1.2 Universal Construction

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

1. (1)

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

2. (2)

   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.

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

1. (1)

   $P \in \obj{\catc}$.

2. (2)

   For each $i \in I$, $\pi_{i} \in \mor{P, A_i}$.

3. (U)

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

1. (1)

   $P \in \obj{\catc}$.

2. (2)

   For each $i \in I$, $\iota_{i} \in \mor{A_i P}$.

3. (U)

   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).label Let $I$ be a set and $\lesssim$ be a relation on $I$, then $(I, \lesssim)$ is a directed set if

1. (1)

   For any $i \in I$, $i \lesssim i$.

2. (2)

   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:

1. (3U)

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

2. (3D)

   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).label 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).label 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:

1. (1)

   $\seqi{A}\subset \obj{\catc}$.

2. (2)

   For each $i \in I$, $f^{i}_{i} = \text{Id}_{A_i}$.

3. (3)

   For each $i, j \in I$ with $i \lesssim j$, $f^{i}_{j} \in \mor{A_i, A_j}$.

4. (4)

   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).label 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:

1. (1)

   For each $i \in I$, $f^{i}_{A} \in \mor{A_i, A}$.

2. (2)

   For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:

   \[\xymatrix{ A_i \ar@{->}[rd]_{f^i_A} \ar@{->}[r]^{f^i_j} & A_j \ar@{->}[d]^{f^j_A} \\ & A }\]

   
   
3. (U)

   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).label 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:

1. (1)

   For each $i \in I$, $f^{A}_{i} \in \mor{A, A_i}$.

2. (2)

   For any $i, j \in I$ with $i \lesssim j$, the following diagram commutes:

   \[\xymatrix{ A_i \ar@{->}[r]^{f^i_j} & A_j \\ A \ar@{->}[u]^{f^A_i} \ar@{->}[ru]_{f^A_j} & }\]

   
   
3. (U)

   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 $i \in I$.