PlanetMath: constant functor

(more info)

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constant functor

(Definition)

Let $\mathcal{C}$ and $\mathcal{D}$ be categories. A constant functor from $\mathcal{C}$ to $\mathcal{D}$ is a functor $k:\mathcal{C\to D}$ such that there is an object $A\in \mathcal{D}$ such that

for all objects in $\mathcal{C}$ , , and
for all morphisms $X\to Y$ in $\mathcal{C}$ , $k(X\to Y)=1_A$ , the identity morphism of .

To see that this is indeed a functor, we merely need to verify that

$\displaystyle k\big((X\to Y)\circ (Y\to Z)\big)=k(X\to Y)\circ k(Y\to Z).$

But this is obvious, as the left hand side is $k(X\to Z)=1_A$ , while the right hand side is $1_A\circ 1_A=1_A$ .

Remarks.

For the constant functor considered above, the object is the fixed value of . To identify with , we often write .
Composing a functor with a constant functor gives us a constant functor. More precisely, let $k_A:\mathcal{C\to D}$ be the constant functor with fixed value . If $F:\mathcal{B\to C}$ is a functor, then $k_A\circ F:\mathcal{B\to D}$ is the constant functor $(k\circ F)_A$ with fixed value at . Moreover, if $G:\mathcal{D\to E}$ is a functor, then $G\circ k_A:\mathcal{C\to E}$ is the constant functor $(G\circ k)_{G(A)}$ valued at .
Given any functor $T:\mathcal{C\to D}$ , any natural transformation $\tau_A: k_A\dot{\to} T$ takes any object $X\in \mathcal{C}$ to a morphism $A\to T(X)$ , and, for any morphism $\alpha: X\to Y$ in $\mathcal{C}$ , a commutative triangle
$\displaystyle \xymatrix@R=7pt@C=1.5cm{ & T(X) \ar[dd]^{T(\alpha)} \ A \ar[dr] \ar[ur] & \ & T(Y) }$
in $\mathcal{D}$ . Similarly, any natural transformation $\sigma_A:T\dot{\to} k_A$ takes any object to $T(X)\to A$ and any morphism $\alpha: X\to Y$ to a commutative triangle
$\displaystyle \xymatrix@R=7pt@C=1.5cm{ T(X) \ar[dd]_{T(\alpha)} \ar[dr] & \ & A \ T(Y) \ar[ur] & }$
If $\alpha:A\to B$ is any morphism in $\mathcal{D}$ , then the natural transformation $\tau:k_A\dot{\to} k_B$ given by sending every object $X\in \mathcal{C}$ to the morphism $\tau_X: A\to B:=\alpha$ can be thought of as a “constant” natural transformation, since, for any morphism $\beta:X\to Y$ in $\mathcal{C}$ , the commutative diagram
$\displaystyle \xymatrix@R=1.25cm@C=2cm{ k_A(X) \ar[d]_{k_A\beta} \ar[r]^{\tau_X... ...& B \ar[d]^{1_B} \ k_A(Y) \ar[r]^{\tau_Y} & k_B(Y) & A \ar[r]^{\alpha} & B }$
reduces to $\alpha$ .
As above, denote $\tau$ by $\tau_{\alpha}$ . Let $\sigma_A: T\dot{\to} k_A$ be a natural transformation. Then the transformation $\tau_{\alpha}\circ \sigma_A:T\dot{\to} k_B$ sends any object in $\mathcal{C}$ to the morphism
$\displaystyle T(X)\to A \stackrel{\alpha}{\to} B$
in $\mathcal{D}$ , and any morphism $\beta:X\to Y$ to the commutative diagram
$\displaystyle \xymatrix@R=7pt@C=1.5cm{ T(X) \ar[dd]_{T(\beta)} \ar[dr] & & \ & A \ar[r]^{\alpha} & B \ T(Y) \ar[ur] & & }$
For any natural transformation $\gamma_B: k_B\dot{\to} T$ , the composition $\sigma_B\circ \tau_{\alpha}$ works similarly.