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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 β and π» be categories with the same objects. The result of the filtering should be a category with the arrows of β, that post-compose with arrows of π»:
{gβπ»1β£βfββ1.cod(f)=dom(g)}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 π».
Ok, I understood my problem better. I can rephrase my question to make it more clear:
Given two functions f:AβB, g:Bβ²βC, construct a function fβg:fβ1(Bβ©Bβ²)βC with fβg(a)=g(f(a)), where fβ1(Bβ©Bβ²)={aβAβ£f(a)βBβ©Bβ²}. I.e. the situation looks like this:
fβ1(Bβ©Bβ²)ΛfβBβ©Bβ²jβ²βBβ²gβCββjAfβBSo the candidate for fβg would be gβjβ²βΛf.
For categories other than Set, we would have to require that Bβ©Bβ² and fβ1(Bβ©Bβ²) are pullbacks and j and jβ² are monomorphisms.
What do you mean by Bβ©Bβ²? This makes no sense outside of a material set theory context. You need some maps BβZβBβ² to get a pullback, and they need to be monomorphisms to get anything like what you are suggesting.
At the moment, I still canβt see what youβre 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 Cat.
You are right, Bβ©Bβ² makes no sense in outside of set theory, but how can I find and object, that acts like Bβ©Bβ² in Cat? In Set an object Bβ©Bβ² with B,Bβ²βZ satisfies the following property:
Bβ©Bβ²iβ²βBβ²βiβjβ²BjβZWhere i,iβ²,j,jβ² are subset inclusion maps and Bβ©Bβ² is the pullback of Z along j and jβ².
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 Set then itβs straightforward:
for f:AβB and g:Bβ²βC two morphism in your category, and if BβZβBβ² 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
AΓZBβ²βΆBΓZBβ²βΆBβ²gβΆCβ(pb)β(pb)βAβΆfBβΆZ.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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