Homotopy Type Theory
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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 of the 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

A category is a precategory such that for all a,b:Aa,b:A, the function idtoiso a,bidtoiso_{a,b} from Lemma 9.1.4 (see precategory) is an equivalence.

The inverse of idtoisoidtoiso is denoted isotoidisotoid.

Examples

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

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

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

  • For any type XX, there is a precategory with XX as its type of objects and with hom(x,y)x=y 0hom(x,y)\equiv \| x= y \|_0. The composition operation

    y=z 0x=y 0y=z 0\|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 pregroupoid of XX.

  • There is a precategory whose type of objects is 𝒰\mathcal{U} and with hom(X,Y)XY 0hom(X,Y)\equiv \| X \to Y\|_0, and composition defined by induction on truncation from ordinary composition of functions. We call this the homotopy precategory of types.

Properties

Lemma 9.1.8

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

Proof. It suffices to show that for any a,b:Aa,b:A, the type a=ba=b is a set. But a=ba=b is equivalent to aba \cong b which is a set. \square

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

Lemma 9.1.9

See also

Category theory

References

Coq code formalizing the concept of 1-categories includes the following:

category: category theory

Last revised on October 11, 2018 at 06:31:20. See the history of this page for a list of all contributions to it.