Categorially a higher inductive type in an extensional type theory is an initial algebra of an endofunctor. In intensional type theory this analogy fails.
In particular $W$-types in an extensional type theory correspond to initial algebras of polynomial functors. Also this is not true for intensional type theories.
This failure of intensional type theory can be (at least for $W$-types and some more general cases) remedied by replacing “initial” by “homotopy initial”. This is the main result (to be found in §2) of
The four standard forms of typing judgements are
(1) $A:Type$
(2) $A=B:Type$ definitional equality for types
(3) $a:A$
(4) $a=b:A$ definitional equality for termes
There is also another notion of equality - namely propositional equality:
(1) $Id$-formation rule
(2) $Id$-introduction rule
(3) $Id$-elimination rule
(4) $Id$-computation rule
One important effect of not having the identity-reflection rule
is that it is impossible to prove that the empty type is an initial object. There are some extensionality principles which are weaker than the identity reflection rule: Streicher’s K-rule, the Uniqueness of Identity Proofs (UIP) which also has benn considered in context of Observational Type Theory. However these constructions seem to be ad hoc and lack a conceptual basis.
The dependent type theory $\mathcal{H}$ is defined to have -in addition to the standard structural rules- the folowing rules:
(1) the rules for identity types.
(2) rules for $\Sigma$-types as presented in Nordstrom-Petterson-Smith §5.8
(3) rules for $\Pi$-types as presented in Garner §3.2
(4) the propositional $\eta$-rule for $\Pi$-types: the rule asserting that for every $f:(\Pi x:A)B(x), the type
is inhabited.
(5) the Function extensionality axiom (FE): the rule asserting that for every $f,g:A\to B$, the type
Comments:
In Voevodsky’s Coq files is shown that the $\eta$-rule for dependent functions and the function extensionality principle imply the corresponding function extensionality principle for $\Sigma$-types. is inhabited.
The function extensionality axiom (FE) is implied by the univalence axiom. The system $\mathcal{H}$ does not have a global extensionality rule such as the identity reflection rule K or UIP
The transport function: An identity term $p:Id(a,b)$ is interpreted as a path between the points $a$ and $b$. More generally an identity term $p(x):Id(a(x),b(x))$ with free variable $x$ is interpreted as a continuous family of paths, i.e. a homotopy between the continuous functions. The identity elimination rule implies that type dependency respects identity: For a dependent type
and an identity term $p:Id_A(a,b)$, there is a transport function
For $x:A$ the function $refl(x)_!:B(x)\to B(x)$ is obtained by $Id$-elimination.
Steve Awodey, Nicola gambino, Kristina Sojakova, inductive types in homotopy type theory, arXiv:1201.3898
S. Awodey, Type theory and homotopy, pdf
T. Streicher, Investigations into intensional type theory, 1993, Habilitation Thesis. Available from the author’s web page.
B. Nordstrom, K. Petersson, and J. Smith, Martin-Löf type theory in Handbook of Logic in Computer Science. Oxford Uni- versity Press, 2000, vol. 5, pp. 1–37
R. Garner, On the strength of dependent products in the type theory of Martin-Löf, Annals of Pure and Applied Logic, vol. 160, pp. 1–12, 2009.
V. Voevodsky, Univalent foundations Coq files, 2010, available from the author’s web page.