The Unapologetic Mathematician

Mathematics for the interested outsider

Limits of sets and Creation of limits

We know that the category \mathbf{Set} has equalizers and all (small) products, so by the existence theorem we know it is complete. However, it will be useful to have an explicit calculation of all small limits at hand. We’ll find this by walking through the existence theorem in this category.

Given any small category \mathcal{J}, a functor F:\mathcal{J}\rightarrow\mathbf{Set} is a collection of sets F(J) and functions F(u):F(J)\rightarrow F(K). We can take the product of all the sets in the collection \prod_JF(J), consisting of all lists of elements x=\{x_J\}_{J\in\mathrm{Ob}(\mathcal{J})} with each x_J\in F(J). It comes with projections \pi_J:\prod_KF(K)\rightarrow F(J), but the triangles formed by these projections might not commute — \pi_K is not equal to F(u)\circ \pi_J in general.

So take the subset L of the product consisting of all those lists \{x_J\}_{J\in\mathrm{Ob}(\mathcal{J})} with \left[F(u)\right](x_J)=x_K for all morphisms u of \mathcal{J}. Then we have functions \lambda_J:L\rightarrow F(J) defined by \lambda_J(x)=x_J which do make the required triangles commute, giving a cone on F. If \mu_J:M\rightarrow F(J) is any other cone on F then each m\in M determines a list x(m)=\{\mu_J(m)\} which lies in L, which determines a unique function x:M\rightarrow L.

Now such a list \{x_J\} is a list of elements of the various F(J), and when it comes right down to it we just don’t like thinking about elements if we don’t have to. Instead of talking about an element x of a set X, we can talk about an arrow x:*\rightarrow X, where {*} is a terminal object in \mathbf{Set} — a set with one point. So what we really have here is a list of arrows x_J:*\rightarrow F(J) with F(u)\circ x_J=x_K, which is a cone on F!

So here’s the result of our calculation: The limit set \varprojlim_\mathcal{J}F is the set \mathrm{Cone}(*,F) of all cones from {*} to F in \mathbf{Set}. Further, we see that this set always exists, so \mathbf{Set} is complete.

We could do a similar calculation in another category like \mathbf{Grp}, but we actually don’t have to reproduce all this work. Instead, we can consider the functor U:\mathbf{Grp}\rightarrow\mathbf{Set} which assigns to every group its underlying set. If F:\mathcal{J}\rightarrow\mathbf{Grp} is a functor, then the composite U\circ F:\mathcal{J}\rightarrow\mathbf{Set} has a limit L=\mathrm{Cone}(*,U\circ F). I claim that there is a unique group structure on this set so that the arrows \lambda_J:L\rightarrow U(F(J)) are actually group homomorphisms to the groups F(J), and that this cone in \mathbf{Grp} is a limiting cone.

Since the projection \lambda_J:L\rightarrow F(J) just reads off the J component from the list, we see that for it to be a homomorphism we have to use the product from F(J). That is, \lambda_J(\{x_J\}\{y_J\})=\lambda_J(\{x_J\})\lambda_J(\{y_J\})=x_Jy_J, so the J component of the product must be x_Jy_J, using the product from F(J). Similarly the J component of \{x_J\}^{-1} must be x_J^{-1}, using the inversion from F(J).

Now if \gamma_J:G\rightarrow F(J) is any cone in \mathbf{Grp} then U(\gamma_J):U(G)\rightarrow U(F(J)) is a cone in \mathbf{Set}, so U(\gamma_J)=\lambda_J\circ h for some unique function G\rightarrow L. We can check that

\lambda_J(h(g_1g_2))=\gamma_J(g_1g_2)=\gamma_J(g_1)\gamma_J(g_2)=\lambda_J(h(g_1))\lambda_J(h(g_2))=\lambda_J(h(g_1)h(g_2))

so h must also be a group homomorphism, proving the required universal property.

In general we say that a functor G:\mathcal{C}\rightarrow\mathcal{D} creates a limit for a functor F:\mathcal{J}\rightarrow\mathcal{C} if for every limiting cone \lambda_J:L\rightarrow G(F(J)) in \mathcal{D} there exists a unique cone \mu_J:M\rightarrow F(J) in \mathcal{C} with L=G(M) and \lambda_J=G(\mu_J), and further this cone is a limiting cone. The above theorem can then be stated that the functor U:\mathbf{Grp}\rightarrow\mathbf{Set} creates all limits. As a consequence, \mathbf{Grp} is complete.

A similar proof to the preceding theorem can be given to show that the “underlying set” functors from \mathbf{Ab}, \mathbf{Ring}, R\mathbf{-mod}, \mathbf{mod-}R, and other algebraic categories all create all (small) limits, and thus those categories are complete. This also shows that the group (ring, module, etc.) structure on a product (equalizer, pullback) is completely determined by the limit on the underlying sets.

June 21, 2007 Posted by | Category theory | 8 Comments

   

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