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I am trying to find category theoretic definitions for different possible notions of a rank of a subset (or a subfamily) of an algebraic structure, and to figure out if any of them would be in some sense dual to one another, and whether there is the "most natural" one among them is some sense. For example, for subsets of vector spaces, there is only one notion of a rank, but it still can be defined in different ways.

Let $A$ be an algebraic structure (call it an algebra) and $S$ be a subset of the underlying set of $A$ (or a family of elements of $A$). I can think of several different notions of a rank of $S$ in $A$:

  1. the minimal cardinality of a set generating the subalgebra generated by $S$,

  2. the minimal cardinality of a subset of $S$ generating the subalgebra generated by $S$,

  3. the maximal cardinality of a subset of $S$ which does not satisfy any nontrivial relation (a trivial relation is a relation satisfied by any $n$ elements of the algebra),

  4. the maximal cardinality of a subset of the subalgebra generated by $S$ which does not satisfy any nontrivial relation.

(The are other possibilities, restricting the notion of a "trivial relation" to only those that are satisfies by all elements of all algebras in a given class.)

I would naively expect these definition to be easy to formulate in category theoretic language, using the adjunction from sets to algebras defined on them, but i only succeed in translating definitions to categoric language "word by word," which makes them not easier but more complicated.

For example, let us denote $\mathbf{Alg}_\Sigma$ the category of algebras of signature $\Sigma$. Let $G$ be the forgetful functor $\mathbf{Alg}_\Sigma\xrightarrow{G}\mathbf{Set}$ and $F$ be the functor $\mathbf{Set}\xrightarrow{F}\mathbf{Alg}_\Sigma$ that builds free algebras on given sets. (Then $F$ and $G$ form a pair of a left adjoint and a right adjoint.) Let $A$ be an algebra (with the underlying set $GA$) and $S\xrightarrow{g}GA$ be a family of elements of $A$. Let $FS\xrightarrow{f}A$ be the morphism in $\mathbf{Alg}_\Sigma$ associated to $g$ (with respect to $S$ and $A$).

Then i may define the rank of $g$ in the sense (1) as the minimal cardinality of a set $P$ such that there exists a commuting diagram \begin{array}{ccc} S & \xrightarrow{g} & GV \\ FS & \xrightarrow{f} & V \\ \text{epi}\searrow && \nearrow\text{mono} \\ & U \\ & \uparrow\text{epi}\hspace{-14pt} \\ & FP \end{array}

Similarly, i may define the rank of $g$ in the sense (4) as the maximal cardinality of a set $Q$ such that there exists a commuting diagram

\begin{array}{ccc} S & \xrightarrow{g} & GV \\ FS & \xrightarrow{f} & V \\ \text{epi}\searrow && \nearrow\text{mono} \\ & U \\ & \uparrow\text{mono}\hspace{-24pt} \\ & FQ \end{array}

These definitions do not look good to me at all.

Question:

Do there exist any standard, or concise, or good in some other ways, category theoretic versions of these or other definitions of rank?

Update: probably it makes sense to restrict my question to Abelian categories. So,

does there exist a category theoretic generalization of the notion of the rank of a subset of a vector space to other Abelian categories?

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So, what is your question? –  Zhen Lin Feb 20 at 16:04
    
Do good category theoretic definitions of rank exist and what are they? –  Alexey Feb 20 at 17:04
    
The upshot is that you should work with the usual notions of rank, and not try to make them category-theoretic unless you really have to. –  Martin Brandenburg Feb 20 at 20:26
    
@MartinBrandenburg, i hoped that Category Theory would help me to decide which definition of the rang of a family of vectors in a vector space is the most natural one. This is for teaching a linear algebra course :). –  Alexey Feb 20 at 21:09
1  
No, the rank of an abelian group $A$ is unambiguously defined as the maximal cardinality of a $\mathbb{Z}$-linear independet subset, or equivalent as the $\mathbb{Q}$-dimension of $A \otimes \mathbb{Q}$. This heavily differs from the minimal cardinality of a generating set. I don't know any serious books which call the latter the rank. –  Martin Brandenburg Feb 20 at 22:57

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