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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2$\begingroup$ What does it mean that an arrow is "injective"? why do you call arrow "maps"? $\endgroup$– YCorCommented May 15, 2020 at 8:03
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$\begingroup$ Okay let me be more specific. I am actually considering categories like Sets, Semigroups, etc. where injective makes sense. $\endgroup$– Jokubas RahmanCommented May 15, 2020 at 8:11
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4$\begingroup$ @YCor, I understand the question about 'injective', but surely it is just a matter of taste whether to refer to arrows, maps, or morphisms in a category. (After all, I've only ever seen $\operatorname{Mor}(C, C')$, never $\operatorname{Arr}(C, C')$.) $\endgroup$– LSpiceCommented May 15, 2020 at 14:46
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1$\begingroup$ @LSpice yes, but in this precise case it doesn't apply, as the question makes sense only for maps in the sense "functions between sets" and hasn't seriously thought about what the right setting for the question is. (I should add that the fact that people use "map" for arrows in arbitrary categories might be misleading to students/beginners, as reflected by this very question.) $\endgroup$– YCorCommented May 15, 2020 at 15:09
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3 Answers
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Suppose your category is generated under colimits by compact objects (this is the case in algebraic categories such as sets, semigroups etc.).
Then if you replace "injective" by "monomorphism" (which again, is the same in algebraic categories such as sets, semigroups etc. ) any sequence of monomorphisms satisfies the desired property.
Indeed, we wish to prove that for any $X$, $\hom(X,C_i)\to \hom(X,C)$ is injective. Writing $X$ as a colimit of compact objects, and using the fact that limits preserve injections in $\mathbf{Set}$, we may assume that $X$ is compact.
But if $X$ is compact, $\hom(X,C) = \varinjlim_j\hom(X,C_j)$, and so if $\alpha,\beta : X\to C_i$ become equal in $C$, they must become equal at some stage : there is $j\geq i$ such that $\alpha, \beta : X\to C_i\to C_j$ are equal, and therefore, since $C_i\to C_j$ is a monomorphism, $\alpha = \beta$.
answered May 15, 2020 at 13:27
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Another situation in which the result is true is that of an Abelian category with enough injectives. Here I take "injective map" to mean monomorphism. In such setting, choose a monomorphism $m_i \colon C_i \to I$, where $I$ is injective. Since $I$ is injective and $C_i \to C_{i + 1}$ is a monomorphism, you can lift $m_i$ to a morphism $m_{i+1} \colon C_{i+1} \to I$, and so on. By putting together all $m_i$, you get a morphism $m \colon C \to I$ such that $m_i$ factors as $C_i \to C \to I$. Since $m_i$ is a monomorphism, the morphism $C_i\to C$ is a monomorphism as well.
I learned this nice argument in Neeman - A counterexample to a 1961 "theorem" in homological algebra, where he also construct an example of an Abelian category and a chain of monomorphism such that the direct limit is $0$ - showing that the analogue of this result is not always true!
answered May 15, 2020 at 13:28
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$\begingroup$ Where does this argument use the assumption that the category is abelian? $\endgroup$ Commented May 15, 2020 at 15:42
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$\begingroup$ Actually it doesn't, it is just that injective objects are more commonly used in that context $\endgroup$ Commented May 15, 2020 at 17:32
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$\begingroup$ $\def\I{\mathcal{I}}\def\C{\mathcal{C}}\def\c{\operatorname{colim}}$Another example is when $\C$ is an abelian category in which direct limits are exact. Let $\{C_i\mid i\in \I\}$ be a direct system in $\C$ such that $C_i\to C_j$ is injective for all $i\leq j$. Let $i\in\I$. We want to see that $C_i\to \c_{j\in\I}C_j$ is injective. We may assume that $i$ is initial in $\I$ (replace $\I$ by the full subcategory $\{j\in\C\mid i\leq j\}$, which is cofinal). The components of the functor morphism $C_i\Rightarrow C$ are all injective. Hence $\c(C_i\Rightarrow C)=C_i\to\c_{j\in\I}C_j$ is injective. $\endgroup$ Commented Jul 9 at 7:31
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$\begingroup$ (Here $C_i\Rightarrow C$ is the obvious natural transformation of functors $\mathcal{I}\to\mathcal{C}$ from the constant functor $j\in\mathcal{I}\mapsto C_i$ to our direct system $C:j\in\mathcal{I}\mapsto C_j$.) $\endgroup$ Commented Jul 9 at 7:35
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Your question is a bit imprecise, but based on what you said in your comment, I think for your situations the answer will be "yes". For Sets, it's an easy exercise to prove that all the $g_i$ are injective. Simply write down the formula for the colimit and work at the level of elements in $C_i$ and $C$. A reference for this fact is Proposition 9.3 in Gaunce Lewis's thesis, which Peter May gave me a long time ago in this question. Lewis also proves the result for compactly generated spaces. You can follow his model to prove it for a wide class of categories built from sets, including semi-groups. Many of these categories are "sets with extra structure" encoded via a monad $T$, and I suspect you could formulate a general result about directed colimits of injections in categories of $T$-algebras over Set.
answered May 15, 2020 at 12:40