EXAMPLES AND EXERCISES IN BASIC CATEGORY THEORY 1
... 1.8. Groupoids. A groupoid is a category in which every morphism is invertible. The notion of a groupoid is very important in algebraic topology. In particular, in §1.9 below I will define the fundamental groupoid of a topological space, which provides the correct way of thinking about fundamental g ...
... 1.8. Groupoids. A groupoid is a category in which every morphism is invertible. The notion of a groupoid is very important in algebraic topology. In particular, in §1.9 below I will define the fundamental groupoid of a topological space, which provides the correct way of thinking about fundamental g ...
NATURAL TRANSFORMATIONS Id −→ Id Here is a categorical way
... It is natural to ask whether or not ζ and ε are inverse isomorphisms. If we start with a natural transformation η, then (ζ ◦ ε)(η)X = λ ◦ (ηI idX ) ◦ λ−1 , and the question is whether or not that coincides with ηIX . To see that that is not true in general, consider the category Ab, regarded as a ...
... It is natural to ask whether or not ζ and ε are inverse isomorphisms. If we start with a natural transformation η, then (ζ ◦ ε)(η)X = λ ◦ (ηI idX ) ◦ λ−1 , and the question is whether or not that coincides with ηIX . To see that that is not true in general, consider the category Ab, regarded as a ...
Compact Categories as †-Frobenius Pseudoalgebras
... Frobenius pseudoalgebra with extra structure provide a finite presentation of exactly the correct kind, and there are compelling reasons to believe that they should at least be close to the correct axioms for constructing the 2-category 3Cob2 , which has circles as objects, 2-manifolds with boundary ...
... Frobenius pseudoalgebra with extra structure provide a finite presentation of exactly the correct kind, and there are compelling reasons to believe that they should at least be close to the correct axioms for constructing the 2-category 3Cob2 , which has circles as objects, 2-manifolds with boundary ...
Topology Qual Winter 2000
... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. a) Let G and H be functors from a category C to a category D. Define a natural transformation from G to H. b) For an admissible pair of topological spaces (X,A) define functors G ...
... Answer eight questions, four from part I and four from part II. Give as much detail in your answers as you can. Part I 1. a) Let G and H be functors from a category C to a category D. Define a natural transformation from G to H. b) For an admissible pair of topological spaces (X,A) define functors G ...
a small observation on co-categories
... 0.6. Definition. A coherent category is a category with all finite limits, and images and unions that are stable under pullback. [Johnstone 2002, A1.3–4] gives various basic results on coherent categories, which we will use here without comment. 0.7. Definition. Coherent logic is the fragment of fir ...
... 0.6. Definition. A coherent category is a category with all finite limits, and images and unions that are stable under pullback. [Johnstone 2002, A1.3–4] gives various basic results on coherent categories, which we will use here without comment. 0.7. Definition. Coherent logic is the fragment of fir ...
Sets with a Category Action Peter Webb 1. C-sets
... which has two objects x and y, a single morphism α from x to y, and the identity morphisms at x and y. We readily see that the transitive (non-empty) C-sets have the form Ωn := n → 1, n ≥ 0 where n = {1, . . . , n} is a set with n elements, the mapping between the two sets sending every element onto ...
... which has two objects x and y, a single morphism α from x to y, and the identity morphisms at x and y. We readily see that the transitive (non-empty) C-sets have the form Ωn := n → 1, n ≥ 0 where n = {1, . . . , n} is a set with n elements, the mapping between the two sets sending every element onto ...
Categories and functors
... is a ring with multiplication given by composition of morphisms. Furthermore, each morphism set HomC (X, Y ) has the structure of an End(Y )End(X)-bimodule with the action of these two rings given by composition. Example 22.3. The category Ab = Z-M od of all additive groups and homomorphisms is a pr ...
... is a ring with multiplication given by composition of morphisms. Furthermore, each morphism set HomC (X, Y ) has the structure of an End(Y )End(X)-bimodule with the action of these two rings given by composition. Example 22.3. The category Ab = Z-M od of all additive groups and homomorphisms is a pr ...
(8 pp Preprint)
... Motivated by the desire to better understand modular tensor categories (or whatever replaces them) in the context of (rational) super conformal 2-dimensional field theories, the following is an attempt to capture the basic axioms of superalgebra in a more “arrow-theoretic” way than commonly done, su ...
... Motivated by the desire to better understand modular tensor categories (or whatever replaces them) in the context of (rational) super conformal 2-dimensional field theories, the following is an attempt to capture the basic axioms of superalgebra in a more “arrow-theoretic” way than commonly done, su ...
Category Theory Example Sheet 1
... (a) Show that the inner automorphisms of C form a normal subgroup of the group of all automorphisms of C. [Don’t worry about whether these groups are sets or proper classes.] (b) If F is an automorphism of a category C with a terminal object 1, show that F (1) is also a terminal object of C (and hen ...
... (a) Show that the inner automorphisms of C form a normal subgroup of the group of all automorphisms of C. [Don’t worry about whether these groups are sets or proper classes.] (b) If F is an automorphism of a category C with a terminal object 1, show that F (1) is also a terminal object of C (and hen ...
Functors and natural transformations A covariant functor F : C → D is
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
... For W a topological space, let [B, W ] be the set of homotopy classes of maps f : B → W . Theorem For each topological group G, there is a space BG called the classifying space of G and an isomorphism of functors between FG and the functor B → [B, BG]. The homotopy class of the identity map BG → BG ...
$doc.title
... •A parallel fragmenta-on naturally occurred in pure math (e.g., algebra, geometry, logic) •Category theory invented as a unifying formalism for pure math. • Category theory is used as a formal underpinni ...
... •A parallel fragmenta-on naturally occurred in pure math (e.g., algebra, geometry, logic) •Category theory invented as a unifying formalism for pure math. • Category theory is used as a formal underpinni ...
Category Theory Example Sheet 1
... 3. Let G be a group viewed as a one-object category. Show that the nat. transformations α : 1G −→ 1G correspond to elements in the centre of the group. 4. A morphism e : A −→ A is called idempotent if ee = e. An idempotent e is said to split if it can be factored as f g where gf is an identity morph ...
... 3. Let G be a group viewed as a one-object category. Show that the nat. transformations α : 1G −→ 1G correspond to elements in the centre of the group. 4. A morphism e : A −→ A is called idempotent if ee = e. An idempotent e is said to split if it can be factored as f g where gf is an identity morph ...
MATH 6280 - CLASS 2 Contents 1. Categories 1 2. Functors 2 3
... Definition 1.1. A category is • a collection of objects obj(C) • for any two objects X, Y ∈ C, a set of morphisms C(X, Y ) or HomC (X, Y ) • For every object X ∈ C, an identity morphism idX = 1X ∈ C(X, X) • For any X, Y, Z ∈ C composition law: ◦ : C(Y, Z) × C(X, Y ) → C(X, Z) that satisfy the follow ...
... Definition 1.1. A category is • a collection of objects obj(C) • for any two objects X, Y ∈ C, a set of morphisms C(X, Y ) or HomC (X, Y ) • For every object X ∈ C, an identity morphism idX = 1X ∈ C(X, X) • For any X, Y, Z ∈ C composition law: ◦ : C(Y, Z) × C(X, Y ) → C(X, Z) that satisfy the follow ...
PDF
... Two instances of this construction are worth noting. If G is a group, we may regard G as a category with one object. Then this construction gives us the group algebra of G. If P is a partially ordered set, we may view P as a category with at most one morphism between any two objects. Then this const ...
... Two instances of this construction are worth noting. If G is a group, we may regard G as a category with one object. Then this construction gives us the group algebra of G. If P is a partially ordered set, we may view P as a category with at most one morphism between any two objects. Then this const ...
An Introduction to Categories.
... We saw that π0 : Top → Set maps a topological space to its set of path components. And, continuous maps f : X → Y induce a map between path components π0 (f ) such that π0 (f g) = π0 (f )π0 (g). Similarly, π1 : Top. → Grp maps a pointed topological space to its fundamental group. We’ve seen that con ...
... We saw that π0 : Top → Set maps a topological space to its set of path components. And, continuous maps f : X → Y induce a map between path components π0 (f ) such that π0 (f g) = π0 (f )π0 (g). Similarly, π1 : Top. → Grp maps a pointed topological space to its fundamental group. We’ve seen that con ...
Category theory
Category theory formalizes mathematical structure and its concepts in terms of a collection of objects and of arrows (also called morphisms). A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. Category theory can be used to formalize concepts of other high-level abstractions such as sets, rings, and groups. Several terms used in category theory, including the term ""morphism"", are used differently from their uses in the rest of mathematics. In category theory, a ""morphism"" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made.