paper, we can concisely state this as:

$(\lambda x.\ e) e' \longrightarrow e[e'/x]$

In addition, function types enjoy a definitional *uniqueness*
rule, which says “any function is a
$\lambda$
expression”. Symbolically, this means that if we have
$f : A \to B$,
the terms
$f$
and
$(\lambda x.\ f\
x)$
are equal, definitionally. This is called
$\eta$-equality,
and when applied right-to-left as a reduction rule,
$\eta$-reduction.

Functions of multiple arguments are defined by iterating the unary
function type. Since functions are types themselves, we can define
functions whose codomain is another function — where we abbreviate
$A \to (B \to C)$
by
$A \to B \to C$.
By the uniqueness rule for function types, we have that any inhabitant
of this type is going to be of the form
$\lambda x.\ \lambda y.\ f\ x\ y$,
which we abbreviate by
$\lambda x\ y.\ f\ x\ y$.
In Agda, we can write this as `λ x y → ...`

, or by mentioning
multiple variables in the left-hand side of a rule, as shown below.

f : Nat → Nat → Nat f x y = x + 3 * y

# Universes🔗

Instead of jumping right into the syntactic definition (and
motivation) for universes, I’m going to take a longer route, through
*topos theory* and eventually *higher* topos theory, which
gives a meaning explanation for the idea of universes by generalising
from a commonly-understood idea: the correspondence between predicates
and subsets. Initially, we work entirely in the category of sets,
assuming excluded middle for simplicity. Then we pass to an arbitrary
elementary topos, and finally to an arbitrary higher topos. **If
you don’t want to read about categories, click here**

First, let’s talk about subsets. I swear I’m going somewhere with this, so bear with me. We know that subsets correspond to predicates, but what even is a subset?

If we regard sets as

*material*collections of objects, equipped with a membership relation $\in$, then $S$ is a subset of $T$ if every element of $S$ is also an element of $T$. However, this notion does not carry over directly to structural set theory (where sets are essentially opaque objects of a category satisfying some properties), or indeed to type theory (where there is no membership relation) — but we can salvage it.Instead of thinking of a subset directly as a subcollection, we define a subset $S \subseteq T$ to be an

*injective function*$f : S \hookrightarrow T$. Note that the emphasis is on the*function*and not on the index set $S$, since we can have many distinct injective functions with the same domains and codomains (**exercise**: give two distinct subsets of $\mathbb{N}$ with the same domain^{4}).To see that this definition is a generalisation of the material definition in terms of $\in$, we note that any injective function $f : S \hookrightarrow T$ can be

*restricted*to a bijective function $f : S \xrightarrow{\sim} \mathrm{im}(f)$, by swapping the original codomain for the image of the function. Since the image is a subset of the codomain, we can interpret any injective $f : S \hookrightarrow T$ as expressing a*particular*way that $S$ can be regarded as a subset of $T$.

Alternatively, we can think of a subset as being a

*predicate*on the larger set $T$, which holds of a given element $x$ when $x$ is in the subset $S$. Rather than directly thinking of a predicate as a formula with a free variable, we think of it as a*characteristic function*$\chi : T \to \{0,1\}$, from which we can recover the subset $S$ as $\{ x \in T : \chi(x) = 1 \}$.This definition works as-is in structural set theory (where the existence of solution sets for equations — called “equalisers” — is assumed), but in a more general setting (one without excluded middle) like an arbitrary elementary topos, or indeed type theory, we would replace the 2-point set $\{0,1\}$ by an object of

*propositions*, normally denoted with $\Omega$.

These two definitions are both appealing in their own right — the “subsets are injective functions” can be generalised to arbitrary categories, by replacing the notion of “injective function” with “monomorphism”, while “subsets are predicates” is a very clean notion conceptually. Fortunately, it is well-known that these notions coincide; For the purposes of eventually generalising to universes, I’d like to re-explain this correspondence, in a very categorical way.

Consider some sets
$A, B$
and an arbitrary function
$f : A \to B$
between them. We define the **fibre of
$f$
over
$y$**
to be the set
$f^*(y) = \{ x \in A : f(x) = y \}$,
that is, the set of points of the domain that
$f$
maps to
$y$
— in the context of set theory, this would be referred to as the
*preimage* of
$y$,
but we’re about to generalise beyond sets, so the more “topological”
terminology will be useful. We can often characterise properties of
$f$
by looking at all of its fibres. What will be useful here is the
observation that *the fibres of an injective function have at most
one element*.

More precisely, let $f : A \hookrightarrow B$ be injective, and fix a point $y \in B$ so we can consider the fibre $f^*(y)$ over y. If we have two inhabitants $a, b \in f^*(y)$, then we know (by the definition of fibre) that $f(a) = y = f(b)$, hence by injectivity of $f$, we have $a = b$ — if $f^*(y)$ has any points at all, then it has exactly one. It turns out that having at most one point in every fibre precisely characterises the injective functions: Suppose $f(a) = f(b)$; Then we have two inhabitants of $f^*(f(a))$, $a$ and $b$, but since we assumed that $f^*(y)$ has at most one inhabitant for any $y$, we have $a = b$, so $f$ is injective.

We call a set (type)
$X$
with at most one element (at most one point) a
**subsingleton**, since it is a subset of the singleton set
(singleton type), or a **proposition**, since a
*choice* of point
$x \in X$
doesn’t give you any more information than merely telling you
“$X$
is inhabited” would. Since there is a 1-to-1 correspondence between
subsingletons
$X$
and propositions
“$X$
is inhabited”, the assignment of fibres
$y \mapsto f^*(y)$
defines a predicate
$\chi_f : B \to \Omega$,
where we have implicitly translated between a subsingleton and its
corresponding proposition.^{5}

Conversely, if we have such an assignment $\chi : B \to \Omega$, we can make a map into $B$ with subsingleton fibres by carefully choosing the domain. We take the domain to be the disjoint union of the family $\chi$, that is, the set $\sum_{(x : B)} \chi(x)$. The elements of this set can be described, moving closer to type-theoretic language, as pairs $(i, p)$ where $i \in B$ and $p \in \chi(i)$ is a witness that $i$ holds.

I claim: the map $\pi_1 : (\sum_{(x : B)} \chi(x)) \to B$ which takes $(i, p) \mapsto i$ has subsingleton fibres. This is because an element of $\pi^*_1(y)$ is described as a pair $(i , q)$, and since it’s in the fibre over $y$, we have $i = y$; Since $\chi$ (by assumption) takes values in subsingletons, concluding that $i = y$ is enough to establish that any two $(i, q)$ and $(i',q') \in \pi_1^*(y)$ must be equal.

The point of the entire preceding discussion is that the notions of
“injections into
$B$”
and “maps
$B \to \Omega$”
are equivalent, and, more importantly, *this is enough to
characterise the object
$\Omega$*^{6}.

Let me break up this paragraph and re-emphasise, since it’ll be the
key idea later: A **type of propositions** is precisely
**a classifier for maps with propositional fibres**.

Since the subobjects of
$B$
are defined to be injections
$X \to B$,
and maps
$B \to \Omega$
represent these subobjects (in a coherent way), we call
$\Omega$
a **subobject classifier**, following in the tradition of
“classifying objects”. A subobject classifier is great to have, because
it means you can talk about logical propositions — and, in the presence
of functions, predicates — as if they were points of a type.

##
**Aside**: More on general classifying objects.

A **classifying object** for some data is an object —
normally written
$B(\text{data})$
— such that maps
$X \to B(\text{data})$
correspond to data “over”
$X$,
in the sense that the data admits some map into
$X$.
Above, we were discussing the *subobject classifier* that one can
find in an elementary topos; Maps
$X \to \Omega$
(the object of propositions) correspond to maps into
$X$
with “propositional data”.

To find another example in category theory, we pass to higher
category theory,
i.e. $n$-category
theory for high values of
$n$,
such as
$n = 2$.
As a 1-dimensional structure, we can consider the data of a category
$E$
*lying over* another category
$B$
to be a functor
$E \to B$
which has certain lifting properties - some sort of categorical
*fibration*.

One would like to find a correspondence between these “categories over categories” and some data that we already know how to conceptualise; It turns out that Grothendieck fibrations over $B$ — i.e., the functors $F : E \to B$ satisfying a certain lifting property — correspond to weak 2-functors into $\mathrm{Cat}$.

Conversely, this equivalence lets us think of a family of categories indexed by $B$ — a 2-dimensional concept — corresponds precisely to 1-dimensional data (a special class of functors between ordinary categories). Furthermore, the equivalence is induced by a construction on (weak 2-)functors — the Grothendieck construction, written $\int F$ — that exactly mirrors the disjoint union $\sum F$ that we used above!

To summarise:

A classifying object $B(\text{data})$ is an object where maps $X \to B(\text{data})$ corresponds to data over (“pointing towards”) $X$.

The kind of classifying objects that we care about here are ubiquitous in all levels of $n$-category theory, but especially in $(n,1)$-category theory, they behave like universes of truncated types.

Since the objects in a $n$-category act like $(n-1)$-categories, the “most complex” classifier that a $n$-topos can have is for the $(n-1)$-category of $(n-2)$-categories, and this all works better in the case where we’re actually talking about $(n,1)$-topoi.

However, we often want to talk about *sets* and *families
of sets* as if they were points of a type, instead of just subsets.
We could pass to a higher category that has a **discrete object
classifier** — this would be a “type of sets” —, but then the
same infelicity would be repeated one dimension higher: we’d have a
classifier for “sets”, but our other objects would look like
“categories”, and we wouldn’t be able to classify those!

##
**Take a moment** to convince yourself that the
interpretation of a discrete object classifier as a type of sets makes
sense.

Note that we defined “proposition” to mean “at most one element”. We
can generalise this to a notion which applies to categories by defining
a **discrete category** to be one that has
*propositional hom-sets*, i.e. one in which the only morphisms
are identities. Note that a discrete category is automatically a
discrete *groupoid*, since the identity map is an
isomorphism.

A **discrete object classifier** (say
$\mathrm{Set}$)
would then be a classifier for functors which have discrete groupoids as
fibres — the fibres have no categorical information, being nothing more
than a set. Given any set
$X$,
we’d have a name for it as the interpretation of the unique map
$\mathrm{Disc}(X) \to *$
as a map
$* \to \mathrm{Set}$.
The name
$\mathrm{Set}$
was chosen because in the
$(2,1)$-category
$\mathfrak{Grpd}$,
the discrete object classifier is exactly the category of sets.

Thus we might hope to find, in addition to a subobject classifier, we
could have a general **object classifier**, an object
$\mathrm{Type}$
such that maps
$B \to \mathrm{Type}$
correspond to *arbitrary* maps
$A \to B$.
Let’s specify this notion better using the language of higher category
theory:

A **subobject classifier** is a representing
object for the functor
$\mathrm{Sub} : \mathscr{E}^{\mathrm{op}} \to \mathrm{Set}$
which takes an object to its poset of subobjects. Since the subobjects
of
$x$
can be described as (isomorphism classes of) objects of a subcategory of
$\mathscr{E}/x$,
we would hope that an **object classifier** would classify
the entire groupoid
$\mathrm{Core}(\mathscr{E}/x)$,
keeping around *all* of the maps into
$x$,
and keeping track of the isomorphisms between them.

##
**Aside**: Slice categories and cores, if you haven’t heard
of them

Giving a *precise* definition of slice categories and cores
would take us further afield than is appropriate here, so let me give a
quick summary and point you towards the nLab for further details. Fix a
category
$C$
that we may talk about:

The core of $C$, written.. well, $\mathrm{Core}(C)$, is the largest groupoid inside $C$. It has all the same objects, but only keeps around the isomorphisms.

The slice over an object
$c$,
written
$C/c$,
is a category where the *objects are arrows*
$a \to c$
in
$C$,
and a morphism between
$f : a \to c$
and
$g : b \to c$
*in the slice over
$c$*
is a morphism
$a \to b$
in
$C$
that makes the following triangle (in
$C$)
commute:

But for us to be able to represent
$\mathrm{Core}(\mathscr{E}/-) : \mathscr{E}^{\mathrm{op}} \to \mathfrak{Grpd}$
as an object
$\mathcal{U}$
of
$\mathscr{E}$,
we need the hom-space
$\mathrm{Hom}_{\mathscr{E}}(x, \mathcal{U})$
to be a groupoid, instead of just a set; Thus
$\mathscr{E}$
was actually a 2-category! We’re not in the clear though, since a slice
of a n-category is another n-category, we now have a 2-category
$\mathscr{E}/x$,
so the core of that is a 2-groupoid, so our
$\mathscr{E}$
was, in reality, a 3-category. This process goes on forever, from which
we conclude that the only category that can have a fully general object
classifier is a
$\infty$-category.
However, since we’re always taking cores, we don’t need mapping
*categories*, only mapping *groupoids*, so we can weaken
this to a
$(\infty,1)$-category:
a
$\infty$-category
where every map above dimension 1 is invertible.

Using the proper higher-categorical jargon, we can define an object classifier $\mathrm{Type}$ in an $(\infty,1)$-category $\mathcal{H}$ to be a representing object for the $(\infty,1)$-presheaf $\mathrm{Core}(\mathcal{H}/-) : \mathcal{H}^{\mathrm{op}} \to \text{$\infty$-\textbf{Grpd}}$.

Again, the importance of object classifiers is that they let us talk
about arbitrary objects as if they were points of an object
$\mathrm{Type}$.
Specifically, any object
$B$
has a *name*, a map
$\ulcorner B \urcorner : * \to \mathrm{Type}$,
which represents the unique map
$B \to *$.
To make this connection clearer, let’s pass back to a syntactic
presentation of type theory, to see what universes look like “from the
inside”.

## Universes, internally🔗

Inside type theory, object classifiers present themselves as types
which contain types, which we call **universes**. Every
type is contained in some universe, but it is not the case that there is
a universe containing all types; In fact, if we did have some magical
universe
$\mathrm{Type}_\infty : \mathrm{Type}_\infty$,
we could reproduce Russell’s paradox, as is done here.

The solution to this is the usual stratification of objects into
“sizes”, with the collection of all “small” objects being “large”, and
the collection of all “large” objects being “huge”, etc. On the semantic
side, a
$(\infty,1)$-topos
has a sequence of object classifiers for maps with
$\kappa$-compact
fibres, where
$\kappa$
is a regular cardinal; Syntactically, this is reflected much more simply
as having a *tower* of universes with
$\mathrm{Type}_i : \mathrm{Type}_{1+i}$.
So, in Agda, we have:

_ : Type _ = Nat _ : Type₁ _ = Type

In Agda, we do not have automatically that a “small” object can be
considered a “large” object. When this is the case, we say that
universes are *cumulative*. Instead, we have an explicit Lifting construction that lets us treat types
as being in higher universes:

_ : Type₂ _ = Lift (lsuc (lsuc lzero)) Type

To avoid having Lift everywhere, universes are indexed by a
(small) type of countable ordinals, called `Level`

, and
constructions can be parametrised by the level(s) of types they deal
with. The collection of all
$\ell$-level
types is
$\mathrm{Type}_{\mathrm{lsuc}\ \ell}$,
which is itself a type in the next universe up, etc. Since levels are
themselves inhabitants of a type, we do in fact have a definable
function
$\lambda \ell.\ \mathrm{Type}_\ell$.
To avoid further Russellian paradoxes, functions out of Level must be
*very big*, so they live in the “first infinite universe”,
$\mathrm{Type}_\omega$.
There is another hierarchy of infinite universes, with
$\mathrm{Type}_{\omega+i}$
inhabiting
$\mathrm{Type}_{\omega+(1+i)}$.
To avoid having even more towers of universes, you can’t quantify over
the indices
$i$
in
$\mathrm{Type}_{\omega+i}$.
To summarize, we have:

$\mathrm{Type} : \mathrm{Type}_1 : \mathrm{Type}_2 : \cdots \mathrm{Type}_{\omega} : \mathrm{Type}_{\omega+1} : \mathrm{Type}_{\omega+2} : \cdots$

To represent a collection of types varying over an value of
$A$,
we use a function type into a universe, called a **type
family** for convenience: A type family over
$A$
is a function
$A \to \mathrm{Type}_i$,
for some choice of level
$i$.
An example of a type family are the *finite types*, regarded as a
family
$\mathrm{Fin} : \mathbb{N} \to \mathrm{Type}$
— where
$\mathrm{Fin}(n)$
is a type with
$n$
elements. Type families are also used to model **type
constructors**, which are familiar to programmers as being the
*generic types*. And finally, if we have a type
$B : \mathrm{Type}_i$,
we can consider the **constant family** at
$B$,
defined to be the function
$\lambda (x : A).\ B$.

To cap off the correspondence between universes and object classifiers, thus of type families and maps, we have a theorem stating that the space of maps into $B$ is equivalent to the space of type families indexed by $B$. As a reminder, you can click on the name Fibration-equiv to be taken to the page where it is defined and explained.

_ : ∀ {B : Type} → (Σ[ E ∈ Type ] (E → B)) ≃ (B → Type) _ = Fibration-equiv

## Universes and size issues🔗

Perhaps one of the most famous paradoxes in the history of formal
logic, Russell’s paradox
reminds us that, for most reasonable kinds of “collection” — be they
sets, types, or categories — considering
*the collection of all collections that do not contain
themselves* is a road that leads to madness. The standard way of
getting around these issues, at least in set-theoretic foundations as
applied to category theory, is to refer to such problematic collections
as “classes”,
and to only axiomatise the *sets* which are not too big.

In the branches of mathematics that Alexander Grothendieck
influenced, a more common approach is to work instead with
*Grothendieck universes*: A Grothendieck universe is a set
$\mathcal{U}$
that forms a transitive model of ZF, closed under
$\mathcal{U}$-small
indexed products. The classification of models of ZF implies that any
Grothendieck universe
$\mathcal{U}$
is equivalent to a segment of the von Neumann
universe
$\mathbf{V}_\kappa$,
where
$\kappa$
is a strongly inaccessible cardinal — essentially, an upper bound on the
size of the sets in
$\mathcal{U}$.

In traditional mathematical writing, it is common to entirely ignore size issues, save perhaps for an offhand mention of Grothendieck universes (or strongly inaccessible cardinals) within the first few sections. Working in a proof assistant forces us to be honest about the size of our constructions: Correspondingly, we try to be precise in our prose as well. As mentioned above, Agda universes are stratified into a pair of hierarchies $\mathrm{Type}_\kappa$ and $\mathrm{Type}_{\omega+n}$, where we’re using $\kappa$ to stand for a variable of Level type. The Level is a built-in type which contains a constant representing $0$, is closed under taking successors, and is closed under taking binary maxima.

We refer to structured type (e.g. a category, or a group) as
**$\kappa$-small**
when its underlying type inhabits the
$\kappa$th
universe; For the specific case of categories, we also use the
terminology **locally
$\kappa$-small**
to refer to the universe in which the family
$\mathbf{Hom}(-,-)$
lands. We use subscript Greek letters in prose to index our structures
by their universe level. For example,
$\mathbf{Sets}_\kappa$
is the
($(1+\kappa)$-small,
locally
$\kappa$-small)
category of
$\kappa$-small
sets.

##
**Aside**: Small (hah!) note on small categories

For those familiar with the notions of *internal* and
*enriched* categories, we might rephrase the *classical*
definitions of
$\kappa$-small
and locally
$\kappa$-small
categories as follows:

A category $\mathcal{C}$ is $\kappa$-small if it is an internal category in $\mathbf{Sets}_\kappa$;

A category $\mathcal{C}$ is locally $\kappa$-small if it is enriched over the Cartesian monoidal category $\mathbf{Sets}_\kappa$.

Because every type is contained in some universe, we note that
*every* category that appears in our development is locally
$\kappa$-small
for *some* universe level
$\kappa$.
In particular, we have no “type of categories”, but instead a type of
“$\kappa$-small,
locally
$\lambda$-small
categories”.

**Warning**: Note that our use of the term
“$\kappa$-small”
is nonstandard. In set-theoretic foundations, where the only objects are
sets anyway, this means “category internal to
$\mathbf{Sets}_\kappa$”,
as mentioned in the infobox above. In the 1Lab, the objects we
manipulate are *higher groupoids* rather than sets, and so very
few categories will be internal to a category of sets. Instead, when we
describe a category
$\mathcal{C}$
is
$\kappa$-small,
we mean that the type of objects of
$\mathcal{C}$
is an inhabitant of the universe
$\mathrm{Type}_\kappa$,
and that the
$\mathbf{Hom}_\mathcal{C}(-,-)$
family is valued in
$\mathbf{Sets}_\kappa$.
Our shorthand for the traditional notion is a “**strict
$\kappa$-small**”
category.

Our policy is to keep the use of universes “as precise as possible”: Definitions are polymorphic over as many levels as they can possibly be, and they are always placed in the smallest universe in which they fit. This is, however, an iterative process: For example, our definition of “sheaf topos” was originally parametrised over 5 levels, but after formalising more desirable properties of topoi, we have found that the definition only really mentions two levels.

# Interlude: Basics of Paths🔗

Since the treatment of paths is the most important aspect of homotopy type theory, rest assured that we’ll talk about it in more detail later. However, before discussing the dependent sum of a type family, we must discuss the fundamentals of paths, so that the categorical/homotopical connections can be made clear. Before we even get started, though, there is something that needs to be made very clear:

**Paths are not
equality!**

The only conceptualisation of paths that is accurate in all cases is that types are $\infty$-groupoids, and hence path spaces are the Hom-$\infty$-groupoids given by the structure of each type. However, this understanding is very bulky, and doesn’t actually help understanding paths all that much — so we present a bunch of other analogies, with the caveat that they only hold for some types, or are only one of the many possible interpretations of paths in those types.

The ideas presented below are grouped in blue
`<details>`

elements, but not because they aren’t
important; You **should** read all of them.

##
**Idea #1**: Some types behave like sets, providing an
embedding of “classical” equality behaviour into homotopy type theory.
While these types may not be as “exciting” as others, they aren’t any
less important!

Contradicting the big bold warning up there, **in a very
restricted class of types**, paths *do* behave like the
equality relation! We call these types **sets**, and their
defining property is that they only have identity paths — hence they
could also be called “discrete types”, by analogy with “discrete
category”, but this terminology is not employed because “discrete”
already has a distinct type-theoretic meaning.^{7}

Types in this class include the natural numbers, the integers, the rationals and the reals; Also included is every finite type, and the powerset of any type at all. The type of functions $A \to B$, where $B$ is a set, is again a set. We can generalise this and say that “most” type formers preserve set-ness.

Note that the meaning of “is a set”/“is not a set” in HoTT is
distinct from how that sentence is used in classical mathematics, where
a collection *is a set* if it’s definable in the language of
whatever foundational system you’re using. For example, in ZFC, the
collection of all sets is not a set — because if it were, we could
reproduce Russell’s paradox.

In HoTT, being a set has to do with *spatial* structure: a set
has only trivial paths. Examples of non-sets would then be given by
types that have interesting paths, like the circle, which has a
non-trivial loop. Coincidentally, the type of all sets is *not* a
set, since its spatial structure is given by the *isomorphisms*
of sets — and sets can have non-trivial automorphism groups.

##
**Idea #2**: Some types are best understood as representing
spaces, with intrinsic homotopic information, which we can investigate
using type-theoretic methods. This interpretation is the core idea of
“synthetic homotopy theory”, a whole field of maths!

We conceptualise some types (like the circle mentioned above) as
being representatives for spaces, though this does not mean that they
have any intrinsically-detectable *topology* in the sense of
“open sets”-topological spaces. Instead, we think of these types as
being an incarnation of spaces *à la* homotopy theory, with
structure similar to what you’d find there: Homology groups, cohomology
groups, but most easily seen are the homotopy groups. In these types,
type-theoretic paths quack like topological paths.

Indeed, any type in homotopy type theory has an associated sequence of homotopy groups, and these behave precisely like the homotopy groups of classical algebraic topology — with the difference that the way we compute them in HoTT has a distinctly more “typey” flavour.

This interpretation of types as spaces leads us to look at the statement “there is a type $X$ with $\pi_1(X) \cong \mathbb{Z}$” and rightly call $X$ “the circle”, even though a topologist might object that our circle is not any subset of $\mathbb{R}^2$. Similarly, we might take the integer $2$ and apply the isomorphism above, to get a path in $X$ which “walks clockwise around the circle twice”.

##
**Idea #3**: HoTT provides a setting for doing category
theory where all constructions are intrinsically isomorphism-invariant.
Under this light, types provide the *underlying groupoids* of
categories.

When doing category theory in HoTT, we have some types that naturally
arise as the type of objects in a category, such as the type of groups
relative to a universe. In category theory done set-theoretically, the
collection of objects in
$C$
has two *different* notions of sameness: The set-theoretic
equality
$a = b$,
with behaviour given by the axiom of extensionality and divorced from
the categorical structure, and the class of *isomorphisms* of
$C$,
$a \cong b$,
which naturally has to do with the categorical structure.

A big part of doing category theory in set-theoretic foundations is learning in which situations we should talk about the set-theoretic equality of objects, and when we should talk about isomorphism — what you end up discovering is that constructions that talk about equality of objects should be done very sparingly, so much so that they’re colloquially called “evil”.

Conversely, the possibility of talking about equality of objects at all means that we have to spend effort verifying that constructions appropriately respect isomorphisms, though of course this verification can be (and often ends up being) handwaved.

In HoTT, we can do better: We add a condition to the definition of
category that says the paths of the type of objects *must*
correspond to the categorical isomorphisms — which we call “univalence”.
While this condition might initially seem hard to fulfill, it’s
surprisingly common to come by univalent categories, since most
constructions on categories preserve univalence! For instance:

If $A$ and $B$ are univalent categories, then so is the product $A \times B$ and the disjoint union $A \uplus B$; The initial and terminal categories are trivially univalent.

If $C$ is a univalent category, then so is the slice $C/c$ and the coslice $c/C$.

If $C$ is a univalent category, then for any $D$, so is the functor category $[D,C]$; This implies that a very large class of categories is univalent, e.g. the models of any Lawvere theory in a univalent category.

Further, since $\mathrm{Set}$ is a univalent category, the Yoneda lemma implies that every $C$ can be embedded into a univalent category - the functor category $[C^{\mathrm{op}},\mathrm{Set}]$. If we restrict to the subcategory of representable presheaves, we have that every $C$ admits a full, faithful and essentially surjective functor into a univalent category, called the Rezk completion of $C$; If $C$ was already univalent, this functor is an equivalence.

In addition to the benefit of univalent categories that will be presented outside the fold, they’re also interesting in their own right because they are better constructively behaved than categories with a set of objects: For instance, in the last bullet point above we appealed to the statement “every fully faithful, essentially surjective functor is an equivalence”. For univalent categories, this is a theorem without any form of choice axiom!

One benefit that HoTT brings to the table is that it very naturally leads to mixing and matching the interpretations above. For example, the underlying groupoid of a univalent category can just as well be interpreted as a space, mixing interpretations #2 and #3. The spatial interpretation of categories really shines when we pass to higher categories, where the notion of “unique” behaves very badly.

When thinking
$n$-categorically,
even for small values of
$n$,
one must first weaken “unique” to “unique up to unique isomorphism”, and
then to something like “unique up to isomorphism, which is unique up to
unique invertible 2-cell”. Reasoning spatially, we see that “unique up
to isomorphism” really means that there is a *contractible space*
of choices for that object — a notion which is well-behaved at any level
of higher categorical complexity.

Every type
$A$
has an associated family of path types — to form a path type, we must
have two inhabitants of
$A$
handy, called the *endpoints*. The type of paths between
$x$
and
$y$
in
$A$
is written
$x \equiv_A y$,
where the subscript
$A$
is omitted when it can be inferred from context. The name *path*
is meant to evoke the topological idea of path, after which the paths in
cubical type theory were modelled. In topology, a path is defined to be
a *continuous function* from the real unit interval
$[0,1] \to X$,
so that we can interpret a path as being a point “continuously varying
over time”.

It would be incredibly difficult (and oversized!) to formalise the
entire real line and its topology to talk about paths, which are meant
to be a fundamental concept in type theory, so we don’t bother! Instead,
we look at which properties of the unit interval
$[0,1]$
give paths their nice behaviour, and only add those properties to our
type theory. The topological interval is **contractible**,
and it has two **endpoint inclusions**, with domain the
one-point space. Those are the properties that characterise it (for
defining paths), so those are the properties we keep!

The **interval type**,
$\mathbb{I}$,
has two introduction rules — but they are so simple that we refer to
them as “closed inhabitants” instead. These are written
$\mathrm{i0}$
and
$\mathrm{i1}$,
and they denote the left- and right- hand side of the unit interval. A
classic result in the interpretation of constructive mathematics says
that any function defined inside type theory is automatically
continuous, and so, we define the type
$x \equiv_A y$
to be the type of functions
$f : \mathbb{I} \to A$,
restricted to those where
$f(\mathrm{i0}) = x$
and
$f(\mathrm{i1}) = y$
*definitionally*.

The introduction rule for paths says that if
$e : \mathbb{I} \to A$,
with
$e(\mathrm{i0}) = x$
and
$e(\mathrm{i1}) = y$
(those are *definitional* equalities), then we can treat it as a
path
$x \equiv_A y$.
Conversely, if we have a term
$p : x \equiv_A y$,
then we can apply it to an argument
$i : \mathbb{I}$
to get a term
$p(i)$
of type
$A$.
The type of paths satisfies the same reduction rule
($\beta$-reduction)
as function types, but with an additional rule that says
$p(\mathrm{i0})$
is definitionally equal to
$x$
(resp.
$\mathrm{i1}$
and
$y$),
even if
$p$
has not reduced to a
$\lambda$-abstraction.

By considering the constant function at a point
$x : A$,
we obtain a term
$(\lambda i.\ x)$
of type
$x \equiv_A x$,
called the trivial path, and referred to (for historical reasons) as
refl, since for the types that are sets, it
expresses *refl*exivity of equality. In the interpretation of
types as
$\infty$-groupoids,
refl is the identity morphism. The rest of
the groupoid structure is here too! Any path
$p : x \equiv y$
has an inverse
$p^{-1} : y \equiv x$,
and for any
$p : x \equiv_A y$
and
$q : y \equiv_A z$,
there is a composite path
$p \bullet q : x \equiv_A z$.
There are paths between paths which express the groupoid identities
(e.g. $p \bullet p^{-1} \equiv \mathrm{refl}$),
and those paths satisfy their own identities (up to a path between paths
between paths), and so on.

This little introduction to paths was written so that we can make
some of the ideas in the following section, about dependent sums,
precise. It’s not meant to be a comprehensive treatment of the path
type; For that, see the section *Paths, in
detail* later on.

# Sums🔗

Recall that in the construction of a map into
$B$
from a predicate
$\chi : B \to \Omega$,
we interpreted
$\chi$
as a family of sets with at most one element, and then took the disjoint
union of that family, written
$\sum_{(x : B)} \chi(x)$,
which admits a *projection* onto
$B$.
That was actually a sneaky introduction of the dependent sum type
former,
$\sum$!
Indeed, the short discussion there also made some mention of the
introduction rule, but let’s reiterate with more clarity here.

If we have some index type
$A : \mathrm{Type}_\ell$,
and a family of types
$B \to \mathrm{Type}_{\ell'}$,
then we can form the **dependent sum type**
$\sum_{(x : A)} B$.
When the index is clear from context, we let ourselves omit the
variable, and write
$\sum B$.
Since
$\sum B$
is a type, it lives in a universe; Since it admits a projection onto
$A$
and a projection onto a fibre of
$B$,
it must be a universe bigger than both
$\ell$
and
$\ell'$,
but it need not be any bigger: The dependent sum of
$B$
lives in the universe indexed by
${\ell \sqcup \ell'}$.

The introduction rule for
$\sum B$
says that if you have an
$a : A$,
and a
$b : B(a)$,
then the pair
$(a,b)$
inhabits the type
$\sum B$.
Thus the inhabitants of the dependent sum are ordered pairs, but where
the second coordinate has a type which *depends on the value* of
the first coordinate. If we take
$B$
to be a constant type family, we recover a type of ordered pairs, which
we call the **product** of
$A$
and
$B$,
which is notated by
$A \times B$.

For the elimination rules, well, pairs would be rather useless if we couldn’t somehow get out what we put in, right? The first elimination rule says that if we have an inhabitant $e : \sum_{(x : A)} B(x)$, then we can project out the first component, which is written $e \mathrm{.fst}$, to get an inhabitant of $A$. The computation rule here says that if you project from a pair, you get back what you put in: $(a,b)\mathrm{.fst} = a$.

Projecting out the second component is trickier, so we start by considering the special case where we’re projecting from a literal pair. We know that in $(a,b)$, the terms have types $a : A$ and $b : B(a)$, so in this case, the second projection is $(a,b)\mathrm{.snd} : B(a)$. When we’re not projecting from a literal pair, though, we’re in a bit of a pickle. How do we state the type of the second projection, when we don’t know what the first component is? Well - we generalise. We know that for a pair, $a = (a,b)\mathrm{.fst}$, so that we may replace the aforementioned typing for $\mathrm{.snd}$ with the wordy $(a,b)\mathrm{.snd} : B((a,b)\mathrm{.fst})$. Now we don’t use any subterms of the pair in the typing judgement, so we can generalise:

$e\mathrm{.snd} : B (e\mathrm{.fst})$

A very common application of the dependent sum type is describing
types equipped with some *structure*. Let’s see an example, now
turning to Agda so that we may see how the dependent sum is notated in
the formalisation. We define a *pointed magma structure* on a
type to be a point of that type, and a binary function on that type. If
we let the type *vary*, then we get a type family of pointed
magma structures:

Pointed-magma-on : Type → Type Pointed-magma-on A = A × (A → A → A)

Taking the dependent sum of this family gives us the type of pointed
magmas, where inhabitants can be described as triples
$(T,i,m)$
where
$T : \mathrm{Type}$
is the *underlying type*,
$i : T$
is the distinguished point, and
$m : T \to T \to T$
is the binary operation. Note that, to be entirely precise, the pair
would have to be written
$(T,(i,m))$,
but since by far the most common of nested pairs is with the nesting on
the right, we allow ourselves to omit the inner parentheses in this
case.

Pointed-magma : Type₁ Pointed-magma = Σ Type Pointed-magma-on

Note that, since we fixed the family to
$\mathrm{Type} \to \mathrm{Type}$,
and recalling that
$\mathrm{Type} : \mathrm{Type}_1$
are the first two universes, the type of all pointed magmas in
$\mathrm{Type}$
lives in the next universe,
$\mathrm{Type}_1$.
If the type of all pointed magmas were in
$\mathrm{Type}$,
then we could *again* reproduce Russell’s paradox.

Just like functions, pairs enjoy a uniqueness principle, which (since
coming up with names is very hard), also goes by
$\eta$-expansion.
The rule says that every inhabitant of a
$\Sigma$-type
is *definitionally* equal to the pairing of its projections:
$p = (p\mathrm{.fst}, p\mathrm{.snd})$.
This essentially guarantees us that the inhabitants of a dependent sum
type can’t have any “extra stuff” hiding from our projection maps; They
are exactly the pairs, no more, and no less.

The dependent sum is a construction that goes by many names. In
addition to the ones mentioned below (which are the names people
actually use), you *could* call it *the Grothendieck
construction*, denote it by
$\int B$,
and have a hearty chuckle — if it weren’t for the fact that the
Grothendieck construction is generally taken for functors between
*categories* rather than just groupoids.

**$\mathbf{\Sigma}$-type**, named after its shape;**Disjoint union**, since if we take $A$ to be the 2-point type, then the dependent sum $\sum B$ of $B : 2 → \mathrm{Type}$ is exactly the disjoint union $B(0) \uplus B(1)$;Thinking homotopically, you can consider a type family $B : A \to \mathrm{Type}$ to be a

*fibration*, since as we shall see later it satisfies the path lifting property; In that sense, $\sum B$ gives the**total space**, and the actual fibration is the first projection map $x \mapsto x\mathrm{.fst}$.

This last correspondence will be particularly important later, and it
inspires much of the terminology we use in HoTT. Since universes are
object classifiers, we know that there is an equivalence between type
families
$F : B \to \mathrm{Type}$
and maps into
$B$.
Dependent sums and paths let us be more precise about this equivalence:
We can turn any type family
$F : B \to \mathrm{Type}$
into a map
$\mathrm{fst} : \sum F \to B$,
and looking at the fibre of first over a point
$y$
recovers
$F(y)$.
We thus blur this distinction a bit and refer to
$F(x)$
for some
$x : B$
as a fibre of
$F$,
and we say that something happens *fibrewise* if it happens for
each
$F(x)$.

One last thing to mention about $\Sigma$-types is that they are the realization of general subsets. This is a special case of the idea of stuff-with-structure, where the structure is merely a witness for the truth of some predicate of the first value. For example, recall that we defined the fibre of $f$ over $y$ as $\{ x \in A : f(x) = y \}$. In type theory, this is rendered with a $\Sigma$-type, as below:

fibre : (A → B) → B → Type fibre f y = Σ[ x ∈ A ] (f x ≡ y)

# Dependent products🔗

Imagine that we have two types, $A$ and $B$. For simplicity, they live in the same universe, which can be any universe of our choice. We know that we can consider $B$ to be a constant type family taking indices in $A$: $\lambda (x : A).\ B$. Furthermore, we know that the total space of this family is the product type $A \times B$, and that it comes equipped with a projection map $\mathrm{fst} : \sum B \to A$.

Given any function
$f : A \to B$,
we define its *graph*
$\mathrm{graph}(f) : A \to A \times B$
to be the function
$\lambda x.\ (x, f(x))$,
which takes each point of the input space to an ordered pair where the
first coordinate is the input, and the second coordinate is the output
of
$f$
*at that input*. The graph of a function is special because of
its interaction with the projection map
$\mathrm{fst} : A \times B \to A$.
In particular, we can see that
$\lambda x.\ \mathrm{graph}(f)\mathrm{.fst}$
is the identity function! This property turns out to precisely
characterise the functions
$A \to A \times B$
which are the graphs of functions
$A \to B$.

By rewriting the equality we noticed above in terms of function
composition, we obtain
$\mathrm{.fst} \circ \mathrm{graph}(f) = \mathrm{id}_{}$.
A function, like
$\mathrm{graph}(f)$,
which composes on the right with some other function
$\mathrm{fst}$
to give the identity function, is called a **section** of
$\mathrm{fst}$.
This lets us characterise the functions
$A \to B$
as the *sections of
$\mathrm{fst} : A \times B \to A$.*
In this sense, we see that a function
$A \to B$
is precisely a choice of values
$f(x)$,
where
$f(x)$
is in the fibre over
$x$
— in this case, the fibres over all points are
$B$!

Given a map $g : A \to A \times B$ that we know is a section of $\mathrm{fst}$, we can recover a function $f : A \to B$. Specifically, given a point $x : A$, we know that the second projection $g(x)\mathrm{.snd}$ has type $B$ — so we can simply define $f$ to be $\lambda x.\ g(x)\mathrm{.snd}$. Since we know that $g$ is a section of $\mathrm{fst}$, we conclude that $\mathrm{graph}(f) = g$.

This idea generalises cleanly from the case where
$B$
is just some type, to the case where
$B$
is a type family indexed by
$A$.
We define a **section of
$B$**
to be a function
$g : A \to \sum B$,
where
$\mathrm{fst} \circ g = \mathrm{id}_{}$.
Note that this terminology conflates the type family
$B$
with the projection map
$\sum B \to A$,
keeping with the trend of identifying type families with fibrations.
Generalising the argument from the previous paragraph, we show that a
section
$g$
gives a rule for assigning a point in
$B(x)$
to each point in
$x$.
We again take the second projection of
$g(x)\mathrm{.snd}$,
but since now
$B$
is a type family, this lives in the type
$B(g(x)\mathrm{.fst})$
— but since we know that
$g$
is a section of
$\mathrm{fst}$,
we can correct this to be
$B(x)$.

We would like to package up such a rule into a function
$A \to B$,
but this is not a well-formed type, since
$B$
is a type *family*. Moreover, this would “forget” that
$x$
is sent to the fibre
$B(x)$.
Thus, we introduce a new type former, the **dependent product
type**, written
$\prod_{(x : A)} B(x)$
or more directly
$(x : A) \to B(x)$.
The inhabitants of this type are precisely the sections of
$B$!
More importantly, the introduction, elimination, computation and
uniqueness rules of this type are exactly the same as for the function type.

The dependent product type takes on another very important role when
universes are involved: Supporting *polymorphism*. In
programming, we say that a function is polymorphic when it works
uniformly over many types; Dependent products where the index type is a
universe reflect this exactly. For example, using polymorphism, we can
write out the types of the first and second projection maps:

project-first : (A : Type) (B : A → Type) → Σ A B → A project-first A B x = x .fst project-second : (A : Type) (B : A → Type) → (p : Σ A B) → B (project-first A B p) project-second A B x = x .snd

In the logical interpretation, the dependent product type implements
the universal quantifier
$\forall$.
For example (keep in mind that in general, **we should not read
path types as equality**), we could read the type
*topological* interpretation!

##
**Worked example**: Interpreting dependent sums as
fibrations and dependent products as sections lets us derive topological
interpretations for types. We show how to do this for the type
$(x\ y : A) \to x \equiv y$,
and show that a space
$A$
admitting an inhabitant of this type has the property of “being
contractible if it is inhabited”.

First, we’ll shuffle the type so that we can phrase it in terms of a
single dependent product, and a single type family. We define a type
$A$
to be a *subsingleton* if there is a term

$f : (xy : A \times A) \to (xy\mathrm{.fst} \equiv xy\mathrm{.snd})$

An $f$ of this type can be seen as a section of the type family $F : A \times A \to \mathrm{Type}$ defined by the rule

$F(x,y) \mapsto (x \equiv y)$

The total space $\sum F$ of this family is the space of all paths in $A$, which will be written as $A^\mathbb{I}$ to emphasise its nature as a path space. Since it is the total space of a type family, it comes equipped with a fibration $\mathrm{fst} : A^\mathbb{I} \to A \times A$. Given that a term in $A^\mathbb{I}$ can be seen to be a pair $((x, y), p)$ where $p : x \equiv y$, we see that this fibration projects both endpoints of a path — so we will write it $(d_0,d_1) : A^\mathbb{I} \to A \times A$, since it is the pairing of the maps which evaluate a path at the left and right endpoints of the interval.

As a very quick aside, there is a map
$r : \lambda x.\ (x, x, \mathrm{refl}$
making the diagram below commute. This diagram expresses that the
diagonal
$A \to A \times A$
can be factored as a weak equivalence followed by a fibration through
$A^\mathbb{I}$,
which is the defining property of a *path space object*.
$A$.

$A \xrightarrow{\~r} A^\mathbb{I} \xrightarrow{(d0,d1)} \to A \times A$

A section of this fibration — that is, a dependent function like
$f$
— is then a *continuous* function
$A \times A \to A^\mathbb{I}$,
with the property that
$(d_0,d_1) \circ f = \mathrm{id}_{}$.
Now assume that our space
$A$
is pointed, say with a point
$y_0 : A$,
and that we have a section
$f$.
We can then define the homotopy
$(x,t) \mapsto f(x,y_0,t)$,
mapping between the identity function on
$A$
and the constant function at
$y_0$,
exhbiting the space
$A$
as contractible.

If you assume the law of excluded middle, every space is either
pointed or empty. In this case, we can describe a space
$A$
equipped with a map like
$f$
as being “either empty or contractible”, but we prefer the terminology
“contractible if inhabited” or **subsingleton**
instead.

# What next?🔗

While the text above was meant to serve as a linearly-structured
introduction to the field of homotopy type theory, the rest of the 1Lab
is not organised like this. It’s meant to be an *explorable*
presentation of HoTT, where concepts can be accessed in any order, and
everything is hyperlinked together. However, we can highlight the
following modules as being the “spine” of the development, since
everything depends on them, and they’re roughly linearly ordered.

## Paths, in detail🔗

open import 1Lab.Path

The module 1Lab.Path develops the theory of path types,
filling in the details that were missing in *Interlude - Basics of paths*
above. In particular, the Path module talks about the structure of the
**interval type**
$\mathbb{I}$,
the definition of **dependent paths**,
**squares**, and the symmetry involution on paths. It then
introduce the **transport** operation, turns a path between
types into an equivalence of types.

It also exposits about the structure of types as Kan complices; In
particular, the Path module defines and talks about **partial
elements**, which generalise the notion of “horn of a simplex” to
any shape of partial cube; Then, it defines
**extensibility**, which exhibits a partial element as
being a subshape of some specific cube, and the **composition and
filling** operations, which express that extensibility is
preserved along paths. Finally, it introduces the **Cartesian
coercion** operation, which generalises transport.

To reiterate, paths are *the most important idea in cubical type
theory*, so you should definitely read through this module!

## Equivalences🔗

The idea of subsingleton type, mentioned in passing in the discussion
about universes and expanded upon in the section on dependent products,
generalises to give the notion of **h-level**, which
measures how much interesting homotopical information a type has. A
h-level that comes up very often is the level of **sets**,
since these are the types where equality behaves like a logical
proposition. Due to their importance, we provide a module defining
equivalent characterisations of sets.

open import 1Lab.HLevel

Contractible types can be used to give a definition of equivalence that is well-behaved in the context of higher category theory, but it is not the only coherent notion. We also have half-adjoint equivalences and bi-invertible maps. Finally, we have a module that explains why these definitions are needed in place of the simpler notion of “isomorphism”.

open import 1Lab.Equiv open import 1Lab.Equiv.Biinv open import 1Lab.Equiv.HalfAdjoint open import 1Lab.Counterexamples.IsIso

## Univalence🔗

open import 1Lab.Univalence

The module 1Lab.Univalence defines and explains the
**Glue** type former, which serves as the Kan structure of
the universe. It then concretises this idea by describing how Glue can
be used to turn any equivalence into a path between types, which leads
to a proof of the **univalence principle**. After, we prove
that the type of equivalences satisfies the same induction principle as
the type of paths, and that universes are object classifiers.

open import 1Lab.Univalence.SIP

While univalence guarantees that equivalence *of types* is
equivalent to paths *in types*, it does not say anything about
types-with-structure, like groups or posets. Fortunately, univalence
implies the **structure identity principle**, which
characterises the paths in spaces of “structured types” as being
homomorphic equivalence.

As usual, we do not have a type of

*all*groups. Instead, we have a family of types of groups that depends on how “big” their underlying type is allowed to be: what universe it lives in.↩︎Consequently, a term of the form $(\lambda x.\ e) e'$ is referred to as a

*$\beta$ redex*(plural*$\beta$ redices*), short for*$\beta$-reducible expression*, but since we’re not really interested in the theory of programming languages here, this terminology will not come up.↩︎A survey paper of POPL proceedings by Guy L. Steele (Steele 2017) identified

**twenty-eight**different notations for substitution, so the word “typical” is.. questionable, at best.↩︎**Solution**: Recall what it means for a map to be injective - that $f(a) = f(b) \to a = b$. If the domain of a function is a set with a single point, then the consequent is automatically satisfied! Since the maps $* \to \mathbb{N}$ are in 1-to-1 correspondence with the elements of $\mathbb{N}$, we can consider*any two numbers*to be distinct subsets with the same domain.↩︎In some foundational systems — including HoTT, the one we are discussing — “subsingleton” is literally taken as the

*definition*of “proposition”, so this translation isn’t merely omitted, it’s not necessary at all.↩︎Let me make this more precise. In any (univalent) category with finite limits $\mathscr{E}$, we have a functor $\mathrm{Sub} : \mathscr{E}^{\mathrm{op}} \to \mathrm{Set}$, which takes $c$ to the full subcategory of $\mathscr{E}/c$ on the injections. We say that $\Omega \in \mathscr{E}$ is a

**subobject classifier**if there is a natural isomorphism $\mathrm{Sub} \cong \mathbf{Hom}(-,\Omega)$, i.e., if $\Omega$ is a representing object for the subobject functor. Since the Yoneda lemma implies that representing objects are unique, this characterises $\Omega$.↩︎*Discreteness*, type-theoretically, has to do with whether we can decide equality in that type by an algorithm. For example, the natural numbers are discrete, because we can compare the numerals they represent in time bounded by the size of the number. On the other hand, sequences of natural numbers are not discrete, because an algorithm can not run forever. Any discrete type is a set, and assuming excluded middle, any set is discrete.↩︎

## References

- Steele, Guy L. 2017. “It’s Time for a New Old Language.”
*SIGPLAN Not.*52 (8): 1. https://doi.org/10.1145/3155284.3018773.