# Equivalent characterizations of group objects

Let $\mathcal C$ be a category. In our lecture, a group object in $\mathcal C$ is defined as

• an object $c ∈ \mathcal C$,
• interpreted by a contravariant functor $L \colon \mathcal C^\mathrm{op} → \mathrm{Grp}$,
• such that$?L = \mathrm{mor}_\mathcal C (–,c)$,

where $? \colon \mathrm{Grp} → \mathrm{Set}$ is the forgetful functor and $\mathrm{mor}_\mathcal C (–,c)$ is the contravariant homfunctor at $c$.

Now, our exercise is to show the equivalence of this definition with the usual one involving an object $G ∈ \mathcal C$, a multiplication morphism $μ \colon G × G → G$, an inversion morphism $ι \colon G → G$ and the neutral morphism $η \colon * → G$, assuming $\mathcal C$ has all products and a terminal $*$.

This is what I got: The definition in our lecture reads that $c$ is a group object if any set of morphisms into $c$ can be interpreted as a group (e.g. by pointwise multiplication for $\mathcal C = \mathrm{Set}$).

So, of course, if I take the usual definition, I can indeed define a pointwise multiplication on any mor-set by postcomposing $μ$ (i.e. $f·g := μ(f×g))$. This seems to be the right way for interpreting a group object by the usual definition as a group object by our definition.

But how, on the other hand, can I interpret a group object by our definition as a group object by the usual definition? My initial idea was to try to somehow lift the group structure of $Lc$ to $c$, but I don’t know how I can do this. I mean, I have to construct arrows $μ$, $η$ and $ι$, but I don’t even know if $L$ is full or not.

Any hints are greatly appreciated.

In the meantime, I actually have found the idea I was looking for myself. (What should now be done to this question?)

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If you've found the answer yourself, you should be able to post an answer your own question. –  Hurkyl May 1 at 21:06
@Hurkyl It may be that others visiting this site still need to do the exercise. On the other hand, … 11 views. –  k.stm May 1 at 21:10
When you asked the question, you did not care much for others that still might need to do the exercise, did you? –  Mariano Suárez-Alvarez May 1 at 21:54
@MarianoSuárez-Alvarez That’s not the point. Homework questions are ideally answered individually, responding to the points where people get struck. Of course, I can address these points pretty well myself now, but nevertheless – it feels a bit pointless to give an individual answer or “hints” to myself. The alternative would be to give a full answer, as it is suggested for answering one’s own question, but this (at least generally) would mean giving away a solution to a pending assignment which I think should be discouraged as well. On the other hand, … who cares? –  k.stm May 1 at 22:06
Well, we care: keeping unanswered questions litters the site. You probably know now what hint could have helped you solve the problem: add that as an answer. You are much less unique than what you think, and there is a great chance that someone in the future may be stuck at the same point and to whom the same hint that would have been useful to you will also be useful. –  Mariano Suárez-Alvarez May 1 at 22:08

Misleadingly, I was vaguely thinking about lifting corresponding $μ$s, $ηs$ and $ι$s in $\mathrm{Grp}$ to $\mathcal C$ by using the functor $L$ in some way.

But that’s not a good way to approach the problem, especially since there are no such morphisms living there naturally (they would be group homomorphisms then).

One needs to create morphisms $μ \colon c × c → c$, $η \colon * → c$ and $ι \colon c → c$ which are, of course, elements of $\mathrm{mor}_{\mathcal C}(c×c,c)$, $\mathrm{mor}_{\mathcal C}(*,c)$ and $\mathrm{mor}_{\mathcal C}(c,c)$, all of which are sets which carry a group structure via $L$ by hypothesis.

Now, one can use the fact that these groups admit morphisms, corresponding to the usual definitions,

• $μ' \colon \mathrm{mor}_{\mathcal C}(c×c,c) × \mathrm{mor}_{\mathcal C}(c×c,c) → \mathrm{mor}_{\mathcal C}(c×c,c)$,
• $η' \colon • → \mathrm{mor}_{\mathcal C}(*,c)$ (where $•$ is an terminal object in $\mathrm{Set}$), and
• $ι' \colon \mathrm{mor}_{\mathcal C}(c,c) → \mathrm{mor}_{\mathcal C}(c,c)$.

By evaluating them (these are in $\mathrm{Set}$, so they are maps) respectiely at suitable elements, one gets morphisms with the correct signatures for $μ$, $η$, and $ι$ as morphisms in $\mathcal C$.

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