- homotopy hypothesis-theorem
- delooping hypothesis-theorem
- periodic table
- stabilization hypothesis-theorem
- exactness hypothesis
- holographic principle

- (n,r)-category
- Theta-space
- ∞-category/∞-category
- (∞,n)-category
- (∞,2)-category
- (∞,1)-category
- (∞,0)-category/∞-groupoid
- n-category = (n,n)-category
- n-poset = (n-1,n)-category
- n-groupoid = (n,0)-category

- categorification/decategorification
- geometric definition of higher category
- algebraic definition of higher category
- stable homotopy theory

In a higher category, invertibility of n-morphisms in its highest dimension is always considered strictly. Thus in an ordinary category, we have 1-morphisms which can be isomorphisms. In a 2-category, it is the 2-morphisms for which, when it comes to invertibility, we always ask for a strict inverse, whereas for 1-morphisms we typically ask only for an equivalence. These 2-morphisms which admit an inverse are known as *2-isomorphisms*.

Let $\mathcal{A}$ be a 2-category. A *2-isomorphism* in $\mathcal{A}$ is a 2-arrow $\phi : f \rightarrow f'$ of $\mathcal{A}$ which admits a (strict) inverse, that is to say, there is a 2-arrow $\phi^{-1}: f' \rightarrow f$ of $\mathcal{A}$ such that $\phi^{-1} \circ \phi = id(f)$ and $\phi \circ \phi^{-1} = id\left(f' \right)$.

A 2-category in which every 2-arrow is a 2-isomorphism is known as a (2,1)-category.

Last revised on July 4, 2020 at 17:50:55. See the history of this page for a list of all contributions to it.