Equivalent characterizations of group objects

Question

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 $*$.

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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.

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

Answer 1

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 $\mathcal C$.