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I find it is hard to understand the "create limit." (You can find it in Mac Lane's Categories for the working mathematician, P112; there it defines: "A functor $V:A→x$ creates limits for a functor $F:J→A$.")

Here are my questions:

(1) Why does it use "a functor $F:J→A$," and how does it match with the common limit in the algebra (before category)?

2) Does the "creates limits" always exist? If not, please give me some examples.

3)How does the forgetful functor create limits? Take the the forgetful functor $U:Grp→Set$, for example; what is "the functor $F:J→A$"?

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What do you mean by "how does it march with the common limit in the algebra"? – t.b. May 26 '11 at 10:44
I think you've parsed the sentence incorrectly. The verb phrase is "create limits for a functor $F : J \to A$", meaning it creates a specific kind of limit. I have no idea how to parse your second question though. – Zhen Lin May 26 '11 at 11:55
I can understand the limit in the algebra,for example in the book Advanced Modern Algebra by Rotman,but in MacLane "s category book,it seems strange,so I want to know if they are the same thing and what is the differences between them. – Strongart May 28 '11 at 10:35
up vote 1 down vote accepted

IMO you are wondering, how limits in analysis relate to limits in category theory. AFAIK they are not related — simple. Instead, limits in category theory are a generalization of the greatest lower bound in a preorder.

At elementary level limits are defined for “diagrams” (you should know what are they). But every functor can be interpreted as a diagram. So people just talk about limits for functors. The functor $F:J\to A$ is interpreted as the diagram in the category $A$. $J$ is the category called a “scheme” of the diagram $F$. E.g. the scheme of binary product diagrams is the free category over the graph consisting of 2 vertexes and no edges.

It is important that functors map not only objects and morphisms, but also diagrams. This is obvious, because a diagram is a bunch of objects and morphisms. $V:A\to X$ maps $F$ to $V(F)$. $F, V(F)$ have the same scheme. Morphisms in $F$ commute, $F$ preserves compositions of morphisms, then morphisms in $V(F)$ commute, then $V(F)$ is a diagram. If you look at $F$ as a functor, then $V(F)$ is actually $V\circ F$.

“$V:A\to X$ creates limits” means that for every diagram $F$ in $A$ and every limit $\tau$ of $V(F)$ there exists a unique limit of $F$ which is a preimage (along $V$) of $\tau$. E.g. if we choose a particular limit $F$ — the binary product diagram of some groups $G_0, G_1$, we choose a particular functor $V$ — the forgetful functor $Grp\to Set$, and $V(G_0)\times V(G_1)$ — the Cartesian product of the carriers (underlying sets) of those groups — is the limit of $V(F)$, then there exists a unique group with the carrier $V(G_0)\times V(G_1)$ (it has this carrier because it is the preimage of the set $V(G_0)\times V(G_1)$), and and that group is the categorical product of $G_0, G_1$ with projections $\pi_0:V(G_0)\times V(G_1)\to V(G_0), \pi_1:V(G_0)\times V(G_1)\to V(G_1)$ understood as group homomorphisms. I.e. that a Cartesian product of carriers can be extended to a categorical product of groups, which is called also a direct product of groups.

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Thank you,that help me a lot.Here is my note,take the forgetful functor U:Grp→Set as an example,J is the graph or diagram or arrow picture,U creating limit means the limit in the Grp can be induced by the limit in the Set,because the graph or diagram or arrow picture are the same. Does it agree with you? – Strongart May 29 '11 at 10:30

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