Limits and Colimits

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Exercises

1. How would you describe a pushout in the category of C++ classes?
2. Show that the limit of the identity functor $\mathbf{Id}:\mathcal{C}\to \mathcal{C}$ is the initial object.
3. Subsets of a given set form a category. A morphism in that category is defined to be an arrow connecting two sets if the first is the subset of the second. What is a pullback of two sets in such a category? What’s a pushout? What are the initial and terminal objects?
4. Can you guess what a coequalizer is?
5. Show that, in a category with a terminal object, a pullback towards the terminal object is a product.
6. Similarly, show that a pushout from an initial object (if one exists) is the coproduct.