Functor - Revision history

DavidB4-bot: clean up

2016-08-18T00:43:51Z

clean up

DavidB4-bot: /* In Pure Mathematics */clean up

2016-06-23T15:04:01Z

In Pure Mathematics: clean up

EJamesW: Undo revision 1031892 by Barack (talk)

2013-01-28T18:24:37Z

Undo revision 1031892 by Barack (talk)

Barack: /* In Pure Mathematics */

2013-01-28T18:23:49Z

In Pure Mathematics

Karajou: Reverted edits by Cukuplz (talk) to last revision by MarkGall

2011-08-31T01:09:21Z

Reverted edits by Cukuplz (talk) to last revision by MarkGall

Cukuplz at 01:05, August 31, 2011

2011-08-31T01:05:51Z

MarkGall: add some fluff to the intro

2009-11-13T16:31:10Z

add some fluff to the intro

MarkGall: add some examples. this article is still completely inaccessible to the uninitiated and probably requires the attentions of an expositor more gifted than i, but i'll look again later

2009-11-13T16:26:41Z

add some examples. this article is still completely inaccessible to the uninitiated and probably requires the attentions of an expositor more gifted than i, but i'll look again later

PatrickD: /* In Pure Mathematics */ Mention contravariance. Mathematical functors are cool; computer language functors less so.

2009-10-01T19:26:17Z

In Pure Mathematics: Mention contravariance. Mathematical functors are cool; computer language functors less so.

PatrickD: Computer science use of the term. Should there be a disambiguation page?

2009-10-01T19:23:03Z

Computer science use of the term. Should there be a disambiguation page?

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 A simple example arises in the category of vector spaces. Fix a vector space <math>V</math>. We can define a functor <math>\textrm{Hom}(V,-) : \textrm{Vect}_k \to \textrm{Vect}_k</math> by
 #An object <math>A</math> (that is a vector space) is sent to <math>\textrm{Hom}(V,A)</math>, the vector space of linear maps from <math>V</math> to <math>A</math>.  We can think of these linear maps as matrices.
-#A linear transformation <math> \phi : A \to B</math> (i.e., an element of <math>\textrm{Hom}_{\textrm{Vect}_k}(A,B)</math> is sent to a linear transformation <math>\textrm{Hom}(V,A) \to \textrm{Hom}(V,B)</math> by setting <math>\textrm{Hom}(V,-)(\phi) = (f \mapsto \phi \circ f)</math>.  I other words, the image of <math>\phi</math> is supposed to be a linear transformation from <math>\textrm{Hom}(V,A)</math> to <math>\textrm{Hom}(V,B)</math>: it is defined by sending a linear transformation <math>f : V \to A</math> to <math>\phi \circ f : V \to B</math>.
+#A linear transformation <math> \phi : A \to B</math> (i.e., an element of <math>\textrm{Hom}_{\textrm{Vect}_k}(A,B)</math> is sent to a linear transformation <math>\textrm{Hom}(V,A) \to \textrm{Hom}(V,B)</math> by setting <math>\textrm{Hom}(V,-)(\phi) = (f \mapsto \phi \circ f)</math>).  In other words, the image of <math>\phi</math> is supposed to be a linear transformation from <math>\textrm{Hom}(V,A)</math> to <math>\textrm{Hom}(V,B)</math>: it is defined by sending a linear transformation <math>f : V \to A</math> to <math>\phi \circ f : V \to B</math>.
 We can similarly obtain a contravariant function <math>\textrm{Hom}(-,V)</math> by

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 In [[category theory]], a '''functor''' is a map between categories satisfying certain relations.  Functors can be thought of as expressing relations between categories, and have a wide range of applications.  For example, given a problem about topological spaces, one might apply some functor (say, taking the homology groups), and obtain an equivalent problem about vector spaces.  Since problems about vector spaces are generally much easier to answer than those about general topological spaces, this may make it easier to understand the original problem.
-Functors, in a sense, provide for categories what group [[homomorphism|homomorphisms]] do for groups.  To be precise, a function <math>F : \mathcal C \to \mathcal D</math> between two categories associates to each object <math>X \in \textrm{Ob } \mathcal C</math> an object <math>F(X) \in \textrm{Ob } \mathcal D</math>, and to each morphism <math>f \in \mathcal C(X,Y)</math> a morphism <math>F(f) \in \mathcal D(FX,FY)</math> such that:
+Functors, in a sense, provide for categories what group [[homomorphism]]s do for groups.  To be precise, a function <math>F : \mathcal C \to \mathcal D</math> between two categories associates to each object <math>X \in \textrm{Ob } \mathcal C</math> an object <math>F(X) \in \textrm{Ob } \mathcal D</math>, and to each morphism <math>f \in \mathcal C(X,Y)</math> a morphism <math>F(f) \in \mathcal D(FX,FY)</math> such that:
 :<math>F(\textrm{id}_X) = id_{F(X)}\,</math>
 :<math>F(g \circ f) = F(g) \circ F(f)</math>.

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 The term '''functor''' has two distinct meanings.
-cumming on my face oh yes sweet milk
+==In Pure Mathematics==
 ==In Computer Science==

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 The term '''functor''' has two distinct meanings.
-==In Pure Mathematics==
+cumming on my face oh yes sweet milk
 ==In Computer Science==

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	The term '''functor''' has two distinct meanings.		The term '''functor''' has two distinct meanings.

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 Functors are the fundamental objects used to relate structures between different categories.
-The above is sometimes called a "covariant functor", to distinguish it from the less common "contravariant functor".  In a contravariant functor, the composition order is reversed:
+The traditional terminology for the above is a "covariant functor", to distinguish it from "contravariant functors", which reverse the direction of compositions, and instead satisfy
 :<math>F(g \circ f) = F(f) \circ F(g)</math>.
 === Examples ===
 Algebraic topology was the first field in which the usefulness of the notion of a functor was recognized.  A basic example is the [[fundamental group]] functor <math>\pi_1 : \textbf{Top}* \to \textbf{Grp}</math>.  The action on objects is defined by sending a topological space to its fundamental group <math>\pi_1(X)</math>.  Recall that a map between two topological spaces <math>f : X \to Y</math> induces a map <math>f_* : \pi_1(X) \to \pi(Y)</math> by <math>f_*([\gamma]) = [f \circ \gamma]</math>.  Set <math>\pi_1(f) = f_*\,</math> so defined.  The functoriality of <math>\pi_1</math> boils down to the fact that <math>F(\textrm{id}_X) = \textrm{id}_{\pi_1(X)}</math>, that is, the identity map on a topological space induces the identity map on its fundamental group, together with the fact that <math>F(g \circ f) = F(g) \circ F(f)</math>, explained at [[fundamental group]].
 ==In Computer Science==