Homotopy Type Theory
category (Rev #3)

Idea

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.

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.

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.

See also

Category theory

References

HoTT Book

Revision on September 4, 2018 at 15:27:21 by Ali Caglayan. See the history of this page for a list of all contributions to it.