Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

I wonder whether the following concept has a name.

Suppose you are given a commutative diagram $\mathcal C$, that we think of a small category where each hom-class (i.e. hom-set) consists of at most one-arrow, and composition of morphisms obeys the obvious laws - we put in every possible composition into the diagram.

Say, we have objects $X_i \in \mathcal C$ and for two objects an arrow $X_i \rightarrow_{\phi_{ij}} X_j$. Let us now choose, if possible, elements $x_i \in X_i$ within this diagram, such that for all $i,j$ we have $x_j = \phi_{ij}( x_i )$.

If such a choice of elements exists, it is a functor from $X_i \rightarrow_{\phi_{ij}} X_j$ to $x_i \rightarrow_{\phi_{ij}} x_j$ (abuse of notation).

Does this construction have a name?

share|improve this question
    
I'm not sure if I understand your question correctly, but it seems to me that the "elements" you're asking about are parameterized by the (inverse=projective) limit $\varprojlim\limits_\mathcal{C}\,X_i$ (if it exists). –  t.b. Jun 23 '11 at 0:21
    
The $x_i$ are, of course, supposed to live in the corresponding $X_i$. I added that - indeed, if the diagram is acyclic, then the choice of elements seems to be an element of the inverse limit as defined in the wikipedia article. I don't know whether there is such a construction if the diagram has cycles. –  Martin Jun 24 '11 at 14:29
1  
The sentence "If such a choice exists..." does not make much sense to me, Martin. Can you expand on it? –  Mariano Suárez-Alvarez Jul 5 '11 at 15:54
add comment

1 Answer

up vote 1 down vote accepted

First you need to embrace the standard way of thinking about diagrams. A diagram $X$ is a functor from some category $\mathcal J$ to $\mathcal C$, $\mathcal J$ defines the type of the diagram $X$. $\mathcal J$ is your “small category”, but it may contain a hom-set with 2 elements, e.g. if the type of diagrams is “equalizer”.

You are talking about elements of an object, which implies one of:

  1. the object is a set;
  2. elements are actually generalized elements.

1 with $\mathcal C=$Set is a particular case of 2, i.e. in Set a set of $1$-valued elements of $S$ is isomorphic to $S$.

Now lets work in Set and talk about your family of elements $x_i$, which I like to call a “coherent fiber of $X$”. For all $i$, we can replace $x_i$ with a $1$-valued element of $X_i$, i.e. with a corresponding function from $1$ to $X_i$. Then the set of coherent fibers of $X$ is isomorphic to the set of cones from $1$ to $X$. The property of being a cone is identical to $\forall i\forall j(x_j = \phi_{ij}(x_i))$. “Cone” is the name you are asking.

Let $L$ be a limit of $X$. The definition of limits says that the set of cones from $1$ to $X$ is isomorphic to $\operatorname{hom}(1,L)$, the set of $1$-valued elements of $L$. See also the standard construction of a limit in Set, an element of such a limit is just that family $x_i$. “Element of a limit” is also the name you are asking.

But you want to think of it as a functor. O'kay, hold your breath. $\operatorname{const}$ is a diagonal functor. Remember that cones from $A$ to $X$ are natural transformations of type $\operatorname{const}(A)\to X$. Those natural transformations are functors from $\mathcal J$ to the comma category $\operatorname{const}(A)\downarrow X$ which are sections for both forgetful functors from $\operatorname{const}(A)\downarrow X$. ($\operatorname{const}(A):\mathcal J\to\mathcal C$.) Because we are interested in the case $A=1$, and $\operatorname{const}(1)\downarrow X$ is similar to the category of elements of $X$, then coherent fibers of $X$ are functors from $\mathcal J$ to the category of elements of $X$ which are sections to the forgetful functor.

To work with $A$-valued elements ($A$ is an object of $\mathcal C$) in place of $1$-valued elements, replace in the above $1$ with $A$ and “the category of elements of $X$” with “the category of $X$-structured morphisms with domain $A$”.

Sincerely, I prefer to stop somewhere near “limit” in the above. :)

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.