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. 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.

C

The theory of categories originated ... with the need to guide complicated calculations involving passage to the limit in the study of the qualitative leap from spaces to homotopical/homological objects. ... But category theory does not rest content with mere classification in the spirit of Wolffian metaphysics (although a few of its practitioners may do so); rather it is the mutability of mathematically precise structures (by morphisms) which is the essential content of category theory.

William Lawvere,

N

[T]he issue of comparing definitions of weak n-category is a slippery one, as it is hard to say what it even means for two such definitions to be equivalent. [...] It is widely held that the structure formed by weak n-categories and the functors, transformations, ... between them should be a weak (n + 1)-category; and if this is the case then the question is whether your weak (n + 1)-category of weak n-categories is equivalent to mine—but whose definition of weak (n + 1)-category are we using here... ?

Tom Leinster, A survey of definitions of n-category

File:Nerve-2-simplex.png

The composition is encoded as a 2-simplex.

References

• A. Joyal, The theory of quasi-categories II (Volume I is missing??)
• Lurie, J., Higher Algebra
• Lurie, J., Higher Topos 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

Further reading

• Groth, M., A Short Course on ∞-categories Template:Webarchive
• Cisinski's notes
• History of topos theory
• Leinster, Higher Operads, Higher Categories, 2003.
• Emily Riehl, A leisurely introduction to simplicial sets
• Categorical Logic lecture notes by Steve Awodey
• (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