4 Corrected a mathematical error.

A great example is found in the famous 1976 Annals (Vol. 103, No. 1) paper of Deligne-Lusztig, "Representations of Reductive Groups over Finite Fields".

This paper begins (in Section 1) with the following:

Suppose that in some category we are given a family $(X_i)$ ($i \in I$) of objects and a compatible system of isomorphisms $\phi_{ji}: X_i \rightarrow X_j$. This is as good as giving a single object $X$, the 'common value' or 'projective limit' of the family. This projective limit is provided with isomorphisms $\sigma: X \rightarrow X_i$ such that $\phi_{ji} \sigma_i = \sigma_j$.

Where this is applied by Deligne-Lusztig, and how it relates to the question, is the following: In a connected reductive algebraic group $G$ over a field $k$, there is no canonical choice of maximal torus $T$. (A maximal torus in $GL_n$ corresponds to a choice of basis vectors up to scaling, so this is very similar to the original question). So Deligne-Lusztig consider the indexing set $I$ consisting of all pairs $(T,B)$ where $T$ is a maximal torus in $G$ and $B$ is a Borel subgroup containing $T$. For each $i = (T,B) \in I$, let $T_i = T$—the first entry in the ordered pair. For each pair $i,j \in I$, there is a unique isomorphism from $T_i$ to $T_j$ given by $\operatorname{Int}(g_{ij})$ (conjugation by $g_{ij}$) for some $g_{ij}$ satisfying $g_{ij} B_i g_{ij}^{-1} = B_j$. (Note that $g_{ij}$ and $\operatorname{Int}(g_{ij})$ are not unique, but the induced isomorphism from $T_i$ to $T_j$ is uniquely determined since maximal tori are normal in Borels, and Borels are self-normalising.$N_G(T) \cap B = T$ when $T \subset B$)

This indexing set $I$ makes all possible choices and the extra data of the Borel subgroup allows for the definition of "THE" maximal torus $T$, the projective limit of the system $(T_i)$.

I think this provides a good example to answer the original question. It also demonstrates how it is possible to make choices with extra data (in this case Borel subgroups in addition to tori), to "rigidify" and eventually define a universal choice.

3 Too much conjugation; why is the isomorphism well determined?; Int -> \operatorname{Int}

A great example is found in the famous 1976 Annals (Vol. 103, No. 1) paper of Deligne-Lusztig, "Representations of Reductive Groups over Finite Fields".

This paper begins (in Section 1) with the following:

Suppose that in some category we are given a family $(X_i)$ ($i \in I$) of objects and a compatible system of isomorphisms $\phi_{ji}: X_i \rightarrow X_j$. This is as good as giving a single object $X$, the 'common value' or 'projective limit' of the family. This projective limit is provided with isomorphisms $\sigma: X \rightarrow X_i$ such that $\phi_{ji} \sigma_i = \sigma_j$.

Where this is applied by Deligne-Lusztig, and how it relates to the question, is the following: In a connected reductive algebraic group $G$ over a field $k$, there is no canonical choice of maximal torus $T$. (A maximal torus in $GL_n$ corresponds to a choice of basis vectors up to scaling, so this is very similar to the original question). So Deligne-Lusztig consider the indexing set $I$ consisting of all pairs $(T,B)$ where $T$ is a maximal torus in $G$ and $B$ is a Borel subgroup containing $T$. For each $i = (T,B) \in I$, let $T_i = T$-- the —the first entry in the ordered pair. For each pair $i,j \in I$, there is a unique isomorphism from $T_i$ to $T_j$ given by $Int(g_{ij})$ \operatorname{Int}(g_{ij})$(conjugation by$g_{ij}$) for some$g_{ij}$satisfying$g_{ij} B_i g^{-1g_{ij}^{-1} = g_{ij} B_j g_{ij}^{-1}$B_j$. (Note that $g_{ij}$ and $Int(g_{ij})$ \operatorname{Int}(g_{ij})$are not unique, but the induced isomorphism from$T_i$to$T_j$is uniquely determined.determined since maximal tori are normal in Borels, and Borels are self-normalising.) This indexing set$I$makes all possible choices and the extra data of the Borel subgroup allows for the definition of "THE" maximal torus$T$, the projective limit of the system$(T_i)$. I think this provides a good example to answer the original question. It also demonstrates how it is possible to make choices with extra data (in this case Borel subgroups in addition to tori), to "rigidify" and eventually define a universal choice. 2 grammar/spelling. A great example is found in the famous 1976 Annals (Vol. 103, No. 1) paper of Deligne-Lusztig, "Representations of Reductive Groups over Finite Fields". This paper begins (in Section 1) with the following: Suppose that in some category we are given a family$(X_i)$($i \in I$) of objects and a compatible system of isomorphisms$\phi_{ji}: X_i \rightarrow X_j$. This is as good as giving a single object$X$, the 'common value' or 'projective limit' of the family. This projective limit is provided with isomorphisms$\sigma: X \rightarrow X_i$such that$\phi_{ji} \sigma_i = \sigma_j$. Where this is applied by Deligne-Lusztig, and how it relates to the question, is the following: In a connected reductive algebraic group$G$over a field$k$, there is no canonical choice of maximal torus$T$. (A maximal torus in$GL_n$corresponds to a choice of basis vectors up to scaling, so this is very similar to the original question). So Deligne-Lusztig consider the indexing set$I$consisting of all pairs$(T,B)$where$T$is a maximal torus in$G$and$B$is a Borel subgroup containing$T$. For each$i = (T,B) \in I$, let$T_i = T$-- the first entry in the ordered pair. For each pair$i,j \in I$, there is a unique isomorphism from$T_i$to$T_j$given by$Int(g_{ij})$(conjugation by$g_{ij}$) for some$g_{ij}$satisfying$g_{ij} B_i g^{-1} = g_{ij} B_j g_{ij}^{-1}$. (Note that$g_{ij}$and$Int(g_{ij})$are not unique, but the induced isomorphism from$T_i$to$T_j$is uniquely determined.) This indexed indexing set$I$includes all maximal tori -- it makes all possible choices -- and the extra data of the Borel subgroup allows for the definition of "THE" maximal torus$T$-- T$, the projective limit of the system $(T_i)$.

I think this provides a good example to answer the original question. It also demonstrates how it is possible to make choices with extra data (in this case Borel subgroups in addition to tori), to "rigidify" and eventually define a universal choice.