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Hello, I am looking for a notion of filtering the arrows of a category, if they can be composed with the arrows of another category, that is:
Let and be categories with the same objects. The result of the filtering should be a category with the arrows of , that post-compose with arrows of :
Is there a way I can state this clearly with category theory instead of relying on this set builder notation?
Thanks, Sven
Since there exist identity morphisms, every object in a category is the source and the domain of some morphism. So the condition in your set builder is empty and the set you write down is just that of morphisms of .
Ok, I understood my problem better. I can rephrase my question to make it more clear:
Given two functions , , construct a function with , where . I.e. the situation looks like this:
So the candidate for would be .
For categories other than , we would have to require that and are pullbacks and and are monomorphisms.
What do you mean by ? This makes no sense outside of a material set theory context. You need some maps to get a pullback, and they need to be monomorphisms to get anything like what you are suggesting.
At the moment, I still can’t see what you’re getting at, and not sure where the conversation is going.
@DavidRoberts, I formulated the problem in terms of sets and functions and now I try to generalize it to .
You are right, makes no sense in outside of set theory, but how can I find and object, that acts like in ? In an object with satisfies the following property:
Where are subset inclusion maps and is the pullback of Z along and .
You had already said it correctly in #4: an intersection of sets is a special kind of fiber product of sets (hence of pullback of sets), namely one where both input morphisms are monomorphisms.
Hence if the diagram in #3 is what you would like to generalize to categories other than then it’s straightforward:
for and two morphism in your category, and if are understood, then, assuming that the respective two pullback squares (pb) exist in your category (for instance if it has all finite limits), then you seem to want to consider the diagram
Here I am using the pasting law law to identify the object in the top left as shown.
@Urs, @DavidRoberts, thanks, this was really helpful. Keep up the good work on ncatlab.
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