nLab (0,1)-category theory

Idea

In the context of higher category theory / (n,r)-categories, a poset is equivalently regarded as a (0,1)-category.

$(0,1)$-categories play a major role in logic, where their objects are interpreted as propositions, their morphisms as implications and limits/products and colimits/coproducts as logical conjunctions and and or, respectively.

Dually, $(0,1)$-categories play a major role in topology, where they are interpreted as categories of open subsets of a topological spaces, or, more generally, of locales.

Clearly, much of category theory simplifies drastically when restricted to $(0,1)$-categories, but it is often most useful to make the parallel explicit.

Related concepts

higher category theory

• (0,1)-category theory

• 1-category theory, (∞,1)-category theory

  enriched category theory

• 2-category theory, (∞,2)-category theory

• (∞,n)-category theory