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, cate ...

mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern 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 ''object ...

.) *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 universes. 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 This is a glossary of properties and concepts in algebraic topology in mathematics. See also: glossary of topology, list of algebraic topology topics, glossary of category theory, glossary of differential geometry and topology, Timeline of mani ...

. 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 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. *''s''Set, 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 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 Positional notation (or place-value notation, or positional numeral system) usually denotes the extension to any base of the Hindu–Arabic numeral system (or ...

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 ''Higher Topos Theory'' is a treatise on the theory of ∞-categories written by American mathematician Jacob Lurie. In addition to introducing Lurie's new theory of ∞-topoi, the book is widely considered foundational to higher category theory ...

'' * * *

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