This is a glossary of properties and concepts in

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

mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...

. (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, a cardinal number is a strongly inaccessible cardinal if it is uncountable, regular, and a strong limit cardinal. A cardinal is a weakly inaccessible cardinal if it is uncountable, regular, and a weak limit cardinal. Since abou ...

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'' ...

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 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 manifo ...

. 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 Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...

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 i ...

: 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 a sequence of sets with internal order structure ( abstract simplices) and maps between them. Simplicial sets are higher-dimensional generalizations of directed graphs. Every simplicial set gives rise to a "n ...

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 (ma ...

, 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. T ...

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

zero 0 (zero) is a number representing an empty quantity. Adding (or subtracting) 0 to any number leaves that number unchanged; in mathematical terminology, 0 is the additive identity of the integers, rational numbers, real numbers, and compl ...

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 ...

'' * * * * 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 *

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