Английская Википедия:Glossary of category theory

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Шаблон:Short description Шаблон: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 <math>i \to j \Leftrightarrow i \le j</math>.)
• 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

Шаблон:Term Шаблон:Defn \times C \to X</math> is the dual of the end of F and is denoted by

<math>\int^{c \in C} F(c, c)</math>.

For example, if R is a ring, M a right R-module and N a left R-module, then the tensor product of M and N is

<math>M \otimes_R N = \int^{R} M \otimes_{\mathbb{Z}} N</math>

where R is viewed as a category with one object whose morphisms are the elements of R.}}

E

Шаблон:Term Шаблон:Defn \times C \to X</math> is the limit

<math>\int_{c \in C} F(c, c) = \varprojlim (F^{\#}: C^{\#} \to X)</math>

where <math>C^{\#}</math> is the category (called the subdivision category of C) whose objects are symbols <math>c^{\#}, u^{\#}</math> for all objects c and all morphisms u in C and whose morphisms are <math>b^{\#} \to u^{\#}</math> and <math>u^{\#} \to c^{\#}</math> if <math>u: b \to c</math> and where <math>F^{\#}</math> is induced by F so that <math>c^{\#}</math> would go to <math>F(c, c)</math> and <math>u^{\#}, u: b \to c</math> would go to <math>F(b, c)</math>. For example, for functors <math>F, G: C \to X</math>,

<math>\int_{c \in C} \operatorname{Hom}(F(c), G(c))</math>

is the set of natural transformations from F to G. For more examples, see this mathoverflow thread. The dual of an end is a coend.}}

I

Шаблон:Term Шаблон:Defn, \mathbf{Set})</math>.}}

Шаблон:Term Шаблон:Defn \times C \to C</math> such that <math>[Y, -]</math> is the right adjoint to <math>- \otimes Y</math> for each object Y in C. For example, the category of modules over a commutative ring R has the internal Hom given as <math>[M, N] = \operatorname{Hom}_R(M, N)</math>, the set of R-linear maps.}}

L

Шаблон:Term Шаблон:Defn \to \mathbf{Set}</math> is

<math>\varprojlim_{i \in I} f(i) = \{ (x_i|i) \in \prod_{i} f(i) | f(s)(x_j) = x_i \text{ for any } s: i \to j \}.</math>}}

Шаблон:Defn \to C</math> is an object, if any, in C that satisfies: for any object X in C, <math>\operatorname{Hom}(X, \varprojlim_{i \in I} f(i)) = \varprojlim_{i \in I} \operatorname{Hom}(X, f(i))</math>; i.e., it is an object representing the functor <math>X \mapsto \varprojlim_i \operatorname{Hom}(X, f(i)).</math>}} Шаблон:Defn

N

Файл:Nerve-2-simplex.png

The composition is encoded as a 2-simplex.

Шаблон:Term Шаблон:Defn([n], C)</math>. For example, if <math>\varphi</math> is a functor in <math>N(C)_2</math> (called a 2-simplex), let <math>x_i = \varphi(i), \, 0 \le i \le 2</math>. Then <math>\varphi(0 \to 1)</math> is a morphism <math>f: x_0 \to x_1</math> in C and also <math>\varphi(1 \to 2) = g: x_1 \to x_2</math> for some g in C. Since <math>0 \to 2</math> is <math>0 \to 1</math> followed by <math>1 \to 2</math> and since <math>\varphi</math> is a functor, <math>\varphi(0 \to 2) = g \circ f</math>. In other words, <math>\varphi</math> encodes f, g and their compositions.}}

P

Шаблон:Term Шаблон:Defn Шаблон:Defn(X_i, Y_i)</math>; the morphisms are composed component-wise. It is the dual of the disjoint union.}}

Шаблон:Term Шаблон:Defn \times C \to \mathbf{Set}</math>.}}

R

Шаблон:Term Шаблон:Defn, \mathbf{Set})</math>; i.e., <math>F \simeq \operatorname{Hom}_C(-, Z)</math> for some object Z. The object Z is said to be the representing object of F.}}

T

Шаблон:Term Шаблон:Defn \to B</math> and <math>G: C \to B</math> is the coend:

<math>F \otimes_C G = \int^{c \in C} F(c) \otimes G(c).</math>}}

Y

Шаблон:Glossary Шаблон:Term Шаблон:Defn, \mathbf{Set}), \, X \mapsto \operatorname{Hom}_C(-, X)</math> is fully faithful and is called the Yoneda embedding. }} Шаблон:Defn Шаблон:Glossary end

References

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

Further reading

• Groth, M., A Short Course on ∞-categories
• Cisinski's notes
• History of topos theory
• http://plato.stanford.edu/entries/category-theory/
• Шаблон:Cite book
• Emily Riehl, A leisurely introduction to simplicial sets
• Categorical Logic lecture notes by Steve Awodey
• Шаблон:Cite arXiv (a detailed discussion of a 2-category)