category theory in nLab
Elementary theory of the category of categories, or ETCC for short, is in a broad sense an appropriate name for any first order theory axiomatizing the ...
The definition of functor in homotopy type theory is a straightforward translation of the ordinary one. However, the notion of univalent ...
geometry of physics – basic notions of category theory in nLab
the left adjoint reflector sends a small groupoid 𝒢 \mathcal{G} to its set of connected components, namely to the set of equivalence classes ...
In the context of higher category theory one sometimes needs, for emphasis, to say 1-category for category. 2. Details. Fix a meaning of ∞ \ ...
For example, one speaks (assuming that any exists) of “the” terminal object of a category, “the” Cartesian product of two objects, “the” left adjoint of a ...
A category is said to be essentially small (or, rarely, svelte) if it is equivalent to a small category. Assuming the axiom of choice, this is ...
1. Idea. An · -category is a higher category such that, essentially: ; 3. Homotopy-theoretic relation. From the point of view of homotopy theory, ...
A displayed category over a category C C is the “classifying map” of a category over C C . That is, it is equivalent to the data of a category D ...
The term ∞ \infty -category broadly refers to higher categories with no bound on the dimension n n of their n n -morphisms. There are two ...
In category theory a limit of a diagram F : D → C F : D \to C in a category C C is an object lim F lim F of C C equipped with morphisms to the ...
Note that this composition is unique by the axioms of category theory. If we instead work in a weak higher category, composition need not be ...
Alternative notions of “1-category” in homotopy type theory include strict categories, whose type of objects is a set, and univalent categories, ...
According to the general pattern on (n,r)-category, an ( ∞ , 1 ) (\infty,1) -category is a (weak) ∞-category in which all n n -morphisms for n ≥ ...
This sort of diagram can be identified with a functor whose domain is a free category, and this is the most common context when we talk about ...
The notion of a test category (Grothendieck 83) is meant to axiomatize common features of categories of shapes used to model homotopy types in homotopy theory.
Recall that Cat is generally used to denote the (or a) category (or 2-category) of categories (and functors and natural transformations).
It is a category equipped with three classes of morphisms, each closed under composition and called weak equivalences, fibrations and cofibrations.
But it is more appropriate in higher category theory to consider these things up to equivalence rather than up to isomorphism; when we do this, ...
The Elementary Theory of the Category of Sets, or ETCS for short, is an axiomatic formulation of set theory in a category-theoretic spirit.
The notion of enriched category is a generalization of the notion of category. Very often instead of merely having a set of morphisms from one ...