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There are many categories whose objects can be consider as categories, intuitively I think there is an embedding. However, the usual 'embedding functor' can only enlarge the category (I think). For example, if $\mathcal{G}$ is a groupoid with only one object, how to 'embed' $\mathcal{G}$ into Grp, both as categories (using functors or other things). Is it possible?

This problem arised when I try to consider Yoneda lemma as generalization of Cayley's theorem. Yoneda lemma only claims that there is a bijection between sets, I'm wondering how to make it become isomorphism between groups using language in category theorem. In this case, we need to get the isomorphism, which is an invertible arrow in Grp, from a fully faithful functor $Y:\mathcal{G} \rightarrow Set^{\mathcal{G}^{op}}$, whose arrow function gives the bijection $hom_\mathcal{G}(\cdot,\cdot)\rightarrow Nat(\mathcal{G}(-,\cdot),\mathcal{G}(-,\cdot))$. Although we can easily see it is an isomporphism by property of functor, how to convert it into Grp? Especially, if we already know $Y:\mathcal{G}_1 \rightarrow \mathcal{G}_2$ is a functor then how to make Y an arrow in Grp?

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  • $\begingroup$ The $\mathcal Y : \mathcal G \to \mathbf{Set}^{\mathcal G^{op}}$ is a functor. Its action on arrows satisfies the functor laws. The action on arrows of a functor between one-object categories is a monoid homomorphism. A monoid homomorphism between groups is a group homomorphism. $\endgroup$
    – Derek Elkins left SE
    Feb 6, 2018 at 10:47
  • $\begingroup$ @DerekElkins I'm just wondering how to express "The action on arrows of a functor between one-object categories is a monoid homomorphism." I want to know how to get this identification explicitly if considering monoid homomorphism as arrows in Mon. $\endgroup$
    – shazitaba
    Feb 6, 2018 at 10:59
  • $\begingroup$ Category theorists don't usually try to avoid the background universe of ordinary set theory. They will usually state it with $\in$ like anybody else. You could always talk about points $1\to\mathbf{Grp}$ instead, but it doesn't get you much. $\endgroup$
    – Malice Vidrine
    Feb 6, 2018 at 11:02

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It is not very clear to me what you mean by

... describe belong to in categorical terms.

It is true that you could define a categorical set theory (see ETCS for instance) that define a membership relation in the language of categories (arrows, composition, source, etc). Nevertheless this notion of membership differs from the one of traditional set theories.

But I am not sure if that is what you are looking for.

About your problem with the Yoneda lemma: what is the generalization of Cayley theorem is actually a corollary of Yoneda lemma, namely that the Yoneda embedding is...an embedding (i.e. a fully faithful functor injective on the objects).

This basically says that you can embed every category in a category of presheaves, as Cayley theorem tells you that you can embed every group in a permutation group.

Hope that this helps.

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