direct limit

by iarnab_kundu, Nov 26, 2018, 2:00 PM

We consider a locally small category $\mathcal{C}$ . Let $I$ be a small category, which is called the "diagram" category.
Given a covariant functor $\alpha:I\to\mathcal{C}$ we call it a "diagram" in $\mathcal{C}$ .
Example- We consider $I$ is the category of natural numbers, that is for each natural number $n\in\mathbb{N}$ there is an object in $I$ , and there is a unique morphism $n\to m$ if and only if $n\le m$ . Then a functor $\alpha:I\to\mathcal{C}$ is said to be a direct system in the category.
In Algebra we are interested in not only the diagrams in $\mathcal{C}$ but also in the "maps" between them. Let $\alpha,\beta:I\to\mathcal{C}$ be two diagrams. Then a map $m:\alpha\Rightarrow\beta$ is the data of a morphism $m(i):\alpha(i)\to\beta(i)$ for each $i\in I$ such that the (DIAGRAM BELOW) commutes.
Remark-Notice that this generalizes our notion of maps between directed systems.
Remark2- Notice that a map between $\alpha,\beta$ above is just a natural transformation between $\alpha,\beta$ .
Therefore we are naturally interested in studying the category $\mathcal{C}^I$ , the category of "diagrams in $\mathcal{C}$ ".
Notice that there are "constant diagrams". Given $A\in\mathcal{C}$ we have a diagram $C_A:I\in\mathcal{C}$ such that $C_A(i)=A$ for all $i\in I$ , and for each morphism $\phi:i\to j$ we have $C_A(\phi)=\text{id}_A$ . We note that we have a functor $C:\mathcal{C}\to\mathcal{C}^I$ given by $A\to C_A$ . In fact it is interesting to note that the map $\text{Hom}(A,B)\to\text{Hom}(C_A,C_B)$ is a bijection. Thus it is fully faithful. Now we fix a diagram $\alpha:I\to\mathcal{C}$ . We have a contravariant functor $f_{\alpha}:\mathcal{C}^{op}\to\text{Sets}$ given by $C\mapsto\text{Hom}(C_A,\alpha)$ .
Definition- An element $\mathfrak{A}$ is said to be the "limit" of $\alpha$ if it represents the functor $f_{\alpha}$ , that is $f_{\alpha}$ is isomorphic to $h_{\mathfrak{A}}$ .

category theory

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Cracking the Nut Open

direct limit

by iarnab_kundu, Nov 26, 2018, 2:00 PM

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