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Post Closed as "Needs details or clarity" by YCor, Jeremy Rickard, user44191, ARG, David Handelman
Fixed typo to this new question
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David White
David White
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I have the following question. Assume you have a category $\mathcal{C}$ such that direct limits exists. Let $(C_n)_{n\in\mathbb{N}}$ be a sequence of objects in $\mathcal{C}$ and consider the following diagram $$ C_1\rightarrow C_2\rightarrow C3\rightarrow\cdots $$$$ C_1\rightarrow C_2\rightarrow C_3\rightarrow\cdots $$ and suppose that all maps $f_i:C_i\to C_{i+1}$ are injective. Let us denote by $C$ the direct limit of this diagram. May we conclude that the maps $g_i:C_i\to C$ are injective as well? Thank you very much in advance.

I have the following question. Assume you have a category $\mathcal{C}$ such that direct limits exists. Let $(C_n)_{n\in\mathbb{N}}$ be a sequence of objects in $\mathcal{C}$ and consider the following diagram $$ C_1\rightarrow C_2\rightarrow C3\rightarrow\cdots $$ and suppose that all maps $f_i:C_i\to C_{i+1}$ are injective. Let us denote by $C$ the direct limit of this diagram. May we conclude that the maps $g_i:C_i\to C$ are injective as well? Thank you very much in advance.

I have the following question. Assume you have a category $\mathcal{C}$ such that direct limits exists. Let $(C_n)_{n\in\mathbb{N}}$ be a sequence of objects in $\mathcal{C}$ and consider the following diagram $$ C_1\rightarrow C_2\rightarrow C_3\rightarrow\cdots $$ and suppose that all maps $f_i:C_i\to C_{i+1}$ are injective. Let us denote by $C$ the direct limit of this diagram. May we conclude that the maps $g_i:C_i\to C$ are injective as well? Thank you very much in advance.

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Jokubas Rahman
Jokubas Rahman
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Injective maps and direct limits

I have the following question. Assume you have a category $\mathcal{C}$ such that direct limits exists. Let $(C_n)_{n\in\mathbb{N}}$ be a sequence of objects in $\mathcal{C}$ and consider the following diagram $$ C_1\rightarrow C_2\rightarrow C3\rightarrow\cdots $$ and suppose that all maps $f_i:C_i\to C_{i+1}$ are injective. Let us denote by $C$ the direct limit of this diagram. May we conclude that the maps $g_i:C_i\to C$ are injective as well? Thank you very much in advance.