This is a glossary of properties and concepts in

category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cat ...

in mathematics. (see also

Outline of category theory The following outline is provided as an overview of and guide to category theory, the area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of ''objec ...

.) *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.If one believes in the existence of

strongly inaccessible cardinal In set theory, an uncountable cardinal is inaccessible if it cannot be obtained from smaller cardinals by the usual operations of cardinal arithmetic. More precisely, a cardinal is strongly inaccessible if it is uncountable, it is not a sum of f ...

s, then there can be a rigorous theory where statements and constructions have references to

Grothendieck universe In mathematics, a Grothendieck universe is a set ''U'' with the following properties: # If ''x'' is an element of ''U'' and if ''y'' is an element of ''x'', then ''y'' is also an element of ''U''. (''U'' is a transitive set.) # If ''x'' and ''y'' a ...

s. 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''= , which is viewed as a category (by writing

i \to j \Leftrightarrow 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

functor In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and ma ...

s. *Fct(''C'', ''D''), the

functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in ...

: the category of

s from a category ''C'' to a category ''D''. *Set, the category of (small) sets. *''s''Set, the category of

simplicial set In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined ...

s. *"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 In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...

, we mean a

quasi-category In mathematics, more specifically category theory, a quasi-category (also called quasicategory, weak Kan complex, inner Kan complex, infinity category, ∞-category, Boardman complex, quategory) is a generalization of the notion of a category. Th ...

, the most popular model, unless other models are being discussed. *The number

zero 0 (zero) is a number representing an empty quantity. In place-value notation such as the Hindu–Arabic numeral system, 0 also serves as a placeholder numerical digit, which works by multiplying digits to the left of 0 by the radix, usuall ...

0 is a natural number.

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

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