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

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.

## 2 Answers

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.

• Thank you very much! This is very helpful and nearly what I am looking for! Is there a way to represent the objects of the coslice category (which are the arrows pointing away from the original object in question) as Morphisms on a category with only one object? Because what I hoped to find is that such a category would then correspond to e.g. the Lie Group in my example above. – exchange Nov 25 '17 at 18:01
• @exchange: A category with one object is precisely a monoid. (And if all its morphisms are isomorphisms, then it's a group, not just a monoid!) In a general setting, I don't believe there are any natural ways of interpreting coslice categories as monoids (let alone groups), although there may be some instances when it is possible, like in the setting you're interested in. – Clive Newstead Nov 26 '17 at 7:24

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

• Thank you very much! This answer is also very helpful! I now accepted Clive's answer because it might be a bit closer to what I was looking for but the ideas to look at the Yoneda Lemma and enriched categories are certainly great too! – exchange Nov 27 '17 at 17:52