Finite Limits and Colimits as Products/Coproducts and Equalizers/Coequalizers
Point of Post: In this post we prove the result that a category has all finite products/coproducts and equalizers/coequalizers if and only if it has all limits/colimits.
In this post we prove a pretty neat fact that, at least for finite sets (we shall state a more general result, but we don’t care about it as much) limits/colimits are the same thing as equalizers/coequalizers and products/coproducts. This is often cool because it reduces having to prove some result holding for all finite limits/colimits to proving some problem holds for equalizers/coequalizers and finite products/coproducts. In particular, it will tell us that a category admits finite limits/colimits if and only if it admits finite products/coproducts.
Finite Limits/Colimits ad Products/Coproducts and Equalizers/Coequalizers
There is really nothing to discuss here, so I’ll just bang this one out:
Theorem: Every finite limit/colimit can be written as the equalizers/coequalizer of a finite product/coproduct of maps.
Proof: Let be a finite category (finite amount of objects AND finite amount of arrows) and a diagram. Consider the two arrows
(where, in case you’ve forgotten, stands for all the morphisms in and if then ) defined projection wise by and where is the projection arrow and is defined similarly (of course you can guess what domain means here).
Let be an equalizer of these two arrows. What we want to show then is that is actually a limit for . To prove this we need to take any possible arrow . Observe then that the set of arrows is a cone for if and only if (check this). From this then we may conclud ethat is a cone. Moreover, we see that any cone is going to have to factorize through since the cone functions of that cone will have to equalize and .
Using the exact same set up (dualizing) one can prove that
Theorem: Every finite colimit can be written as the coequalizer of a finite coproduct.
Corollary: If a category admits all equalizers (resp. coequalizers) and finite products (resp. coproducts) then the category admits all finite limits (resp. colimits).
Corollary: An abelian category admits all finite limits and colimits.
Proof: Recall that an abelian category, by definition, includes all kernels an cokernels. And, since and we also know that it admits all equalizers and coequalizers. Since abelian categories also admit all finite products and coproducts the conclusion follows.
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 Herrlich, Horst, and George E. Strecker. Category Theory: An Introduction. Lemgo: Heldermann, 2007. Print.