TITLE: Is there a fundamental distinction between objects and its types?
QUESTION [10 upvotes]: My question is related to the formal presentation of type theory as stated in the context of Homotopy Type Theory. Every formalization is grounded on typing judgements like
$$ a: A $$
where mostly it is said that $a$ is an object that can be built due to several formation or term-forming rules and $A$ is a Type. But are types and objects different layers or is every type in some sense also a term. So in conclusion: Is there only one base class of mathematical entities in general? 
I'm aware that the objects above are proofs. What confuses me mostly is that for the universe hierarchy postulated one has
$$ U_n : U_m $$ for $n<m$. Statements of the form $$A : B$$ with $A$ and $B$ types suggest me to believe that there is no distinction between proofs and Types. Is this the correct point of view?
Addendum, 01.01.2015:
I think the "homotopy perspective" is also a clue for that - if one assumes that points of a space can be regarded as spaces itself in some sense. 
Update, 02.01.2015
I have found a very nice answer on CS SE. Roughly spoken the answer is: Depending on the type theory used, there is no distinction (or there is). To quote the answer there: 

There are type systems that go further and completely mix types and base terms, so that there is no distinction between the two. Such type systems are said to be higher-order. 

So my question reduces to: Is it true that in Martin-Löf Type Theory, which HoTT is based on, there is no distinction between terms and types?

REPLY [10 votes]: "Martin-Lof type theory" is, unfortunately, not a completely unambiguous designator of a type theory.  However, in the type theory used in the HoTT book, it is true that types are particular terms, namely terms whose type is a universe.  (Of course, not every term is a type.)  In other closely related type theories, types are not particular terms; instead there is an operation "El" (for "elements of") which maps terms-whose-type-is-a-universe to types.  The first version is sometimes referred to as a "universe a la Russell", and the second (with "El") as a "universe a la Tarski".  Russell universes are often more convenient to use, but Tarski universes have some formal advantages, due to the fact that they don't "mix levels" between types and terms.