In a way of defining a category called as single-sorted definition of a category which uses only one collection (representing the collection of morphisms) and thus is formulated as an untyped (or 1-sorted) first-order theory, the basic idea is that an object can be identified with its identity morphism.
Definition
A category (single-sorted version) is a collection C, whose elements are called morphisms, together with two functions s,t:C→C and a partial function ∘:C×C→C, such that:
- s(s(x))=s(x)=t(s(x))
- t(t(x))=t(x)=s(t(x))
The elements of their common image (the x such that s(x)=x, or equivalently t(x)=x) are called identities or objects.
https://ncatlab.org/nlab/show/single-sorted+definition+of+a+category
At the same time, the nLab article also mentions:
Specializations
A monoid is a single-sorted category in which s is a constant function (hence so is t, and they are equal). This works up to isomorphism of categories, not merely equivalence, so single-sorted categories may seem to be a more direct oidification of monoids than the usual categories.
I understood s and t are identity morphism s,t:C→C, but here it claims they are constant functions, and I feel it does not make sense.
Does this exactly mean, function composition such as
(C→5)∘(C→C)??
I am still confused.
In addition, for the mention:
This works up to isomorphism of categories, not merely equivalence, so single-sorted categories may seem to be a more direct oidification of monoids than the usual categories.
I don't quite understand the meaning.
Can someone who is familiar with single-sorted definition of a category explain? Thanks.
Perhaps, this topic is related to a Q&A: The $2$-category of monoids
I don't know what you mean by this. The intended interpretation is that $C$ is the collection of all morphisms in a category and $s, t$ send a morphism $f : x \to y$ in the ordinary sense to the identity endomorphism of its source $s(f) = \text{id}_x$ and target $t(f) = \text{id}_y$ respectively.
If we have a monoid $M$, interpreted as a one-object category $BM$ (so there is one object $\bullet$ and it has endomorphisms $M$), then both $s$ and $t$ are necessarily constant, and their constant value is the identity endomorphism $\text{id}_{\bullet}$ of the unique object $\bullet$ of $BM$. Is it clear now?