1.2 Universal Construction

Definition 1.2.1 (Universal Object). Let $\catc$ be a category and $P \in \obj{\catc}$, then $P$ is...

universally attracting if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{A, P}$.
universally repelling if for every $A \in \obj{\catc}$, there exists a unique $f \in \mor{P, A}$.

If $P$ is universally attracting or repelling, then $P$ is a universal object.

If $P, Q \in \obj{\catc}$ are both universally attracting/repelling, then they are isomorphic.

Proof. By assumption, there exists morphisms $f \in \mor{P, Q}$ and $g \in \mor{Q, P}$. Since $f \circ g \in \mor{Q, Q}$ and $g \circ f \in \mor{P, P}$ are unique, $f \circ g = \text{Id}_{Q}$ and $g \circ f = \text{Id}_{P}$. Thus $f$ is an isomorphism.$\square$

Definition 1.2.2 (Product). Let $\catc$ be a category and $\seqi{A}\subset \obj{\catc}$. A product of $\seqi{A}$ is a pair $(P, \seqi{\pi})$ where

$P \in \obj{\catc}$.
For each $i \in I$, $\pi_{i} \in \mor{P, A_i}$.
For any pair $(C, \seqi{f})$ satisfying (1) and (2), there exists a unique $f \in \mor{C, P}$ such that the following diagram commutes
\[\xymatrix{ C \ar@{->}[rd]^{f_i} \ar@{->}[d]_{f} & \\ P \ar@{->}[r]_{\pi_i} & A_i }\]
for all $i \in I$.

Definition 1.2.3 (Coproduct). Let $\catc$ be a category and $\seqi{A}\subset \obj{\catc}$. A product of $\seqi{A}$ is a pair $(P, \seqi{\iota})$ where

$P \in \obj{\catc}$.
For each $i \in I$, $\iota_{i} \in \mor{A_i P}$.
For any pair $(C, \seqi{f})$ satisfying (1) and (2), there exists a unique $f \in \mor{P, C}$ such that the following diagram commutes
\[\xymatrix{ & C \\ A_i \ar@{->}[r]_{\iota_i} \ar@{->}[ru]^{f_i} & P \ar@{->}[u]_{f} }\]
for all $i \in I$.

Definition 1.2.4 (Directed Set). Let $I$ be a set and $\lesssim$ be a relation on $I$, then $(I, \lesssim)$ is a directed set if

For any $i \in I$, $i \lesssim i$.
For any $i, j, k \in I$ such that $i \lesssim j$ and $j \lesssim k$, $i \lesssim k$.

and one of the following holds:

For any $i, j \in I$, there exists $k \in I$ with $i, j \lesssim k$.
For any $i, j \in I$, there exists $k \in I$ with $k \lesssim i, j$.

The directed set is upward-directed if it satisfies (3U), and downward-directed if it satisfies (3D).

Definition 1.2.5 (Cofinal). Let $(I, \lesssim)$ be a upward/downward directed set, then $J \subset I$ is cofinal if for every $\alpha \in I$, there exists $\beta \in J$ with $\beta \gtrsim \alpha$/$\beta \lesssim \alpha$.

Definition 1.2.6 (Directed System). Let $\catc$ be a category and $(I, \lesssim)$ be a directed set. A directed system is a pair $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ such that:

$\seqi{A}\subset \obj{\catc}$.
For each $i \in I$, $f^{i}_{i} = \text{Id}_{A_i}$.
For each $i, j \in I$ with $i \lesssim j$, $f^{i}_{j} \in \mor{A_i, A_j}$.
For each $i, j, k \in I$ with $i \lesssim j \lesssim k$, $f^{j}_{k} \circ f^{i}_{j} = f^{i}_{k}$.

If $I$ is upward/downward-directed, then $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ is upward/downward-directed.

Definition 1.2.7 (Direct Limit). Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system, then a direct limit of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longrightarrow}A_{i}$, is a pair $(A, \bracsn{f^i_A}_{i \in I})$ such that:

For each $i \in I$, $f^{i}_{A} \in \mor{A_i, A}$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ A_i \ar@{->}[rd]_{f^i_A} \ar@{->}[r]^{f^i_j} & A_j \ar@{->}[d]^{f^j_A} \\ & A }\]
For any pair $(B, \bracsn{g^i_B}_{i \in I})$ satisfying (1) and (2), there exists a unique $g \in \mor{A, B}$ such that the following diagram commutes
\[\xymatrix{ A_j \ar@{->}[d]_{f^j_A} \ar@{->}[rd]^{g^i_B} & \\ A \ar@{->}[r]_{g} & B }\]
for all $i \in I$.

Definition 1.2.8 (Inverse Limit). Let $\catc$ be a category and $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$ be an downward-directed system, then an inverse limit of $(\seqi{A}, \bracsn{f^i_j| i, j \in I, i \lesssim j})$, denoted $\lim_{\longleftarrow}A_{i}$, is a pair $(A, \bracsn{f^A_i}_{i \in I})$ such that:

For each $i \in I$, $f^{A}_{i} \in \mor{A, A_i}$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ A_i \ar@{->}[r]^{f^i_j} & A_j \\ A \ar@{->}[u]^{f^A_i} \ar@{->}[ru]_{f^A_j} & }\]
For any pair $(B, \bracsn{g^B_i}_{i \in I})$, there exists a unique $g \in \mor{B, A}$ such that the following diagram commutes
\[\xymatrix{ & A_i \\ B \ar@{->}[r]_{S} \ar@{->}[ru]^{g^B_i} & A \ar@{->}[u]_{f^A_i} }\]
for all $i \in I$.

Proposition 1.2.9. Let $R$ be a ring and $(\seqi{A}, \bracsn{T^i_j| i, j \in I, i \lesssim j})$ be an upward-directed system of $R$-modules, then there exists $(A, \bracsn{T^i_A}_{i \in I})$ such that:

For each $i \in I$, $T^{i}_{A} \in \hom({A_i, A})$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ A_i \ar@{->}[rd]_{T^i_A} \ar@{->}[r]^{T^i_j} & A_j \ar@{->}[d]^{T^j_A} \\ & A }\]
For any pair $(B, \bracsn{S^i_B}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom({A, B})$ such that the following diagram commutes
\[\xymatrix{ A_i \ar@{->}[d]_{T^i_A} \ar@{->}[rd]^{S^i_B} & \\ A \ar@{->}[r]_{g} & B }\]
for all $i \in I$.

Proof. Let $M = \bigoplus_{i \in I}A_{i}$. For any $i, j \in I$ with $i \lesssim j$ and $x \in A_{i}$, let $x_{i, j}\in M$ such that for any $k \in I$,

\[\pi_{k}(x_{i, j}) = \begin{cases}x&k = i \\ T^{i}_{j} x&k = j \\ 0&k \ne i, j\end{cases}\]

Let $N \subset M$ be the submodule generated by $\bracs{x_{i, j}|i, j \in I, i \lesssim j, x \in A_i}$, $A = M/N$, and $\pi: M \to M/N$ be the canonical map.

(1): For each $i \in I$, let

\[T^{i}_{M}: A_{i} \to M \quad \pi_{k} T^{i}_{M} x = \begin{cases}x &k = i \\ 0 &k \ne i\end{cases}\]

and $T^{i}_{A} = \pi \circ T^{i}_{M}$.

(2): Let $i, j \in I$ with $i \lesssim j$, then for any $x \in A_{i}$, $T^{i}_{M}x - T^{j}_{M} T^{i}_{j} x \in N$. Hence $T^{i}_{A}x = T^{j}_{A} T^{i}_{j}x$.

(U): Let

\[S_{0}: M \to B \quad x \mapsto \sum_{i \in I}S^{i}_{B} \pi_{i} x\]

then $S_{0}$ is the unique linear map such that $S_{0} \circ T^{i}_{M} = S^{i}_{B}$ for all $i \in I$. For any $i, j \in I$ with $i \lesssim J$, $S^{i}_{B} x = S^{j}_{B} T^{i}_{j} x$, so $\ker S_{0} \supset N$. By the first isomorphism theorem, there exists a unique $S \in \hom(A; B)$ such that $S_{0} = S \circ \pi$.$\square$

Proposition 1.2.10. Let $R$ be a ring and $(\seqi{A}, \bracs{T^i_j|i, j \in I, i \lesssim j)}$ be a downward-directed system of $R$-modules, then there exists $(A, \bracsn{T^A_i}_{i \in I})$ such that:

For each $i \in I$, $T^{A}_{i} \in \hom(A; A_{i})$.
For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:
\[\xymatrix{ A_i \ar@{->}[r]^{T^i_j} & A_j \\ A \ar@{->}[u]^{T^A_i} \ar@{->}[ru]_{T^A_j} & }\]
For any pair $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2), there exists a unique $S \in \hom(B; A)$ such that the following diagram commutes
\[\xymatrix{ & A_i \\ B \ar@{->}[r]_{S} \ar@{->}[ru]^{S^B_i} & A \ar@{->}[u]_{T^A_i} }\]
for all $i \in I$.

Proof. Let

\[A = \bracs{x \in \prod_{i \in I}A_i \bigg | \pi_j(x) = T^i_j\pi_i(x) \forall i, j \in I, i \lesssim j}\]

For each $i \in I$, let $T^{A}_{i} = \pi_{i}$, then $(A, \bracsn{T^A_i}_{i \in I})$ satisfies (1) and (2) by definition of $A$.

(U): Let $(B, \bracsn{S^B_i}_{i \in I})$ satisfying (1) and (2). Let

\[S: B \to \prod_{i \in I}A_{i} \quad \pi_{i}(Sx) = S^{B}_{i}\]

then for any $x \in B$ and $i, j \in I$ with $i \lesssim j$,

\[\pi_{j} (Sx) = S^{B}_{j}x = T^{i}_{j} S^{B}_{i}x = T^{i}_{j} \pi_{i}(S x)\]

so $S \in \hom(B; A)$, and the diagram commutes. Since any map $f: B \to A$ is uniquely determined by its composition with the projections, $S$ is unique.$\square$

Jerry's Digital Garden

1.2 Universal Construction

Jerry's Digital Garden