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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...
In category theory, a dashed arrow typically represents a morphism that is not fully defined or partially defined, while a solid arrow represents a fully defined morphism. Dashed arrows are often used to indicate that some properties of the morphism are not known or do not exist.
In category theory diagrams, dashed arrows are typically used to represent functors or natural transformations between categories. These structures may not be fully defined or may only exist partially, thus the use of dashed arrows to indicate this.
Yes, dashed arrows can be composed in category theory just like solid arrows. However, the resulting dashed arrow may still not be fully defined or may only exist partially, depending on the properties of the original dashed arrows being composed.
One limitation of using dashed arrows in category theory is that they may not be associative or commutative in the same way that solid arrows are. This is because dashed arrows may not have all the properties required for these properties to hold.
Dashed arrows can be used to represent various structures in category theory, such as monads, monoids, and adjoints. These structures may not always be fully defined or may only exist partially, thus the use of dashed arrows to represent them.