Identity elements for monoidal categories

An identity element for a monoidal category depends of course on both the category and the monoidal structure.

Here we look at several examples.

(Just for linguistic variety, sometimes we will call an identity element a "neutral element".)

$\newcommand\calJ{{\mathcal J}}$

Most familiar probably is the category of sets.

For its coproduct structure, i.e. disjoint union, the identity is the empty set $\emptyset : X \cup \emptyset = X$ for all $X\in\Set$.

I.e. $X$ $+$ nothing is still $X$.

For its categorical product, the cartesian product, the neutral element is any one-element set: $1\times X \cong X$, where $1$ is any one-element set and $X$ is an arbitrary set.

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More complex examples are given by (the three monoidal structures on $(\Set\downarrow\N) \simeq [\N,\Set]$), 

namely (the coproduct), (cartesian product), and (the $\circ$ binary operator defined by Kelly in his MVFC).

As in any functor category, the first two identity elements are just the functions (functors) $\N\to\Set$ which are constant respectively at $\emptyset\in\Set$ and $1\in\Set$, where $1$ can be any one-element set (often $1$ is chosen to be $\{\emptyset\} = \{0\}$).

The identity element for (the $\circ$ binary operator), given as an overset, has a single element over $1\in\N$, i.e. of arity or rank $1\in\N$ and nothing over the rest of $\N$.

There clearly is a choice for what to call that element; Kelly, with a view towards his intended application, denotes it $\bfone$ (bold face 1).

We should view it as (a formal symbol) which will be interpreted as the identity function (or arrow) in the semantics of structures.

As to the overset, the identity for $\circ$, Kelly in MVFC denotes it $\calJ$ (calligraphic J).