Is there a notion in Category Theory that refers to the structure of the arrows which point away from an object?

Question

In a category, a lot of structure can be encoded in the set of arrows that point away from an object. For example, if I let the objects be tensors on some vector space and the Morphisms actions of a Lie Group on that vector space under which the tensors transform, then all the arrows that point away from a tensor encode the information of the whole Lie Group (e.g. all possible ways in which a tensor could be transformed).

However, the "arrows that point away from an object" can not necessarily be comprehended as the elements of a group, it could be elements of some other structure or they could not admit any structure. Therefore I am looking for a category theoretical definition that can help to capture what structure is encoded in the arrows pointing away from an object.

Furthermore, such a notion shall be suitable, if possible, to compare structure of away-pointing arrows of objects of different categories, i.e. to check whether two such sets are equivalent in a structure preserving sense. For example if I would construct a functor from the above mentioned tensor category into another category, I would like to be able to check whether the group structure of away-pointing arrows can be preserved or not.

Any ideas or references to a correlated notion would be helpful. Thanks.

Answer 1

Given a category $\mathcal{C}$ and an object $A$, the 'arrows pointing out of $A$' form a category $A/\mathcal{C}$, called the coslice category of $\mathcal{C}$ under $A$.

The objects of $A/\mathcal{C}$ are morphisms $x : A \to X$ in $\mathcal{C}$, and a morphism $f : (x : A \to X) \to (y : A \to Y)$ in $A/\mathcal{C}$ is a morphism $f : X \to Y$ in $\mathcal{C}$ such that $f \circ x = y$: $$\begin{matrix} &&A&&\\ &{^x}\swarrow && \searrow{^y}&\\ X&&\underset{f}{\to}&&Y \end{matrix}$$

There are lots of examples of coslice categories that arise naturally, for example the category $\mathbf{Top}_*$ of pointed topological spaces is equivalent to the coslice $1/\mathbf{Top}$, where $1$ is the one-point space.

Answer 2

Firstly, let me mention that the fact that an object can be described by the arrows pointing away from it is essentially the Yoneda Lemma : $A\mapsto Hom(A, -)$ is an embedding.

Secondly, maybe what you're looking for is the notion of a slice (or comma) categories, in which case you can look at Clive Newstead's answer (or the first comment)

But I think you may also be interested in enriched categories: the way you mention the structure of the arrows pointing out of $A$ makes me think of defining $Hom$ not as a set, but for instance as a group, or a vector space,...; which is precisely what enriched categories are about : see https://en.wikipedia.org/wiki/Enriched_category for instance