Category Theory: Inverse Limit in Sets

[Mandelbroth]

I think this looks like a homework problem, so I'll just put it here.

Homework Statement

Demonstrate that, for any index category ##\mathscr{J}## and any diagram ##\mathcal{F}:\mathscr{J}\to\mathbf{Sets}##,

$$\varprojlim_{\mathscr{J}}A_j=\left\{a\in \prod_{j\in \operatorname{obj}( \mathscr{J})}A_j~\vert~(i,k\in\operatorname{obj}(\mathscr{J}),~a_i\in A_i,~a_k\in A_k, \text{ and } \phi_{ik}\in \hom_{\mathscr{J}}(i,k))~\implies~a_k=\mathcal{F}(\phi_{ik})(a_i) \right\}=A$$
along with the obvious projections, which I'll denote ##l_i: A\to A_i##.

Homework Equations

I don't know how to make diagrams in TeX, so I'll just link to the universal property.

The Attempt at a Solution

Suppose ##W## has morphisms ##w_i: W\to A_i## that satisfy ##w_k=\mathcal{F}(\phi_{ik})\circ w_i##. We wish to show the existence of a unique morphism ##w: W\to A## such that ##w_i=l_i\circ w##.

My thought is that both ##W## and ##A## clearly map into the product ##\prod A_j##. We even have the unique map from ##A## to ##\prod A_j##, set inclusion, satisfying the universal property for the product. However, I don't know how to proceed. Any nudges in the right direction would be helpful. Thank you!

 

[micromass]

The maps ##w_i:W\rightarrow A_i## induce a unique map ##w:W\rightarrow \prod_{i\in I} A## such that ##l_i\circ w = w_i##. This is essentially by definition of the product in the category.

Now show that ##w## actually maps into ##A##, that is, that ##w(W)\subseteq A## and that it satisfies the universal property.

 

[Mandelbroth]

The maps ##w_i:W\rightarrow A_i## induce a unique map ##w:W\rightarrow \prod_{i\in I} A## such that ##l_i\circ w = w_i##. This is essentially by definition of the product in the category.

Now show that ##w## actually maps into ##A##, that is, that ##w(W)\subseteq A## and that it satisfies the universal property.

Let me see if I understand what you're saying. Let ##\pi_i: \prod A_j\to A_i## be the natural projection maps from the universal property defining the product. Then, ##\pi_i\circ w=w_i##. But, since the ##w_i## commute with the induced maps ##\mathcal{F}(\phi_{ik})##, and ##A## is defined as precisely the subset of ##\prod A_j## that does this, ##w(W)\subseteq A##. Let ##\rho: \prod A_j\to A## be a left inverse of the inclusion map ##i:A\to\prod A_j##, and let ##w'=\rho\circ w:W\to A##. This map ##w'## is unique because ##w(W)\subseteq A##, so the left inverse would take any element of ##w(W)## to its corresponding element of ##A##.

Is this correct?

 

[micromass]

Yes, seems right!