∗-AUTONOMOUS CATEGORIES: ONCE MORE
... monoidal categories. All but one are categories of models of a commutative theory and get their closed monoidal structure from that (see 3.7 below). The theory of Banach balls is really different from first six and is treated in detail in [Barr, Kleisli, to appear]. What we do here is provide a unif ...
... monoidal categories. All but one are categories of models of a commutative theory and get their closed monoidal structure from that (see 3.7 below). The theory of Banach balls is really different from first six and is treated in detail in [Barr, Kleisli, to appear]. What we do here is provide a unif ...
Finitely generated abelian groups, abelian categories
... The category A of abelian groups has some special properties. Namely, for every pair of objects C, D of A, the morphisms M orA(C, D) = HomA(C, D) are an abelian group. Furthermore, each f ∈ HomA(C, D) has a kernel and a cokernel, which are objects of A. Each such f can be factored as a composition o ...
... The category A of abelian groups has some special properties. Namely, for every pair of objects C, D of A, the morphisms M orA(C, D) = HomA(C, D) are an abelian group. Furthermore, each f ∈ HomA(C, D) has a kernel and a cokernel, which are objects of A. Each such f can be factored as a composition o ...
The separated extensional Chu category.
... of extensional objects by Chue = Chue (A, ⊥). We follow Pratt in denoting the full subcategory of objects that are both separated and extensional by chu = chu(A, ⊥). Since Chus is evidently the dual of Chue , it is immediate that chu is self-dual. It is useful to ask if chu is also ∗-autonomous. In ...
... of extensional objects by Chue = Chue (A, ⊥). We follow Pratt in denoting the full subcategory of objects that are both separated and extensional by chu = chu(A, ⊥). Since Chus is evidently the dual of Chue , it is immediate that chu is self-dual. It is useful to ask if chu is also ∗-autonomous. In ...
Contents - Harvard Mathematics Department
... The standard examples of categories are the ones above: structured sets together with structurepreserving maps. Nonetheless, one can easily give other examples that are not of this form. Example 1.4 (Groups as categories) Let G be a finite group. Then we can make a category BG where the objects just ...
... The standard examples of categories are the ones above: structured sets together with structurepreserving maps. Nonetheless, one can easily give other examples that are not of this form. Example 1.4 (Groups as categories) Let G be a finite group. Then we can make a category BG where the objects just ...
Lecture 11
... objects, indexed by Z, together with coboundary maps di : Ai −→ Ai+1 , such that the composition of any two is zero. The ith cohomology of the complex, is obtained in the usual way: hi (A• ) = Ker di / Im di+1 . A morphism of complexes f : A• −→ B • is simply a collection of morphisms f i : Ai −→ B ...
... objects, indexed by Z, together with coboundary maps di : Ai −→ Ai+1 , such that the composition of any two is zero. The ith cohomology of the complex, is obtained in the usual way: hi (A• ) = Ker di / Im di+1 . A morphism of complexes f : A• −→ B • is simply a collection of morphisms f i : Ai −→ B ...
Categories
... (ii) Dual: all retractions are epic; (iii) The composition of two sections is a section; (iv) Dual: the composition of two retractions is a retraction; (v) If fg is a section then f is a section; (vi) Dual: if fg is a retraction then g is a retraction; (vii) All isomorphisms are bijic; (viii) Compos ...
... (ii) Dual: all retractions are epic; (iii) The composition of two sections is a section; (iv) Dual: the composition of two retractions is a retraction; (v) If fg is a section then f is a section; (vi) Dual: if fg is a retraction then g is a retraction; (vii) All isomorphisms are bijic; (viii) Compos ...
The universal extension Let R be a unitary ring. We consider
... Considering the base change of the above diagram to R0 we obtain therefore the universal extension over R0 . Q.E.D. Let us consider a surjection of rings π : S → R with kernel a. We assume that p is nilpotent in S and that a is endowed with divided powers γ, but we do not assume that the divided pow ...
... Considering the base change of the above diagram to R0 we obtain therefore the universal extension over R0 . Q.E.D. Let us consider a surjection of rings π : S → R with kernel a. We assume that p is nilpotent in S and that a is endowed with divided powers γ, but we do not assume that the divided pow ...
Locally cartesian closed categories and type theory
... given by the type theory of Martin-Lof [5], since one of the features of Martin-Lof's type theory is that it formalizes ' ambiguities' of this sort. However, to the best of my knowledge, no one has ever worked out the details of the relationship, and when the question again arose in the McGill Categ ...
... given by the type theory of Martin-Lof [5], since one of the features of Martin-Lof's type theory is that it formalizes ' ambiguities' of this sort. However, to the best of my knowledge, no one has ever worked out the details of the relationship, and when the question again arose in the McGill Categ ...
Lecture 8
... Note that one often works with several types of mathematical objects such as groups, abelian groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family of all topological spaces. These entities are huge and do not qualify to be sets. We shall call them fam ...
... Note that one often works with several types of mathematical objects such as groups, abelian groups, vector spaces and topological spaces. Thus one talks of the family of all groups or the family of all topological spaces. These entities are huge and do not qualify to be sets. We shall call them fam ...
sheaf semantics
... Topos theory may to a large extent be developed within a constructive higher order logic (see BELL[l]). However the very definition of an elementary topos relies on a nonpredicativity: the axiom for the subobject classifier. Fortunately, the more restricted class of Grothendieck topoi (see [4]), i. ...
... Topos theory may to a large extent be developed within a constructive higher order logic (see BELL[l]). However the very definition of an elementary topos relies on a nonpredicativity: the axiom for the subobject classifier. Fortunately, the more restricted class of Grothendieck topoi (see [4]), i. ...
REVIEW OF MONOIDAL CONSTRUCTIONS 1. Strict monoidal
... pass from topological spaces to simplicial sets. Conversely, there is a construction that allows us to pass from simplicial sets to topological spaces. Under this construction, we take one copy (u, ∆n ) of the simplex ∆n for each element u ∈ Kn . And then identify them by means of the equalities (di ...
... pass from topological spaces to simplicial sets. Conversely, there is a construction that allows us to pass from simplicial sets to topological spaces. Under this construction, we take one copy (u, ∆n ) of the simplex ∆n for each element u ∈ Kn . And then identify them by means of the equalities (di ...
Composing functors Horizontal composition (functors): C D E If F, G
... The term monad Fix Σ and E. Consider the functor T : Set → Set given by T (V) = TermsΣ,E(V). This is a monad: • Functor: for f : V → W, define T (f) : TermsΣ,E(V) → TermsΣ,E(W) by “renaming” all the variables in a term. • Unit: ηV : V → TermsΣ,E(V) maps a variable x to the term x. • Multiplication: ...
... The term monad Fix Σ and E. Consider the functor T : Set → Set given by T (V) = TermsΣ,E(V). This is a monad: • Functor: for f : V → W, define T (f) : TermsΣ,E(V) → TermsΣ,E(W) by “renaming” all the variables in a term. • Unit: ηV : V → TermsΣ,E(V) maps a variable x to the term x. • Multiplication: ...
A convenient category for directed homotopy
... U-initial lift of a cone (fi : X → UAi )i∈I is given by putting a ≤ b on X if and only if fi (a) ≤ fi (b) for each i ∈ I. (2) An ordered set is a preordered set (A, ≤) where ≤ is also antisymmetric, i.e., if it satisfies (∀x, y)(x ≤ y ∧ y ≤ x → x = y). The category of ordered sets is not topological ...
... U-initial lift of a cone (fi : X → UAi )i∈I is given by putting a ≤ b on X if and only if fi (a) ≤ fi (b) for each i ∈ I. (2) An ordered set is a preordered set (A, ≤) where ≤ is also antisymmetric, i.e., if it satisfies (∀x, y)(x ≤ y ∧ y ≤ x → x = y). The category of ordered sets is not topological ...
Axiomatic Topological Quantum Field Theory
... Rham, simplicial, singular) which are important for applications. However the purely formal properties are best studied independently of any geometric realisation. The same applies to TQFT. In fact, producing a rigorous geometric or analtic construction may not yet be possible in all cases, hence ha ...
... Rham, simplicial, singular) which are important for applications. However the purely formal properties are best studied independently of any geometric realisation. The same applies to TQFT. In fact, producing a rigorous geometric or analtic construction may not yet be possible in all cases, hence ha ...
Two-dimensional topological field theories and Frobenius - D-MATH
... In particular, the map φ 7→ cφ is a contravariant functor from the category of orientation preserving diffeomorphisms of oriented closed (n − 1)manifolds to nCob. Moreover, two isotopic diffeomorphisms induce the same cobordism class. For a proof see (Milnor 1965, Theorems 1.6 and 1.9). A first con ...
... In particular, the map φ 7→ cφ is a contravariant functor from the category of orientation preserving diffeomorphisms of oriented closed (n − 1)manifolds to nCob. Moreover, two isotopic diffeomorphisms induce the same cobordism class. For a proof see (Milnor 1965, Theorems 1.6 and 1.9). A first con ...
Composition of linear transformations and matrix multiplication Math
... includes ‘morphisms’ (also called maps or arrows) between the objects. In the case of the category of vector spaces, the morphisms are the linear transformations. We’ll start with the formal definition of categories. Category theory was developed by Eilenberg and Mac Lane in the 1940s. ...
... includes ‘morphisms’ (also called maps or arrows) between the objects. In the case of the category of vector spaces, the morphisms are the linear transformations. We’ll start with the formal definition of categories. Category theory was developed by Eilenberg and Mac Lane in the 1940s. ...
Math 8211 Homework 1 PJW
... (1) Show that a functor F : G → H is ‘the same as’ a group homomorphism f : G → H when G and H are regarded as groups in the usual way. (2) Show that a functor F : G → G is naturally isomorphic to the identity functor 1G : G → G if and only if F is an isomorphism of categories of the form cg : G → G ...
... (1) Show that a functor F : G → H is ‘the same as’ a group homomorphism f : G → H when G and H are regarded as groups in the usual way. (2) Show that a functor F : G → G is naturally isomorphic to the identity functor 1G : G → G if and only if F is an isomorphism of categories of the form cg : G → G ...
Bicartesian closed categories and logic
... Example .. The category of all sets, denoted by Set, has as its objects every set, and its arrows are just functions between sets. One of the most pervasive concepts in category theory is the concept of a universal property. I will not attempt to give a definition here, but I will give some importa ...
... Example .. The category of all sets, denoted by Set, has as its objects every set, and its arrows are just functions between sets. One of the most pervasive concepts in category theory is the concept of a universal property. I will not attempt to give a definition here, but I will give some importa ...
groups and categories
... and join x ∨ y operations, with terminal object 1 and initial object 0 as units, respectively (assuming P has these structures), as well as the poset End(P ) of monotone maps f : P → P , ordered pointwise, with composition g ◦ f as ⊗ and 1P as unit. A discrete monoidal category, i.e. one with a disc ...
... and join x ∨ y operations, with terminal object 1 and initial object 0 as units, respectively (assuming P has these structures), as well as the poset End(P ) of monotone maps f : P → P , ordered pointwise, with composition g ◦ f as ⊗ and 1P as unit. A discrete monoidal category, i.e. one with a disc ...
Algebraic Models for Homotopy Types EPFL July 2013 Exercises 1
... with respect to acyclic fibrations (resp., fibrations) is a cofibration (resp., acyclic cofibration). (Hint: Factor and apply the lifting property to the factored map.) 2. (Homotopy pushouts I) In a closed model category, show that in a pushout square, if one arrow is an acyclic cofibration, so is t ...
... with respect to acyclic fibrations (resp., fibrations) is a cofibration (resp., acyclic cofibration). (Hint: Factor and apply the lifting property to the factored map.) 2. (Homotopy pushouts I) In a closed model category, show that in a pushout square, if one arrow is an acyclic cofibration, so is t ...
Model categories - D-MATH
... commutes. We note that R can be described as a category with only one element, where the elements of R are given by the morphisms and multiplication of elements is the composition of arrows (naturally there is the additional structure of an abelian group on the morphisms, thus this is actually a so ...
... commutes. We note that R can be described as a category with only one element, where the elements of R are given by the morphisms and multiplication of elements is the composition of arrows (naturally there is the additional structure of an abelian group on the morphisms, thus this is actually a so ...
Defining Gm and Yoneda and group objects
... between the Z-points of X and Y for each scheme Z and these maps satisfy certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, wh ...
... between the Z-points of X and Y for each scheme Z and these maps satisfy certain naturality conditions, then they arise from a map from X to Y . This correspondence also helps us make sense of a group object in a category, which is an object G and morphisms m : G × G → G, i : G → G and e : ? → G, wh ...
Commutative Algebra Fall 2014/2015 Problem set III, for
... for every open set U associates (a set, an abelian group, a vector space, etc) S(U ). By default, we will assume that S(∅) is the terminal object in the category. Moreover for every pair of open sets U ⊆ V we have a morphism rV U : S(V ) → S(U ) such that for any triple U ⊆ V ⊆ W it holds rV U ◦ rW ...
... for every open set U associates (a set, an abelian group, a vector space, etc) S(U ). By default, we will assume that S(∅) is the terminal object in the category. Moreover for every pair of open sets U ⊆ V we have a morphism rV U : S(V ) → S(U ) such that for any triple U ⊆ V ⊆ W it holds rV U ◦ rW ...
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.