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category theory

Contents

Definition

An endomorphism of an object $x$ in a category $C$ is a morphism $f : x \to x$.

An endomorphism that is also an isomorphism is called an automorphism.

Properties

Given an object $x$, the endomorphisms of $x$ form a monoid under composition, the endomorphism monoid of $x$:

$End_C(x) = Hom_C(x,x) ,$

which may be written $End(x)$ if the category $C$ is understood. Up to equivalence, every monoid is an endomorphism monoid; see delooping.

An endomorphism monoid is a special case of a monoid structure on an end construction. Let $d:D\to C$ be a diagram in $C$, where $C$ is a monoidal category (in the case above the monoidal structure is the cartesian product and $d$ is a constant diagram from the initial category). One defines $End(d)$ as an object in $C$, equipped with a natural transformation $a: End(d) \otimes d \to d$ which is universal in the sense that for all objects $Z \in C$, and any natural transformation $f: Z \otimes d \to d$ there exists a unique morphism $g: Z \to End(d)$ such $a \circ (g \otimes d) = f: Z \otimes d \to d$.

Proposition

If the universal object $(End(d),a)$ exists then there is a unique structure of an internal monoid $\mu: End(d) \otimes End(d) \to End(d)$, such that the map $a: End(d) \otimes d \to d$ is an action.

Proposition

In a cartesian monoidal category $C$, if an endomorphism monoid $End(c)$ for an object $c: 1 \to C$ exists and is commutative, then $c$ is a subterminal object.

Proof

Let $k: c \to End(c)$ correspond to first projection $\pi_1: c \times c \to c$. Then the composition

$c \times c \stackrel{k \times k}{\to} End(c) \times End(c) \stackrel{comp}{\to} End(c)$

(where $comp$ denotes internal composition) may be computed to be $k \pi_1$, corresponding to first projection $\pi_1: c \times c \times c \to c$. Thus, assuming commutativity of $End(c)$ and letting $\sigma$ generally denote a symmetry map, consideration of the diagram

$\array{ c \times c & \stackrel{k \times k}{\to} & End(c) \times End(c) & \stackrel{comp}{\to} & End(c) \\ \mathllap{\sigma} \downarrow & & \mathllap{\sigma} \downarrow & & \downarrow \mathrlap{id} \\ c \times c & \stackrel{k \times k}{\to} & End(c) \times End(c) & \stackrel{comp}{\to} & End(c) }$

leads to the conclusion that $k \pi_1 = k \pi_1 \sigma$, or $\pi_1 = \pi_2: c \times c \times c \to c$. We easily conclude $\pi_1 = \pi_2: c \times c \to c$, which forces equality $f = g$ for any two maps $f, g: d \to c$, so that the unique map $!: c \to 1$ is a monomorphism.

Last revised on July 4, 2018 at 06:51:45. See the history of this page for a list of all contributions to it.