nLab Rezk completion

Idea

There is a canonical way to turn any category in homotopy type theory into a weakly equivalent univalent category. This can be thought of as an analogue of univalence but for isomorphisms instead of equivalence.

Intuitively, the Rezk completion is a “strictification via Yoneda” type result, in the style of strictification for bicategories via Yoneda. One starts with a category that may not have nice strictness properties, embeds it into the category of presheaves, which does have the nice strictness properties, and then restricts to representables, which gives something equivalent to the original category, but retains the nice strictness properties.

Another intuition is that the Rezk completion is a vertical categorification of the construction of a quotient of a type by an equivalence relation by taking equivalence classes. That is, we can think of a type $X$ equipped with an equivalence relation as a boolean-enriched groupoid, and a boolean-valued presheaf is equivalently a predicate on $X$ that respects the equivalence relation. Then the representable presheaves are those predicates that are furthermore inhabited, i.e., precisely equivalence classes.

Construction

We work in a dependent type theory where UIP or axiom K cannot be proven. These results are from UFP13. Note: UFP13 calls a category a “precategory” and a univalent category a “category”, but here we shall refer to the standard terminology of “category” and “univalent category” respectively.

Theorem

(Theorem 9.9.5 in UFP13) For any category $A$, there is a univalent category $\hat{A}$ and a weak equivalence $A\to \hat{A}$.

Proof

Let $\hat{A}_0$ be the type of representable objects of $\mathit{Set}^{A^{\mathrm{op}}}$, with hom-sets inherited from $\mathit{Set}^{A^{\mathrm{op}}}$. Then the inclusion $\hat{A} \to \mathit{Set}^{A^{\mathrm{op}}}$ is fully faithful and an embedding on objects. Since $\mathit{Set}^{A^{\mathrm{op}}}$ is a category by Theorem 9.2.5 (see functor category), $\hat{A}$ is also a category. Let $A \to \hat{A}$ be the Yoneda embedding. This is fully faithful by corollary 9.5.6 (See Yoneda embedding), and essentially surjective by definition of $\hat{A}_0$. Thus it is a weak equivalence.

• $\hat{A}_0\equiv \sum_{(F : \mathit{Set}^{A^{\mathrm{op}}})} \sum_{(a:A)} \mathbf{y} a \cong F$

This has the unfortunate side effect of raising the universe level. If $A$ is a univalent category in a universe $\mathcal{U}$, then in this proof $\mathit{Set}$ must at least be as large as $\mathit{Set}_{\mathcal{U}}$. Hence the univalent category $\mathit{Set}_{\mathcal{U}}$ is in a higher universe than $\mathcal{U}$ hence $\mathit{Set}^{A^{\mathrm{op}}}$ must also be in a higher universe and finally $\hat{A}$ is also in a higher universe than $\mathcal{U}$.

Now this can all be avoided by constructing a higher inductive type $\hat{A}$ with constructors:

• A function $i : A_0 \to \hat{A}_0$.
• For each $a,b:A$ and $e: a \cong b$, an equality $j e : i a = i b$
• For each $a : A$, an equality $j(1_a)=refl_{i a}$.
• For each $a,b,c:A$, $f : a \cong b$, and $g : b \cong c$, an equality $j(g \circ f)=j(f) \cdot j(g).$
• 1-truncation: for all $x,y:\hat{A}_0$ and $p,q: x = y$ and $r,s : p = q$, an equality $r=s$.

If we ignore the last constructor we could also write the above as $\| \hat{A}_0 \|_1$.

We now go on to build a univalent category $\hat{A}$ with a weak equivalence $A \to \hat{A}$ by taking the type of objects as $\hat{A}_0$ and defining hom-sets by double induction. The advantage of this approach is that it preserves universe levels, there are a lot of things to check but it is an easy proof. The kind of proof that is well suited to a proof assistant. For the complete proof see Theorem 9.9.5 of the HoTT book.

Related concepts

• Segal completion

• category theory

• Yoneda lemma

• weak equivalence

• higher inductive type

References

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

• Urs Schreiber, Segal completenes and Univalence (2012)

This was developed 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$]$

See also:

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

• Egbert Rijke, The join construction, 26 Jan 2017, (arXiv:1701.07538)

category: category theory