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Two singleton sets give canonically isomorphic representations of the identity functor on $S e t$

Example

For distinct sets $s_{0}$ and $s_{1}$ , the singleton sets $S_{0} = {s_{0}}$ and $S_{1} = {s_{1}}$ both represent the identity functor on $S e t$ . Their universal elements are $s_{0} \in S_{0}$ and $s_{1} \in S_{1}$ , and the unique function

$i : S_{0} ⟶ S_{1}, i (s_{0}) = s_{1},$

is the canonical isomorphism of these representations.

Facts & Assumptions

Given: The category $S e t$ , the singletons $S_{0}, S_{1}$ , and its identity functor.

[F1]

Sets and functions form a category under ordinary identity functions and composition (Sets and functions form the large locally small category $S e t$ ).

[F2]

A covariant representation of $F$ is a natural isomorphism $S e t (R, -) ≅ F$ (Presheaves, covariantly and contravariantly representable functors, and representations).

[L1]

Two covariant universal elements for the same functor have a unique isomorphism $i$ satisfying $F (i) (u) = u^{'}$ (Representing objects are unique up to a unique isomorphism compatible with their universal elements).

[F3]

A function assigns exactly one value to each domain element, and two functions with the same domain and codomain are equal exactly when their values agree everywhere (A function is a relation $f$ with $(a, b) \in f$ and $(a, c) \in f$ implying $b = c$ ; $f : A \to B$ , the value $f (a)$ , domain and codomain, Functions $f$ and $g$ are equal if and only if $dom f = dom g$ and $f (x) = g (x)$ for every $x$ in that common domain).

Verification

technique · constructive

1.1

For $k \in {0, 1}$ and every set $X$ , define $E_{X}^{k} : S e t (S_{k}, X) \to X$ by $E_{X}^{k} (f) = f (s_{k})$ . Its inverse sends $x \in X$ to the function $S_{k} \to X$ with value $x$ at $s_{k}$ .

F3construct

2.1

The two formulas in step 1.1 are inverse by [F3]. If $g : X \to Y$ , then $E_{Y}^{k} (g \circ f) = g (E_{X}^{k} (f))$ , so the bijections are natural by [F1].

step 1.1F1F3

3.1

By [F2], both $S_{0}$ and $S_{1}$ represent the identity functor; their universal elements are the values of the identity functions, namely $s_{0}$ and $s_{1}$ . The conclusion remains valid at $X = \emptyset$ , where both sides of each component bijection are empty.

step 1.1step 2.1F2

4.1

The function $i : S_{0} \to S_{1}$ with $i (s_{0}) = s_{1}$ carries the first universal element to the second. By [L1], it is the unique compatible isomorphism; explicitly its inverse is the unique map $S_{1} \to S_{0}$ .

step 3.1L1F3

5.1

Thus the word canonical refers to compatibility with the chosen universal points, not merely to the fact that the underlying singleton sets happen to be isomorphic.

step 4.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 43 results over 19 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

Emily Riehl, Category Theory in Context, Example 2.1.5(i) and Corollary 2.3.2 (standard reference, not scraped)

Two singleton sets give canonically isomorphic representations of the identity functor on SetSet