Glossary of category theory

Template:Short description Template:Use American English This is a glossary of properties and concepts in category theory in mathematics. (see also Outline of category theory.)

• Notes on foundations: In many expositions (e.g., Vistoli), the set-theoretic issues are ignored; this means, for instance, that one does not distinguish between small and large categories and that one can arbitrarily form a localization of a category. Like those expositions, this glossary also generally ignores the set-theoretic issues, except when they are relevant (e.g., the discussion on accessibility.)

Especially for higher categories, the concepts from algebraic topology are also used in the category theory. For that see also glossary of algebraic topology.

The notations and the conventions used throughout the article are:

• [n] = {0, 1, 2, …, n}, which is viewed as a category (by writing $i\to j\iff i\le j$.)
• Cat, the category of (small) categories, where the objects are categories (which are small with respect to some universe) and the morphisms functors.
• Fct(C, D), the functor category: the category of functors from a category C to a category D.
• Set, the category of (small) sets.
• sSet, the category of simplicial sets.
• "weak" instead of "strict" is given the default status; e.g., "n-category" means "weak n-category", not the strict one, by default.
• By an ∞-category, we mean a quasi-category, the most popular model, unless other models are being discussed.
• The number zero 0 is a natural number.

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<dt id="" >2-category Template:Defn Template:Defn

A

<dt id="" >abelian Template:Defn <dt id="" >accessible Template:Defn Template:Defn <dt id="" >additive Template:Defn <dt id="" >adjunction Template:Defn <dt id="" >algebra for a monad Template:Defn <dt id="" >amnestic Template:Defn <dt id="" >anodyne Template:Defn

B

<dt id="" >balanced Template:Defn <dt id="" >Beck's theorem Template:Defn <dt id="" >bicategory Template:Defn <dt id="" >bifunctor Template:Defn <dt id="" >bimonoidal Template:Defn <dt id="" >bimorphism Template:Defn <dt id="" >Bousfield localization Template:Defn

C

<dt id="" >calculus of functors Template:Defn <dt id="" >calculus of fractions Template:Defn <dt id="" >cartesian closed Template:Defn <dt id="" >cartesian functor Template:Defn <dt id="" >cartesian morphism Template:Defn Template:Defn <dt id="" >Cartesian square Template:Defn <dt id="" >categorical logic Template:Defn <dt id="" >categorical probability Template:Defn <dt id="" >categorification Template:Defn

The theory of categories originated ... with the need to guide complicated calculations involving passage to the limit in the study of the qualitative leap from spaces to homotopical/homological objects. ... But category theory does not rest content with mere classification in the spirit of Wolffian metaphysics (although a few of its practitioners may do so); rather it is the mutability of mathematically precise structures (by morphisms) which is the essential content of category theory.

William Lawvere,

<dt id="" >category Template:Defn <dt id="" >category of Template:Defn Template:Defn <dt id="" >classifying space Template:Defn <dt id="" >co- Template:Defn <dt id="" >codensity monad Template:Defn <dt id="" >coend Template:Defn <dt id="" >coequalizer Template:Defn <dt id="" >coherator Template:Defn <dt id="" >coherence theorem Template:Defn <dt id="" >coherent Template:Defn Template:Defn <dt id="" >cohesive Template:Defn <dt id="" >coimage Template:Defn <dt id="" >colored operad Template:Defn <dt id="" >comma Template:Defn <dt id="" >comonad Template:Defn <dt id="" >compact Template:Defn <dt id="" >complete Template:Defn <dt id="" >completeness Template:Defn <dt id="" >composition Template:Defn Template:Defn Template:Defn <dt id="" >computad Template:Defn <dt id="" >concrete Template:Defn <dt id="" >cone Template:Defn <dt id="" >connected Template:Defn <dt id="" >conservative functor Template:Defn <dt id="" >constant Template:Defn <dt id="" >contravariant functor Template:Defn <dt id="" >coproduct Template:Defn <dt id="" >core Template:Defn <dt id="" >cubical Template:Defn

D

<dt id="" >Day convolution Template:Defn <dt id="" >Dendroidal Template:Defn <dt id="" >density theorem Template:Defn <dt id="" >diagonal functor Template:Defn <dt id="" >diagram Template:Defn Template:Defn <dt id="" >diagrammatic set Template:Defn <dt id="" >differential graded category Template:Defn <dt id="" >direct limit Template:Defn <dt id="" >discrete Template:Defn <dt id="" >distributor Template:Defn <dt id="" >double Template:Defn <dt id="" >Dwyer–Kan equivalence Template:Defn

E

<dt id="" >Elementary Theory of the Category of Sets Template:Defn <dt id="" >Eilenberg–Moore category Template:Defn <dt id="" >Eilenberg–Zilber category Template:Defn <dt id="" >empty Template:Defn <dt id="" >end Template:Defn <dt id="" >endofunctor Template:Defn <dt id="" >enriched category Template:Defn <dt id="" >epimorphism Template:Defn <dt id="" >equalizer Template:Defn <dt id="" >equivalence Template:Defn Template:Defn <dt id="" >equivalent Template:Defn <dt id="" >essentially surjective Template:Defn <dt id="" >evaluation Template:Defn <dt id="" >exact Template:Defn Template:Defn Template:Defn Template:Defn <dt id="" >exit Template:Defn

F

<dt id="" >faithful Template:Defn <dt id="" >fundamental category Template:Defn <dt id="" >fundamental groupoid Template:Defn <dt id="" >fibered category Template:Defn <dt id="" >fiber product Template:Defn <dt id="" >fibrant Template:Defn <dt id="" >filtered Template:Defn Template:Defn <dt id="" >final Template:Defn <dt id="" >finitary monad Template:Defn <dt id="" >finite Template:Defn <dt id="" >forgetful functor Template:Defn <dt id="" >free category Template:Defn <dt id="" >free completion Template:Defn <dt id="" >free functor Template:Defn <dt id="" >Frobenius category Template:Defn <dt id="" >Fukaya category Template:Defn <dt id="" >full Template:Defn Template:Defn <dt id="" >functor Template:Defn <dt id="" >functor category Template:Defn

G

<dt id="" >Gabriel–Popescu theorem Template:Defn <dt id="" >Galois category Template:Defn Template:Defn <dt id="" >generator Template:Defn <dt id="" >generalized Template:Defn <dt id="" >Gray Template:Defn Template:Defn <dt id="" >gros topos Template:Defn <dt id="" >Grothendieck's Galois theory Template:Defn <dt id="" >Grothendieck category Template:Defn <dt id="" >Grothendieck construction Template:Defn <dt id="" >Grothendieck fibration Template:Defn <dt id="" >groupoid Template:Defn Template:Defn

H

<dt id="" >Hall algebra of a category Template:Defn <dt id="" >heart Template:Defn <dt id="" >higher category theory Template:Defn <dt id="" >higher stack Template:Defn <dt id="" >homological dimension Template:Defn <dt id="" >homotopy category Template:Defn <dt id="" >homotopy colimit Template:Defn <dt id="" >homotopy hypothesis Template:Defn

I

<dt id="" >idempotent Template:Defn <dt id="" >identity Template:Defn Template:Defn Template:Defn <dt id="" >image Template:Defn <dt id="" >ind-limit Template:Defn <dt id="" >inductive limit Template:Defn <dt id="" >∞-category Template:Defn <dt id="" >(∞, n)-category Template:Defn <dt id="" >∞-sheaf Template:Defn <dt id="" >initial Template:Defn Template:Defn <dt id="" >injective Template:Defn Template:Defn <dt id="" >internal Hom Template:Defn <dt id="" >inverse Template:Defn Template:Defn <dt id="" >Isbell Template:Defn Template:Defn Template:Defn <dt id="" >isomorphic Template:Defn Template:Defn <dt id="" >isomorphism Template:Defn

K

<dt id="" >Kan Template:Defn Template:Defn Template:Defn <dt id="" >Kan extension Template:Defn Template:Defn <dt id="" >Ken Brown's lemma Template:Defn <dt id="" >Kleisli category Template:Defn

L

<dt id="" >lax Template:Defn <dt id="" >length Template:Defn <dt id="" >limit Template:Defn Template:Defn Template:Defn <dt id="" >localization of a category Template:Defn

M

<dt id="" >Mittag-Leffler condition Template:Defn <dt id="" >modification Template:Defn <dt id="" >modular Template:Defn <dt id="" >monad Template:Defn <dt id="" >monadic Template:Defn Template:Defn <dt id="" >monoidal category Template:Defn <dt id="" >monoid object Template:Defn <dt id="" >monomorphism Template:Defn <dt id="" >multicategory Template:Defn

N

<dt id="" >n-category

[T]he issue of comparing definitions of weak n-category is a slippery one, as it is hard to say what it even means for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak n-categories and the functors, transformations, ... between them should be a weak (n + 1)-category; and if this is the case then the question is whether your weak (n + 1)-category of weak n-categories is equivalent to mine—but whose definition of weak (n + 1)-category are we using here... ?

Tom Leinster, A survey of definitions of n-category

Template:Defn <dt id="" >natural Template:Defn Template:Defn

File:Nerve-2-simplex.png

The composition is encoded as a 2-simplex.

<dt id="" >nerve Template:Defn Template:Defn <dt id="" >normal Template:Defn

O

<dt id="" >object Template:Defn Template:Defn <dt id="" >op-fibration Template:Defn <dt id="" >opposite Template:Defn

P

<dt id="" >perfect Template:Defn <dt id="" >pointed Template:Defn <dt id="" >polygraph Template:Defn <dt id="" >polynomial Template:Defn <dt id="" >pre-abelian Template:Defn <dt id="" >preadditive Template:Defn <dt id="" >presentable Template:Defn Template:Defn <dt id="" >presheaf Template:Defn <dt id="" >product Template:Defn Template:Defn <dt id="" >profunctor Template:Defn <dt id="" >projective Template:Defn Template:Defn <dt id="" >PROP Template:Defn <dt id="" >pseudoalgebra Template:Defn <dt id="" >pseudo-abelian envelope Template:Defn

Q

<dt id="" >Q Template:Defn <dt id="" >Quillen Template:Defn <dt id="" >quasi-abelian Template:Defn <dt id="" >quasitopos Template:Defn

R

<dt id="" >Reedy Template:Defn <dt id="" >reflect Template:Defn Template:Defn <dt id="" >regular Template:Defn <dt id="" >representable Template:Defn <dt id="" >retraction Template:Defn <dt id="" >rig Template:Defn

S

<dt id="" >section Template:Defn <dt id="" >Segal Template:Defn Template:Defn <dt id="" >semi-abelian Template:Defn <dt id="" >semisimple Template:Defn <dt id="" >Serre functor Template:Defn <dt id="" >sieve Template:Defn <dt id="" >simple object Template:Defn <dt id="" >simplex category Template:Defn <dt id="" >simplicial category Template:Defn <dt id="" >Simplicial localization Template:Defn <dt id="" >simplicial object Template:Defn <dt id="" >Simpson Template:Defn <dt id="" >simplicial set Template:Defn <dt id="" >site Template:Defn <dt id="" >skeletal Template:Defn Template:Defn <dt id="" >slice Template:Defn <dt id="" >small Template:Defn Template:Defn <dt id="" >species Template:Defn <dt id="" >stable Template:Defn <dt id="" >strict Template:Defn Template:Defn Template:Defn <dt id="" >strict n-category Template:Defn <dt id="" >strictification Template:Defn <dt id="" >subcanonical Template:Defn <dt id="" >subcategory Template:Defn <dt id="" >subobject Template:Defn <dt id="" >subquotient Template:Defn Template:Defn <dt id="" >subterminal object Template:Defn <dt id="" >symmetric monoidal category Template:Defn <dt id="" >symmetric sequence Template:Defn <dt id="" >syntactic category Template:Defn

T

<dt id="" >t-structure Template:Defn <dt id="" >Tannakian duality Template:Defn <dt id="" >tensor category Template:Defn <dt id="" >tensor triangulated category Template:Defn <dt id="" >tensor product Template:Defn <dt id="" >terminal Template:Defn Template:Defn <dt id="" >Theta category Template:Defn <dt id="" >thick subcategory Template:Defn <dt id="" >thin Template:Defn <dt id="" >tiny Template:Defn <dt id="" >topological topos Template:Defn <dt id="" >transfinite Template:Defn <dt id="" >triangulated category Template:Defn

U

<dt id="" >universal Template:Defn Template:Defn

V

<dt id="" >Verdier localization Template:Defn

W

<dt id="" >Waldhausen category Template:Defn <dt id="" >wellpowered Template:Defn

Y

<dt id="" >Yoneda Template:Defn Template:Defn

Z

<dt id="" >zero Template:Defn

References

• A. Joyal, The theory of quasi-categories II (Volume I is missing??)
• Lurie, J., Higher Algebra
• Lurie, J., Higher Topos Theory

• Pierre Deligne, Voevodsky’s lectures on cross functors, Unpublished, available at https://www.math.ias.edu/vladimir/node/94 (2001).
• J. Lurie, Kerodon, https://kerodon.net

Further reading

• Groth, M., A Short Course on ∞-categories
• Cisinski's notes
• History of topos theory
• Leinster, Higher Operads, Higher Categories, 2003.
• Emily Riehl, A leisurely introduction to simplicial sets
• Categorical Logic lecture notes by Steve Awodey
• (a detailed discussion of a 2-category)
• Lawvere, Categories of spaces may not be generalized spaces as exemplified by directed graphs
• Category Theory in Stanford Encyclopedia of Philosophy