|
|
| Line 1: |
Line 1: |
| {{Unreferenced|date=December 2009}}
| | The author's name is Christy Brookins. What I adore doing is soccer but I don't have the time recently. Kentucky is where I've always been living. He functions as a bookkeeper.<br><br>Also visit my web page authentic psychic readings ([http://kjhkkb.net/xe/notice/374835 click the up coming web site]) |
| In [[category theory]], for any object <math>a</math> in any [[category (mathematics)|category]] <math>\mathcal{C}</math> where the [[product (category theory)|product]] <math>a\times a</math> exists, [[there exists]] the '''diagonal morphism'''
| |
| | |
| :<math>\delta_a : a \rightarrow a \times a</math>
| |
| | |
| satisfying
| |
| | |
| :<math>\pi_k \circ \delta_a = id_a</math> for <math>k \in \{ 1,2 \}</math>,
| |
| | |
| where <math>\pi_k</math> is the [[canonical projection morphism]] to the <math>k</math>-th component. The existence of this morphism is a consequence of the [[universal property]] which [[characterization (mathematics)|characterize]]s the product ([[up to]] [[isomorphism]]). The restriction to binary products here is for ease of notation; diagonal morphisms exist similarly for arbitrary products. The [[image (category theory)|image]] of a diagonal morphism in the [[category of sets]], as a [[subset]] of the [[Cartesian product]], is a [[relation (mathematics)|relation]] on the [[domain of a function|domain]], namely [[equality (mathematics)|equality]].
| |
| | |
| For [[concrete categories]], the diagonal morphism can be simply described by its action on elements <math>x</math> of the object <math>a</math>. Namely, <math>\delta_a(x) = \langle x,x \rangle</math>, the [[ordered pair]] formed from <math>x</math>. The reason for the name is that the [[Image (mathematics)|image]] of such a diagonal morphism is diagonal (whenever it makes sense), for example the image of the diagonal morphism <math>\mathbb{R} \rightarrow \mathbb{R}^2</math> on the [[real line]] is given by the line which is a [[graph of a function|graph]] of the equation <math>y=x</math>. The diagonal morphism into the [[infinite product]] <math>X^\infty</math> may provide an [[Injective function|injection]] into the [[space of sequences]] valued in <math>X</math>; each element maps to the constant [[sequence]] at that element. However, most notions of sequence spaces have [[convergent series|convergence]] restrictions which the image of the diagonal map will fail to satisfy.
| |
| | |
| In particular, the [[category of small categories]] has products, and so one finds the '''diagonal functor''' <math>\mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C}</math> given by <math>\Delta(a) = \langle a,a \rangle</math>, which maps objects as well as morphisms. This [[functor]] can be employed to give a succinct alternate description of the product of objects ''within'' the category <math>\mathcal{C}</math>: a product <math>a \times b</math> is a universal arrow from <math>\Delta</math> to <math>\langle a,b \rangle</math>. The arrow comprises the projection maps.
| |
| | |
| More generally, in any [[functor category]] <math>\mathcal{C}^\mathcal{J}</math> (here <math>\mathcal{J}</math> should be thought of as a [[small category|small]] [[index category]]), for each object <math>a</math> in <math>\mathcal{C}</math>, there is a [[constant functor]] with fixed object <math>a</math>: <math>\Delta(a) \in \mathcal{C}^\mathcal{J}</math>. The diagonal functor <math>\Delta : \mathcal{C} \rightarrow \mathcal{C}^\mathcal{J}</math> assigns to each object of <math>\mathcal{C}</math> the functor <math>\Delta(a)</math>, and to each morphism <math>f: a \rightarrow b</math> in <math>\mathcal{C}</math> the obvious [[natural transformation]] <math>\eta</math> in <math>\mathcal{C}^\mathcal{J}</math> (given by <math>\eta_j = f</math>). In the case that <math>\mathcal{J}</math> is a discrete category with two objects, the diagonal functor <math>\mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C}</math> is recovered.
| |
| | |
| Diagonal functors provide a way to define [[limit (category theory)|limits]] and [[colimit]]s of functors. The limit of any functor <math>\mathcal{F} : \mathcal{J} \rightarrow \mathcal{C}</math> is a [[universal arrow]] from <math>\Delta</math> to <math>\mathcal{F}</math> and a [[colimit]] is a universal arrow <math>F \rightarrow \Delta</math>. If every functor from <math>\mathcal{J}</math> to <math>\mathcal{C}</math> has a limit (which will be the case if <math>\mathcal{C}</math> is complete), then the operation of taking limits is itself a functor from <math>\mathcal{C}^\mathcal{J}</math> to <math>\mathcal{C}</math>. The limit functor is the [[adjoint functors|right-adjoint]] of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor <math>\mathcal{C} \rightarrow \mathcal{C} \times \mathcal{C}</math> described above is the left-adjoint of the binary [[product (category theory)|product functor]] and the right-adjoint of the binary [[coproduct|coproduct functor]].
| |
| | |
| {{Functors}}
| |
| | |
| {{DEFAULTSORT:Diagonal Functor}}
| |
The author's name is Christy Brookins. What I adore doing is soccer but I don't have the time recently. Kentucky is where I've always been living. He functions as a bookkeeper.
Also visit my web page authentic psychic readings (click the up coming web site)