Instantiate spaces in commutative diagram by "appropriate" elements - name of this idea? 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?
 A: 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:


*

*the object is a set;

*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. :)
