module index where

# 1lab🔗

A formalised, cross-linked reference resource for cubical methods in
Homotopy Type Theory. Unlike the HoTT book, the 1lab is
not a “linear” resource: Concepts are presented as a directed graph,
with links indicating *dependencies*. For instance, the statement
of the univalence principle depends on universes, identifications and equivalences. In addition to the
hyperlinked “web of concepts” provided by the Agda code, there is a
short introduction to homotopy type theory: **Start here**.

_ : ∀ {ℓ} {A B : Type ℓ} → is-equiv (path→equiv {A = A} {B}) _ = univalence

The purpose of the “web of concepts” approach is to let each reader
approach the 1lab at their own pace: If you already know what all of the
code above means, you can click on univalence
to be taken directly to the construction of the equivalence — but if you
*don’t*, you can click on other definitions like is-equiv and path→equiv, and in turn explore the
dependencies of *those* concepts, and so on.

The 1lab is a community project: we use GitHub for source control and
talk on Discord. Our purpose
is to make cubical methods in homotopy type theory accessible to, and
inclusive of, everyone who is interested, regardless of cultural
background, age, ability, ethnicity, gender identity, or expression.
Correspondingly, interactions in those forums are governed by the Contributor
Covenant Code of Conduct. **We believe HoTT is for everyone,
and are committed to fostering a kind, inclusive
environment.**

Mathematics is, fundamentally, a social activity. Correspondingly, we have a page dedicated to letting authors introduce and talk a bit themselves and their other work:

open import Authors

Similarly, we maintain this list of related resources which serve as an introduction to HoTT in general:

The “canonical” reference is the HoTT Book, written by a variety of mathematicians at the IAS Special Year for Univalent Mathematics, held between 2012-2013 and organised by Steve Awodey, Thierry Coquand, and the late Vladimir Voevodsky.

The Book is often referred to on this site - with those words - so if you don’t know which book “The Book” is, it’s the HoTT book! It’s split into two parts: Type Theory, which introduces the concepts of Homotopy Type Theory with no previous knowledge of type theory assumed; and Mathematics, which develops some mathematics (homotopy theory, category theory, set theory, and real analysis) in this theory.

Prof. Martín Escardó, at Birmingham, has done a great service to the community by

*also*formalising a great deal of univalent mathematics in Literate Agda, in his Introduction to Univalent Foundations of Mathematics with Agda.Prof. Escardó’s notes, unlike the 1lab, are done in base Agda, with univalence assumed explicitly in the theorems that need it. This is a principled decision when the goal is introducing univalent mathematics, but it is not practical when the goal is to

*practice*univalent mathematics in Agda.Even still, that document is

*much better*than this site will*ever*be as an introduction to the subject! While many of the pages of the 1lab have introductory*flavour*, it is not meant as an introduction to the subject of univalent mathematics.

Prof. Favonia has kindly uploaded the outline, videos and lecture notes for their 2020 course on higher-dimensional type theory, which also serves as an introduction to cubical methods in homotopy type theory, aimed at graduate students. You can find the course page here, the videos here on their YouTube, and the notes here (though heed the warning in the README).

Another comprehensive, formalised Agda resource is the agda-unimath project, though unlike us (and like prof. Escardó’s lecture notes) they make use of

*axiomatic*HoTT: Univalence is a postulate, and thus does not have computational content.Regardless, they have formalised a great deal of “ordinary” mathematics in the univalent context: elementary number theory, group theory and combinatorics being the most prominent projects.

## Technology🔗

The 1Lab uses Julia Mono as its monospace typeface. Julia Mono is licensed under the SIL Open Font License, v1.1, a copy of which can be found here. As the sans-serif typeface, we use the Inria Sans webfont, and as a serif typeface, EB Garamond. These fonts are both open-source, though rather than rehosting them, we use them from Google Fonts.

Mathematics is rendered using KaTeX, and as so, the 1Lab redistributes KaTeX’s fonts and stylesheets, even though the rendering is done entirely at build-time. KaTeX is licensed under the MIT License, a copy of which can be found here.

Our favicon is Noto Emoji’s ice cube (cubical type theory - get it?), codepoint U+1F9CA. This is the only image from Noto we redistribute. Noto fonts are licensed under the Apache 2.0 License, a copy of which can be found here.

Commutative diagrams appearing in body text are created using quiver, and rendered to SVG using a combination of LaTeX and pdftocairo, part of the Poppler project. No part of these projects is redistributed.

And, of course, the formalisation would not be possible without Agda.

# Type Theory🔗

Most of the modules in the 1Lab assume a baseline knowledge of type
theory. For this, **read the
introduction here**.

The first things to be explained are the foundational constructions
in (cubical) type theory - things like types themselves, universes, paths, equivalences, glueing and the univalence “axiom”. These
are developed under the `1Lab`

namespace. Start here:

-- All of these module names are links you can click! open import 1Lab.Type -- Universes open import 1Lab.Path -- Path types open import 1Lab.Path.Groupoid -- Groupoid structure of types open import 1Lab.Path.Reasoning -- Combinators for reasoning with path composition open import 1Lab.Path.IdentitySystem -- Families R for which R(x,y) ≃ (x ≡ y) open import 1Lab.Equiv -- “Contractible fibres” equivalences open import 1Lab.Equiv.Biinv -- Biinvertible maps open import 1Lab.Equiv.FromPath -- Transport is an equivalence, cubically open import 1Lab.Equiv.Embedding -- Embeddings open import 1Lab.Equiv.Fibrewise -- Fibrewise equivalences open import 1Lab.Equiv.HalfAdjoint -- Half-adjoint equivalences open import 1Lab.HLevel -- h-levels open import 1Lab.HLevel.Retracts -- Closure of h-levels under retractions/isos open import 1Lab.HLevel.Universe -- The type of n-types is a (n+1)-type open import 1Lab.Univalence -- Equivalence is equivalent to identification open import 1Lab.Univalence.SIP -- Univalence + preservation of structure open import 1Lab.Univalence.SIP.Auto -- Derive is-univalent for families of types open import 1Lab.Type.Pi -- Properties of dependent products open import 1Lab.Type.Sigma -- Properties of dependent coproducts open import 1Lab.HIT.Truncation -- Propositional truncation open import 1Lab.Counterexamples.IsIso -- Counterexample: is-iso is not a prop open import 1Lab.Counterexamples.Russell -- Counterexample: Russell's paradox open import 1Lab.Counterexamples.Sigma -- Counterexample: Sigma is not prop

## Data types🔗

The `Data`

namespace contains definitions of oft-used data
types, which are fundamental to the rest of the development but not
“basic type theory”. These modules contain (or re-export) the types
themselves, useful operations on them, characterisation of their path
spaces, etc.

open import Data.Nat -- The natural numbers open import Data.Int -- The integers open import Data.Sum -- Coproduct types open import Data.Bool -- The booleans open import Data.List -- Finite lists open import Data.Power -- Power sets open import Data.Power.Lattice -- Power sets form a lattice open import Data.Set.Coequaliser -- Set coequalisers

# Category Theory🔗

In addition to providing a framework for the synthetic study of higher groupoids, HoTT also provides a natural place to develop constructive, predicative category theory, while still being compatible with classicality principles like the axiom of choice and/or the law of excluded middle. Here, we do not assume any classicality principles.

## Basics🔗

The main modules in the `Cat`

namespace provide the
foundation for the rest of the development, defining basic constructions
like precategories themselves, functors, natural transformations,
etc.

open import Cat.Base -- Precategories, functors, natural transformations open import Cat.Solver -- Automatic solver for associativity problems open import Cat.Morphism -- Important classes of morphisms open import Cat.Reasoning -- Categorical reasoning combinators

### Diagrams🔗

For convenience, we define a plethora of “concrete” universal diagrams, unpacking their definitions as limits or colimits. These are simpler to work with since they provide the relevant data with fewer layers of indirection.

open import Cat.Diagram.Congruence -- Internal equivalence relations -- Colimits: open import Cat.Diagram.Initial -- Initial objects open import Cat.Diagram.Pushout -- Pushouts open import Cat.Diagram.Coproduct -- Binary coproducts open import Cat.Diagram.Coequaliser -- Coequalisers open import Cat.Diagram.Colimit.Base -- Conical colimits open import Cat.Diagram.Coproduct.Indexed -- Indexed coproducts open import Cat.Diagram.Coequaliser.RegularEpi -- Regular epimorphisms open import Cat.Diagram.Duals -- Dualisation of co/limits open import Cat.Diagram.Image -- Image factorisations open import Cat.Diagram.Idempotent -- Idempotent morphisms -- Limits open import Cat.Diagram.Product -- Binary products open import Cat.Diagram.Pullback -- Fibred products open import Cat.Diagram.Terminal -- Terminal objects open import Cat.Diagram.Equaliser -- Equalisers open import Cat.Diagram.Limit.Base -- Conical limits open import Cat.Diagram.Limit.Finite open import Cat.Diagram.Limit.Product open import Cat.Diagram.Limit.Pullback open import Cat.Diagram.Limit.Equaliser open import Cat.Diagram.Product.Indexed open import Cat.Diagram.Equaliser.Kernel -- Kernels open import Cat.Diagram.Pullback.Properties -- Properties of fibred products open import Cat.Diagram.Equaliser.RegularMono -- Regular monomorphisms open import Cat.Diagram.Monad -- Monads open import Cat.Diagram.Monad.Limits -- Limits in Eilenberg-Moore categories open import Cat.Diagram.Zero -- Zero objects

## Functors🔗

This namespace has definitions of properties functors can have, utility modules for working with functors, the definition of full subcategories, and adjoint functors.

open import Cat.Functor.Hom -- Hom functor, Yoneda embedding, Coyoneda lemma open import Cat.Functor.Base -- Compendium of functor properties open import Cat.Functor.Pullback -- Base change, dependent sum, Σf ⊣ f* open import Cat.Functor.Bifunctor -- Functors out of product categories open import Cat.Functor.Equivalence -- Equivalences of (pre)categories open import Cat.Functor.Conservative -- Functors which reflect isomorphisms open import Cat.Functor.FullSubcategory -- Full subcategories open import Cat.Functor.Equivalence.Complete -- Completeness respects equivalence

About adjoint functors, and their associated monads:

open import Cat.Diagram.Monad -- Definition of monads open import Cat.Functor.Adjoint -- Unit-counit adjunctions and universal arrows open import Cat.Functor.Adjoint.Monad -- Monad from an adjunction open import Cat.Functor.Adjoint.Monadic -- Monadic adjunctions open import Cat.Functor.Adjoint.Compose -- Adjunctions compose open import Cat.Functor.Adjoint.Continuous -- Right adjoints preserve limits open import Cat.Functor.Adjoint.Reflective -- Reflective subcategories

Monadicity theorems:

open import Cat.Functor.Monadic.Beck -- Beck's coequalisers open import Cat.Functor.Monadic.Crude -- The crude monadicity theorem

About Kan extensions:

open import Cat.Functor.Kan -- Left Kan extensions open import Cat.Functor.Kan.Right -- Right Kan extensions open import Cat.Functor.Kan.Nerve -- The nerve/realisation adjunction

## Univalent categories🔗

In HoTT/UF, the word “category” is reserved for the precategories (what the rest of the world refers to as just “category”) in which isomorphic objects are indistinguishable, i.e. the categories which satisfy a version of the univalence axiom. Sometimes we also refer to these as “univalent categories” to make the distinction clear.

open import Cat.Univalent -- Basic properties of categories open import Cat.Univalent.Rezk -- Free category on a precategory open import Cat.Univalent.Instances.Algebra -- Eilenberg-Moore categories preserve univalence

## Category instances🔗

Here’s where we actually build some categories and prove that they have desirable properties.

-- Comma categories: open import Cat.Instances.Comma open import Cat.Instances.Comma.Univalent open import Cat.Instances.Delooping -- Delooping a monoid to give a category open import Cat.Instances.Discrete -- Discrete categories open import Cat.Instances.Elements -- Category of elements of a presheaf -- Functor categories: open import Cat.Instances.Functor open import Cat.Instances.Functor.Limits -- Co/limits in functor categories open import Cat.Instances.Functor.Duality -- 2-cell duality in Cat -- Completion of a category under splitting idempotents open import Cat.Instances.Karoubi open import Cat.Instances.Lift -- Lifting a category to higher universes open import Cat.Instances.Product -- Product categories -- The category of sets open import Cat.Instances.Sets -- is univalent open import Cat.Instances.Sets.Complete -- is complete open import Cat.Instances.Sets.Cocomplete -- is cocomplete, with disjoint coproducts open import Cat.Instances.Sets.Congruences -- has effective congruences open import Cat.Instances.Sets.CartesianClosed -- and is locally cartesian closed -- Diagram shapes: open import Cat.Instances.Shape.Join open import Cat.Instances.Shape.Cospan open import Cat.Instances.Shape.Interval open import Cat.Instances.Shape.Isomorphism open import Cat.Instances.Shape.Parallel open import Cat.Instances.Shape.Terminal -- Slice categories: open import Cat.Instances.Slice open import Cat.Instances.Slice.Presheaf -- PSh(C)/X ≅ PSh(∫ X) -- Strict categories open import Cat.Instances.StrictCat open import Cat.Instances.StrictCat.Cohesive -- ^ Strict category structure is a sort of "spatial" structure on a category

## Thin categories🔗

Strict thin categories are a presentation of pre-ordered sets,
i.e. sets equipped with a transitive and reflexive relation — so we call
them “prosets”. When this relation is antisymmetric, we additionally
have a *univalent* thin strict category — so we call these
“posets”.

open import Cat.Thin -- Basics of thin categories open import Cat.Thin.Limits -- Limits in thin categories open import Cat.Thin.Completion -- Free poset on a proset

## Displayed categories🔗

We also have a work-in-progress formalisation of Foundations of Relative Category Theory, in which the core idea is thinking of “categories over categories”.

open import Cat.Displayed.Base -- Displayed categories open import Cat.Displayed.Total -- Total category of a displayed category open import Cat.Displayed.Fibre -- Fibre categories of a displayed category open import Cat.Displayed.Cartesian -- Cartesian lifts, cartesian fibrations open import Cat.Displayed.Univalence -- Univalence for displayed categories open import Cat.Displayed.Reasoning -- Reasoning combinators for displayed categories open import Cat.Displayed.Instances.Slice -- Canonical self-indexing open import Cat.Displayed.Instances.Family -- Family fibration open import Cat.Displayed.Instances.Pullback -- Pullback of a displayed category by a functor open import Cat.Displayed.Instances.Scone -- The displayed category of scones

## Bicategories🔗

The modules under `Cat.Bi`

are very much
work-in-progress.

open import Cat.Bi.Base -- Prebicategories, lax functors, pseudofunctors, lax transformations, -- pseudonatural transformations, modifications, and the bicategory of -- categories. open import Cat.Bi.Instances.Spans -- The prebicategory of spans in a precategory with pullbacks open import Cat.Bi.Instances.Discrete -- The locally discrete prebicategory associated to a precategory

### Diagrams in bicategories🔗

open import Cat.Bi.Diagram.Monad open import Cat.Bi.Diagram.Adjunction

## Homological algebra🔗

The theory of abelian categories.

open import Cat.Abelian.Base open import Cat.Abelian.Images open import Cat.Abelian.Limits open import Cat.Abelian.Functor

# Topos theory🔗

Grothendieck topos theory developed constructively and predicatively.

open import Topoi.Base -- Topoi, properties of topoi, geometric morphisms open import Topoi.Reasoning -- Exactness properties of topoi (cont'd), reasoning open import Topoi.Classifying.Diaconescu -- ^ Presheaf topoi classify flat functors on their site

# Algebra🔗

In the `Algebra`

namespace, the theory of universal
algebra is developed from a univalent perspective. Specifically, every
definition of an algebraic structure comes with an associated proof that
it is univalent — concretely, identification of groups is group
isomorphism (for instance).

open import Algebra.Magma -- Binary operations open import Algebra.Magma.Unital -- Operations with two-sided units open import Algebra.Magma.Unital.EckmannHilton -- The Eckmann-Hilton argument open import Algebra.Semigroup -- Semigroups (associative magmas) open import Algebra.Monoid -- Monoids as unital semigroups open import Algebra.Monoid.Category -- The category of monoids open import Algebra.Lattice -- Lattices open import Algebra.Semilattice -- Semilattices open import Algebra.Ring -- Rings open import Algebra.Group -- Groups as monoids with inverses open import Algebra.Group.Free -- Free groups open import Algebra.Group.Action -- Group actions open import Algebra.Group.Cayley -- Cayley's theorem open import Algebra.Group.Cat.Base -- The category of groups open import Algebra.Group.Cat.Monadic -- ... is monadic over Sets open import Algebra.Group.Cat.FinitelyComplete -- Finite limits in Groups open import Algebra.Group.Subgroup -- Subgroups, images and kernels open import Algebra.Group.Homotopy -- Homotopy groups open import Algebra.Group.Homotopy.BAut -- Delooping groupoids of automorphism groups open import Algebra.Group.Ab -- Abelian groups, and the category Ab open import Algebra.Group.Ab.Sum -- Direct sum abelian groups open import Algebra.Group.Ab.Free -- The free abelian group on a group