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[parent] 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

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 $ k$ considered above, the object $ A$ is the fixed value of $ k$. To identify $ k$ with $ A$, we often write $ k_A:=k$.
  • 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 $ A$. 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 $ A$. 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 $ G(A)$.
  • 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 $ X$ 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 $ X$ 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.

Bibliography

1
S. Mac Lane, Categories for the Working Mathematician (2nd edition), Springer-Verlag, 1997.



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Cross-references: composition, transformation, commutative diagram, triangle, commutative, natural transformation, fixed, right hand side, left hand side, obvious, identity, morphisms, object, functor, categories
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This is version 6 of constant functor, born on 2007-10-24, modified 2007-10-24.
Object id is 10012, canonical name is ConstantFunctor.
Accessed 415 times total.

Classification:
AMS MSC18-00 (Category theory; homological algebra :: General reference works )

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