Category Theory
Zulip Server
Archive

You're reading the public-facing archive of the Category Theory Zulip server.
To join the server you need an invite. Anybody can get an invite by contacting Matteo Capucci at name dot surname at gmail dot com.
For all things related to this archive refer to the same person.

Stream: learning: questions

Topic: opposite category

Gyanendra Aggarwal (Mar 07 2021 at 18:45):

We have a category C (2-category) with 2 objects. Object x {-2,-1,0,1,2} and object y {0,1,4) and there is a morphism f: x -> y s.t.
f(-2) = 4, f(-1) = 1, f(0) = 0, f(1) = 1 and f(2) = 4. What will be opposite category for C?

Joshua Meyers (Mar 07 2021 at 19:13):

The opposite category for $C$ has objects $x=\{-2,-1,0,1,2\}$ and $y=\{0,1,4\}$ and a morphism $f:y\to x$, which is not a function (morphisms don't have to be functions)

Joshua Meyers (Mar 07 2021 at 19:20):

Insofar as $C$ is considered a category, the only data that can be accessed is

• the class of objects
• the class of morphisms
• what morphisms go from what objects to what objects
• what morphisms are the identities on what objects
• what morphisms compose to give what morphisms

For some categories, the objects are sets, the morphisms are functions, and composition is function composition, though this is not required of categories. For categories where this is the case, such as $C$, you still cannot access the elements of the objects or what functions the morphisms are, if you are considering $C$ as a category.

Joshua Meyers (Mar 07 2021 at 19:21):

If you have experience with OOP, you can think of "category" as an interface that simply doesn't have any methods that will tell you about the elements of objects or where morphisms send those elements.