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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.
A category is a precategory such that for all $a,b:A$, the function $idtoiso_{a,b}$ from Lemma 9.1.4 (see precategory) is an equivalence.
The inverse of $idtoiso$ is denoted $isotoid$.
Note: All precategories given can become categories via the Rezk completion.
There is a precategory $\mathit{Set}$, 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 precategory of set-level structures such as groups, rings topologicial spaces, etc. is also a category.
For any 1-type $X$, there is a 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 precategory with $X$ as its type of objects and with $hom(x,y)\equiv \| x= y \|_0$. The composition operation
is defined by induction on truncation? from concatenation of the identity type. We call this the fundamental pregroupoid of $X$.
There is a precategory 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 precategory of types.
In a 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. $\square$
There is a canonical way to turn a precategory into a category via the Rezk completion.
Benedikt Ahrens, Chris Kapulkin, Michael Shulman, Univalent categories and the Rezk completion, Mathematical Structures in Computer Science 25.5 (2015): 1010-1039 (arXiv:1303.0584)
Univalent Foundations Project, chapter 9 of Homotopy Type Theory – Univalent Foundations of Mathematics, IAS 2013
Coq code formalizing the concept of 1-categories includes the following:
Revision on September 7, 2018 at 08:29:46 by Urs Schreiber. See the history of this page for a list of all contributions to it.