Dashed arrow and solid arrow in category theory

[navigator]

In category theory,what is the difference between dashed arrow and solid arrow?I am just curious why there is not any textbook which I could find mention about it formally .heh...

 

[morphism]

You mean something like this?

The contexts I've seen it used in usually imply the existence and uniqueness of a certain morphism.

 

[navigator]

yeh.That is what i mean.so solid arrow imply "forall"($$\forall$$),dashed arrow imply "exists"($$\exists$$),all right?

 

[morphism]

I don't quite understand what you mean!

 

[navigator]

In your example,If the arrow U(g) is solid,it means
$$\forall U(g)[U(g) \circ \phi = f]$$,and when the arrow U(g) is dashed,then it means $$\exists U(g)[U(g) \circ \phi = f]$$,right?

 

[LukeD]

A solid arrow signifies that the morphism exists. A dashed arrow signifies that the morphism exists and is the unique morphism that makes the diagram commute.

For instance, in this diagram http://upload.wikimedia.org/wikipedia/commons/b/b2/CategoricalProduct-03.png"
(this is the definition of a product), the object X1 x X2 along with the morphisms Pi1 and Pi2 from X1 x X2 to X1 and to X2 respectively is called the product of X1 and X2 if for every object Y and morphisms f1:Y->X1 and f2:Y->X2, there exists a unique (meaning of dashed arrow) morphism f that makes the diagram commute.

Any time any arrow or object is drawn on the diagram, it has to exist. Any time an arrow is dashed, the morphism is unique. Some times other decorations are used on the arrows to signify some algebraic properties of the morphisms. For instance, a "split tail" on an arrow signifies that it is monic (Definition and picture (both a bit hidden, sorry) here: http://dlxs2.library.cornell.edu/cgi/t/text/pageviewer-idx?c=math;cc=math;q1=monic;rgn=full%20text;idno=gold010;didno=gold010;view=image;seq=54;page=root;size=S;frm=frameset" ) while both of these together signifies that it is an isomorphism.

Diagrams are useful tools to help show what the the author is trying to convey, but by themselves, they mean nothing. I can use the same diagram with a bunch of arrows to help illustrate a proof about a diagram commuting, or I can use it to illustrate a definition or a proof that a certain morphism exists and has some property. A good book will present a diagram as an aide but still present a proof and just refer back to the diagram to help the reader understand what morphism is doing what.

 

[navigator]

Thanks for your helpful answers.

 

[stevensenger]

Thanks so much for this post. It was very helpful!