nLab
0-category

Skip the Navigation Links | Home Page | All Pages | Recently Revised | Authors | Feeds | Export |

Context

Higher category theory

higher category theory

Basic concepts

Basic theorems

Applications

Models

Morphisms

Functors

Universal constructions

Extra properties and structure

1-categorical presentations

Contents

Definition

A 0-category (or (0,0)-category) is simply a set (or class).

This terminology may seem strange at first, it simply follows the logic of n-categories (and (n,r)-categories). To understand these, it is very helpful to see sets as the beginning of a sequence of concepts: sets, categories, 2-categories, 3-categories, etc. Doing so reveals patterns such as the periodic table; it also sheds light on the theory of homotopy groups and n-stuff.

For example, there should be a 1-category of 0-categories; this is the category of sets. Then a category enriched over this is a 1-category (more precisely, a locally small category). Furthermore, an enriched groupoid is a groupoid (or 1-groupoid), so a 0-category is the same as a 0-groupoid.

To some extent, one can continue to define a (−1)-category to be a truth value and a (−2)-category to be a triviality (that is, there is exactly one). These don't fit the pattern perfectly; but the concepts of (−1)-groupoid and (−2)-groupoid for them do work perfectly, as does the concept of 0-poset for a truth value.

Revised on November 10, 2010 09:50:29 by Urs Schreiber (87.212.203.135)
Edit | Back in time (7 revisions) | See changes | History | Views: Print | TeX | Source | Linked from: vertical categorification, higher category theory, omega-category, oriental, globe, globular set, groupoid cardinality, Simpson's conjecture, category, equivalence, (-1)-category, infinity-groupoid, semi-strict infinity-category, infinity-stack, Cat, double category, bicategory, geometric definition of higher categories, geometric shape for higher structures, omega-groupoid, crossed complex, Nonabelian Algebraic Topology, (infinity,1)-category, 0-category, (-2)-category, 0-groupoid, subobject classifier, strict 2-category, periodic table, (n,r)-category, infinity-category, n-category, Kan complex, n-groupoid, k-tuply monoidal n-category, Gray-category, 2-category, strict omega-category, Crans-Gray tensor product, strict omega-groupoid, algebraic definition of higher categories, n-fold category, crossed module, regular epimorphism, tricategory, (-1)-groupoid, 0-poset, generalized universal bundle, k-surjective functor, simplicial T-complex, strict n-category, A-infinity-category, 2-group, homotopy hypothesis, pseudofunctor, (infinity,n)-category, regular monomorphism, (infinity,n)-category of cobordisms, Trimble n-category, simplicial object in Cat, simplicial groupoid, string diagram, cat-n-group, 2-vector space, 2-crossed module, crossed square, double groupoid, Dold-Kan correspondence, delooping hypothesis, Pursuing Stacks, k-tuply monoidal (n,r)-category, 3-category, connection on a cubical set, n-fibration, (-2)-groupoid, 1-category, 2-groupoid, 2-poset, n-poset, 3-groupoid, T-complex, cubical T-complex, point, Higher Topos Theory, contractible space, (infinity,0)-category, (infinity,1)-topos, Godement product, Eckmann-Hilton argument, complicial set, weak complicial set, (infinity,1)-category of (infinity,1)-categories, strict 2-functor, microcosm principle, simplicial model for weak omega-categories, higher category theory and physics, negative thinking, (2,1)-category, (n,1)-category, replete subcategory, tetracategory, omega-nerve, computad, n-group, transfor, k-morphism, n-functor, (infinity,2)-category, n-fold complete Segal space, opetope, weak inverse, exchange law, essential image, Gray-groupoid, lax functor, decategorification, simplicial resolution, Batanin omega-category, center, differential crossed module, higher topos theory, (0,1)-category, graph of a function, cograph of a functor, higher category theory - contents, cellular set, order theory, Theta-space, Theta category, full image, nPOV, 0-dimensional TQFT, companion pair, smooth loop space, coherence theorem for monoidal categories, covering, Grothendieck-Maltsiniotis infinity-category, (infinity,n)-module, applications of (higher) category theory, 2Cat, Homotopy Theory of Higher Categories, semicategory, (infinity,1)-category theory, homotopy coherent diagram, hypergroupoid, coherence theorem, hypercrossed complex, vertical composition, globular operad, pasting diagram, associator, unitor, algebraic Kan complex, algebraic quasi-category, 2-category theory, automorphism 2-group, embedding, effective monomorphism, Duskin nerve, cosmic cube, symmetric monoidal (infinity,n)-category, Gray group, hom-groupoid, horizontal composition, holographic principle of higher category theory, infinity-anafunctor, 3-morphism, coherence law, associativity, 1-morphism, (1,1)-category, pentagonator, hexagonator, strict 2-groupoid, bigroupoid, 0-functor, nGrpd, (infinity,n)-presheaf, blob n-category, (infinity,n)Cat, integral transforms on sheaves, integral transform, (infinity,n)-category of spans, fully dualizable object, (infinity,2)-sheaf, natural equivalence, (infinity,n)-category with duals, looping, truncated object, contents of contents, outer automorphism infinity-group, (infinity,2)-topos, 0-morphism, model structure on cellular sets, parity complex, higher Segal space, Cartesian fibration of dendroidal sets, atom, (n,r)-congruence, Segal n-category, homotopy 2-category, (infinity,n)-sheaf, coherence theorem for symmetric monoidal categories, (infinity,n)-category with adjoints, TwoVect, coherence theorem for braided monoidal categories, coherence theorem for monoidal bicategories, coherence theorem for braided monoidal bicategories, coherence theorem for symmetric monoidal bicategories, (infinity,n)-topos, symmetric monoidal 2-category, globular theory, double Lie groupoid, Lie algebroid-groupoid