Glossary of category theory

Template:Short description Template:Use American English This is a glossary of properties and concepts in category theory in mathematics, including those in topos theory. (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.

N

The composition is encoded as a 2-simplex.

References

Basic category theory

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• (SGA4-1) Théorie des topos et cohomologie étale des schémas. Tome 1: Théorie des topos, Séminaire de Géométrie Algébrique du Bois Marie 1963–1964 (SGA 4). Dirigé par M. Artin, A. Grothendieck, et J.-L. Verdier. Avec la collaboration de N. Bourbaki, P. Deligne et B. Saint-Donat. Lecture Notes in Mathematics 269, Springer 1972.
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• Template:Cite arXiv

Higher category theory

• A. Joyal, The theory of quasi-categories II Template:Webarchive (Volume I is missing??)
• Lurie, J., Higher Algebra
• Lurie, J., Higher Topos Theory
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• Pierre Deligne, Voevodsky’s lectures on cross functors, Unpublished, available at https://www.math.ias.edu/vladimir/node/94 Template:Webarchive (2001).
• J. Lurie, Kerodon, https://kerodon.net Template:Webarchive
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Further reading

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