geometric type theory in nLab
So in a very broad sense, quality types are intended as ingredients to a synthetic homotopy theory where the (homotopy) contraction of a space ( ...
Terminology 2.1. We will say a manifold k k -diagram ( ℝ k , f ) (\mathbb{R}^k,f) is a stratum k k -type if the diagram is of the form ...
in differential geometry/linear algebra: tensor ... in formal logic and type theory: the contraction rule is one of the structural rules,.
homotopy type theory is the formal language for which elementary (infinity,1)-toposes are the semantics. For more on this see at categorical ...
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However, in type theory, one cannot quantify over families of elements x : A ⊢ b ( x ) : B x:A \vdash b(x):B , which are the analogue of ...
2. Related entries · group, Grp · combinatorial group theory, geometric group theory · discrete group, finite group · permutation group, symmetric ...
This is analogous to the definition of 'a real number' as an equivalence class of Cauchy sequences. However, as usual in homotopy theory, merely ...
If 𝒞 \mathcal{C} is the classifying topos of some geometric theory T T , then an object of 𝒞 \mathcal{C} may be called a “ T T -structured ...
Synthetic topology, like synthetic domain theory, synthetic differential geometry, and synthetic computability, are part of synthetic mathematics.
nLab category-theoretic approaches to probability theory
Category theory was first developed to model particular structures in algebraic topology, and subsequently algebraic geometry, algebra, logic and computer ...
Category theory can provide a common model theory to compare various set theories. Although only structural set theories like ETCS treat the ...
It has only partial internal unary parametricity, rather than full internal unary parametricity, so that it has semantics in any ( ∞ , 1 ) (\ ...
The notion of a “generalized the” can be formalized and treated uniformly in homotopy type theory. Here one may define an introduction rule for the as follows:.
One can also say that classical algebraic geometry often provides a testing ground for more general developments in model theory. (For the most ...
The category whose objects are topological spaces and whose morphisms are homotopy equivalence-classes of continuous functions is also called ...
The systematic use of the tensor product structure here goes back to (Balmer 02) and the concept of the spectrum of a tensor triangulated ...
This is of course the same as a model for the “empty theory” in that signature, which has the same types, operations, and relations, but no ...
Often it is either a monad or comonad, and often this monad or comonad is idempotent, but not always. Similarly, in homotopy type theory, the ...
geometric model for elliptic cohomology in nLab
One expects that this encodes actually the differential refinements of the corresponding cohomology theories, such as differential K-theory.