I'm studying cones, limits and their dual versions in category theory, but some doubts remain. This is about equalizer and perhaps co-equalizer as well in a Monoid seen as a category.

Consider the category named NAd, which is the monoid < $\mathbb{N}$, +, 0 > considered as a category, where there is only one object, let's name it $*$, and the morphisms are all the natural numbers, with composition being the sum and identity the morphism 0.

1) Are there equalizers for any pair of morphisms in this category? It seems to me that there will always be an equalizer, because there is only one object ($*$), thus all morphisms will be $* \rightarrow *$. But I don't know if the composition plays any tricks here making my thinking fail.

2) Regarding the answer to the first question, is there any special case when considering the co-equalizer?

Thanks in advance,


Equalizers in $\mathrm{NAd}$ only exist when you consider the same arrow twice, and then they must be isomorphisms, and thus it must be $0$. Indeed, if $e$ is an equalizer for two natural numbers $m$ and $n$, then $e+m=e+n$, and thus $m=n$; and then, since $0+m=m=n=0+n$, there must be some natural $t$ such that $0=t+e$, and in particular $e=0$.

As for coequalizers, since addition in $\mathbb{N}$ is commutative, $\mathrm{NAd}$ is isomorphic to $\mathrm{NAd}^{op}$, and thus the same result holds.

However this relies a lot on the properties of $\mathbb{N}$ as a monoid (namely the fact that it is cancellative); (co-)equalizers could very well exist in other monoids seen as categories.


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