Glossary of category theory
Template:Short description Template:Use American English This is a glossary of properties and concepts in category theory in mathematics. (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.[1] 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 .)
- 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.
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Notes
References
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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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- Template:Cite arXiv
Further reading
- Groth, M., A Short Course on ∞-categories Template:Webarchive
- Cisinski's notes
- History of topos theory
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- Leinster, Higher Operads, Higher Categories, 2003.
- Emily Riehl, A leisurely introduction to simplicial sets
- Categorical Logic 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
- Category Theory in Stanford Encyclopedia of Philosophy
- ↑ If one believes in the existence of strongly inaccessible cardinals, then there can be a rigorous theory where statements and constructions have references to Grothendieck universes.