nLab internal category in homotopy type theory

Idea

One may consider internal categories in homotopy type theory. Under the interpretation of HoTT in an (infinity,1)-topos, this corresponds to the concept of a category object in an (infinity,1)-category. The general idea is presented there at Homotopy Type Theory Formulation.

For internal 1-categories in HoTT (as opposed to more general internal (infinity,1)-categories) a comprehensive discussion was given in (Ahrens-Kapulkin-Shulman-13).

In some literature, the “Rezk-completeness” condition on such categories is omitted from the definition, and categories that satisfy it are called saturated or univalent.

Similarly to the univalence axiom we make two notions of sameness the same. This leads to some nice concequences.

Definition

In homotopy type theory, there are two notions of “sameness” for objects of a precategory. On one hand we have an isomorphism between objects $a$ and $b$, on the other hand we have equality of objects $a$ and $b$.

There is a special kind of category called a univalent category where these two notions of equality coincide and some very nice properties arise.

Lemma

If $A$ is a category and $a,b:A$, then there is a map

$$idtoiso : (a=b) \to (a \cong b)$$

Proof

By induction on identity, we may assume $a$ and $b$ are the same. But then we have $1_a:hom_A(a,a)$, which is clearly an isomorphism. $\square$

The inverse of $idtoiso$ is denoted $isotoid$.

Examples

Note: All categories given can become univalent via the Rezk completion.

• There is a category $\mathit{Set}$, whose type of objects is $Set$, and with $hom_{\mathit{Set}}(A,B)\equiv (A \to B)$. Under univalence this becomes a category. One can also show that any category of set-level structures such as groups, rings topologicial spaces, etc. is also univalent.

• For any 1-type $X$, there is a univalent category with $X$ as its type of objects and with $hom(x,y)\equiv(x=y)$. If $X$ is a set we call this the discrete category on $X$. In general, we call this a groupoid.

• For any type $X$, there is a category with $X$ as its type of objects and with $hom(x,y)\equiv \| x= y \|_0$. The composition operation

  $$\|y=z\|_0 \to \|x=y\|_0 \to \|y=z\|_0$$

  is defined by induction on truncation from concatenation of the identity type. We call this the fundamental groupoid of $X$.

• There is a category whose type of objects is $\mathcal{U}$ and with $hom(X,Y)\equiv \| X \to Y\|_0$, and composition defined by induction on truncation from ordinary composition of functions. We call this the homotopy category of types.

Properties

Lemma

In a univalent category, the type of objects is a 1-type.

Proof

It suffices to show that for any $a,b:A$, the type $a=b$ is a set. But $a=b$ is equivalent to $a \cong b$ which is a set.

There is a canonical way to turn a category into a univalent category via the Rezk completion.

Related pages

• type-theoretic definition of category

• Rezk completion (for displayed categories)

• homotopy groups of spheres in homotopy type theory

• cohomology in homotopy type theory

• Hopf construction in homotopy type theory

• synthetic $(\infty,1)$-category theory

References

General

The relation between Segal completeness (now often “Rezk completeness”) for internal categories in HoTT and the univalence axiom had been pointed out in

• Urs Schreiber, Segal completenes and Univalence (2012)

A comprehensive discussion for 1-categories then appeared in:

• Benedikt Ahrens, Chris Kapulkin, Michael Shulman, Univalent categories and the Rezk completion, Mathematical Structures in Computer Science 25 5 (From type theory and homotopy theory to Univalent Foundations of Mathematics), (2015) 1010-1039 $[$arXiv:1303.0584, doi:10.1007/978-3-319-21284-5_14$]$

Exposition:

• Mike Shulman, Category Theory in Homotopy Type Theory, 2013

Implementation in Coq:

• Jason Gross, Adam Chlipala, David Spivak, Experience Implementing a Performant Category-Theory Library in Coq, In: Interactive Theorem Proving. ITP 2014 Lecture Notes in Computer Science, 8558 Springer (2014) $[$arXiv:1401.7694, doi:10.1007/978-3-319-08970-6_18$]$

Coq code formalizing this includes the following:

• github.com/benediktahrens/rezk_completion

Generalization to (n,1)-categories is discussed in

• Paolo Capriotti, Nicolai Kraus, Univalent Higher Categories via Complete Semi-Segal Types (arXiv:1707.03693)

and, by different means, in

• Emily Riehl, Michael Shulman, A type theory for synthetic $\infty$-categories (arXiv:1705.07442)

• Emily Riehl, The synthetic theory of ∞-categories vs the synthetic theory of ∞-categories, talk at Vladimir Voevodsky Memorial Conference 2018 (pdf)

Formalization of bicategories:

• Benedikt Ahrens, Dan Frumin, Marco Maggesi, Niels van der Weide, Bicategories in Univalent Foundations (arXiv:1903.01152)

Lemmas and proofs are taken from

• Univalent Foundations Project, Homotopy Type Theory – Univalent Foundations of Mathematics (2013)

By coinduction

A formalization in HoTT-Agda of general (n,r)-categories for $n,r \in \mathbb{N} \sqcup \{\infty\}$, defined as coinductive types of infinity-graphs, with operations defined by induction-coinduction, is implemented in

• Darin Morrison, agda-nr-cats, agda-infinity-categories