91Ó°ÊÓ

A zero object in a category \(\mathcal{C}\) is an object that is both an initial object and a terminal object. (i) Prove the uniqueness to isomorphism of initial, terminal, and zero objects, if they exist. (ii) Prove that \(\\{0\\}\) is a zero object in \({ }_{R}\) Mod and that \(\\{1\\}\) is a zero object in Groups. (iii) Prove that neither Sets nor Top has a zero object. (iv) Prove that if \(A=\\{a\\}\) is a set with one element, then \((A, a)\) is a zero object in Sets \(_{*}\), the category of pointed sets. If \(A\) is given the discrete topology, prove that \((A, a)\) is a zero object in Top \(_{*}\), the category of pointed topological spaces.

(i) Let \(Y\) be a set, and let \(\mathcal{P}(Y)\) denote its power set; that is, \(\mathcal{P}(Y)\) is the partially ordered set of all the subsets of \(Y\). As in Example 1.3(iii), view \(\mathcal{P}(Y)\) as a category. If \(A, B \in\) \(\mathcal{P}(Y)\), prove that the coproduct \(A \sqcup B=A \cup B\) and that the product \(A \sqcap B=A \cap B\). (ii) Generalize part (i) as follows. If \(X\) is a partially ordered set viewed as a category, and \(a, b \in X\), prove that the coproduct \(a \sqcup b\) is the least upper bound of \(a\) and \(b\), and that the product \(a \sqcap b\) is the greatest lower bound. (iii) Give an example of a category in which there are two objects whose coproduct does not exist.

(i) Let \(K\) be a cofinal subset of a directed index set \(I\) (that is, for each \(i \in I\), there is \(k \in K\) with \(i \preceq k\) ). Let \(\left\\{M_{i}, \varphi_{j}^{i}\right\\}\) be a direct system over \(I\), and let \(\left\\{M_{i}, \varphi_{j}^{i}\right\\}\) be the subdirect system whose indices lie in \(K\). Prove that the direct limit over \(I\) is isomorphic to the direct limit over \(K\). (ii) Let \(K\) be a cofinal subset of a directed index set \(I\), let \(\left\\{M_{i}, \varphi_{j}^{i}\right\\}\) be an inverse system over \(I\), and let \(\left\\{M_{i}, \varphi_{j}^{i}\right\\}\) be the subinverse system whose indices lie in \(K\). Prove that the inverse limit over \(I\) is isomorphic to the inverse limit over \(K\). (iii) A partially ordered set \(I\) has a top element if there exists \(\infty \in I\) with \(i \preceq \infty\) for all \(i \in I\). If \(\left\\{M_{i}, \varphi_{j}^{i}\right\\}\) is a direct system over \(I\), prove that $$ \lim _{\longrightarrow} M_{i} \cong M_{\infty} $$ (iv) Show that part (i) may not be true if the index set is not directed.

We call \(\lim _{I} F\) or \(\lim _{I} F\) finite if the index set \(I\) is finite. Prove that if \(\mathcal{A}\) is an additive category having kernels and cokernels, then \(\mathcal{A}\) has all finite inverse limits and direct limits. Conclude that \(\mathcal{A}\) has pullbacks, pushouts, equalizers, and coequalizers.

If \(\mathcal{A}\) is an abelian category, prove that a morphism \(f=\left(f_{n}\right)\) in \(\operatorname{Comp}(\mathcal{A})\) [i.e., a chain map] is monic (or epic) if and only if each \(f_{n}\) is monic (or epic) in \(\mathcal{A}\).