Glossary Of Category Theory
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This is a glossary of properties and concepts in category theory 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 ''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 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. 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, an ...
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
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, an ...
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, we mean a quasi-category, 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, usu ...
0 is a natural number.


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Notes


References

* * * * 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 ...
'' * * *


Further reading

* Groth, M.
A Short Course on ∞-categories

Cisinski's notes
*
History of topos theory This article gives some very general background to the mathematical idea of topos. This is an aspect of category theory, and has a reputation for being abstruse. The level of abstraction involved cannot be reduced beyond a certain point; but on the ...
*http://plato.stanford.edu/entries/category-theory/ * *Emily Riehl
A leisurely introduction to simplicial sets

Categorical Logic
lecture notes by
Steve Awodey Steven M. Awodey (; born 1959) is an American mathematician and logician. He is a Professor of Philosophy and Mathematics at Carnegie Mellon University. Biography Awodey studied mathematics and philosophy at the University of Marburg and the ...
* (a detailed discussion of a 2-category) {{DEFAULTSORT:Glossary Of Category Theory * Category theory Wikipedia glossaries using description lists