natural deduction metalanguage, practical foundations
type theory (dependent, intensional, observational type theory, homotopy type theory)
computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory
There are two main styles of definition of category in the literature: one which immediately generalises to the usual definition of internal category and one which immediately generalises to the usual definition of enriched category. Here we consider the latter definition, and show how it may naturally be expressed in dependent type theory.
Note that in a type theory without identity types, where types are presets (without an inherent equality predicate), it is not manifestly possible to express concepts that violate the principle of equivalence; in that categories are not necessarily strict. In homotopy type theory, we have identity types, but they are not extensional; thus it is also possible to define non-strict categories there (but there is a subtlety; see below).
In a dependent type theory with dependent record types? we can define a type of categories as follows.
We use a two-dimensional syntax, which is convenient to allow inference of implicit parameters?, and to signify notation. We read the horizontal line as a rule, so for instance the second line means that whenever $a$ and $b$ have type $\mathrm{Obj}$, then we have a type $\hom(a,b)$ (with equality).
The notation $p\coloneq P$ signifies that $p$ is a proof of the proposition $P$ (under propositions as types or propositions as some types, this may be the same as $p\colon P$, but many type theories treat propositions as distinct from types).
Here, $\mathrm{Type}$ is a type of types, and $\mathrm{Type}_=$ is a type of types with equality predicates (we may or may not have $\mathrm{Type}=\mathrm{Type}_=$). Specifically:
In extensional type theory (with extensional identity types), we have $\mathrm{Type} = \mathrm{Type}_=$, and the above definition simply makes no use of the equality predicate on the type of objects. In this case we obtain strict categories, although that is not immediately visible from the definition.
In dependent type theory without identity types, basically the only option for $\mathrm{Type}_=$ is the type of setoids. In this case we obtain a notion of non-strict category, since the type of objects has no equality predicate at all.
In homotopy type theory, it is natural to take $\mathrm{Type}_=$ to be the type of h-sets: types whose identity/path types behave extensionally. We should also restrict the homotopy level of the type of objects, however, since a true 1-category should have no more than a 1-groupoid of objects; thus $\mathrm{Type}$ in the definition above is really the type of h-groupoids instead of the type of all small types. That is, we should take $\mathrm{Obj}$ to be $1$-truncated in addition to taking each $\hom(a,b)$ to be $0$-truncated.
This gives a notion of non-strict category (since there is no equality predicate on a $1$-truncated type other than isomorphism). However, it is not quite the right definition of “$1$-category” in homotopy type theory, because nothing requires that the paths in $\mathrm{Obj}$ are the same as the isomorphisms defined categorically. We need to impose a version of the “completeness” condition on a complete Segal space; in other words, we require that the core of the category be the equivalent as a groupoid to the original type of objects.
The defined type of categories cannot itself be a member of $\mathrm{Type}$, otherwise we run into Girard's paradox. This is related to the size issues for categories.