some category theory 1

The main reference: Methods of Homological Algebra.

  1. Suppose C\mathcal{C} is a category, C\mathcal{C}^\circ is its dual category. Thus any contravariant functor φ:CD\varphi:\mathcal{C}\to D is a (covariant) functor φ:CD\varphi:\mathcal{C}^\circ\to \mathcal{D} .

  2. Suppose C\mathcal{C} and D\mathcal{D} are categories. We can construct a category of functors form C\mathcal{C} to D\mathcal{D} by the follow steps. Firstly, the objects in Funct(C,D)\textit{Funct}(\mathcal{C},\mathcal{D}) are functors form C\mathcal{C} to D\mathcal{D} . Then, a morphism ff of functors from FF to GG (notation f:FGf:F\to G ) is a family of morphisms in D\mathcal{D} that f(X):F(X)G(X), f(X):F(X)\to G(X), one for each XCX\in \mathcal{C} , satisfying the following condition: for all morphism φ\varphi in C\mathcal{C} the following diagram \[\require{AMScd} \begin{CD} F(X) @>f(X)>> G(X)\\ @V{F(\varphi)}VV @VV{G(\varphi)}V \\ F(Y) @>f(Y)>> G(Y) \end{CD} \] is commutative. If FF is isomorphic to GG (notation FGF\cong G ), we should have morphisms ff and gg such that f(X)g(X)=idG(X)f(X)\circ g(X)=\operatorname{id}_{G(X)} and g(X)f(X)=idF(X)g(X)\circ f(X)=\operatorname{id}_{F(X)} for all XCX\in \mathcal{C} .

  3. Suppose C^=Funct(C,Set)\hat{\mathcal{C}}=\textit{Funct}(\mathcal{C}^\circ,\textit{Set}) , XC\forall X\in \mathcal{C} , we can associate a X^C^\hat{X}\in \hat{\mathcal{C}} by X^(Y)=HomC(Y,X)\hat{X}(Y)=\operatorname{Hom}_\mathcal{C}(Y,X) and Z^(φ)f=fφ\hat{Z}(\varphi)f=f\circ \varphi for φHomC(Y,X)\varphi \in \operatorname{Hom}_\mathcal{C}(Y,X) and fHomC(X,Z)f\in \operatorname{Hom}_\mathcal{C}(X,Z) . Then Z^(φ):Z^(X)Z^(Y)\hat{Z}(\varphi):\hat{Z}(X)\to \hat{Z}(Y) .

Proposition 1: Suppose XX is a set in the category Set, it can be identified with the set X^(e)=HomSet(e,X)\hat{X}(e)=\operatorname{Hom}_{\textit{Set}}(e,X) , where ee is a one-point set.

Proof. Obviously.

Indeed, in an arbitrary category C\mathcal{C} an analogue of one-point set does not necessarily exist. However, by considering HomC(Y,X)\operatorname{Hom}_{\mathcal{C}}(Y,X) for all YCY\in \mathcal{C} simultaneously, we can recover complete information about an object XCX\in \mathcal{C} . This is the idea of representation theory.

Definition 1: A functor FC^F\in \hat{\mathcal{C}} is said to be representable if FX^F\cong \hat{X} for some XCX\in \mathcal{C} . One says also that the object XX represents the functor FF .

Then

Theorem 1: HomC(X,Y)HomC^(X^,Y^)\operatorname{Hom}_\mathcal{C} (X,Y)\cong \operatorname{Hom}_{\hat{\mathcal{C}}} (\hat{X},\hat{Y}) .

Proof. Let φ:XY\varphi:X\to Y be a morphism in C\mathcal{C} . We associate with φ\varphi the morphism of functors φ^:X^Y^\hat{\varphi}:\hat{X}\to \hat{Y} by φ^(Z):θφθY^(Z), \hat{\varphi}(Z):\theta\mapsto\varphi\circ\theta\in \hat{Y}(Z), where θX^(Z)\theta\in \hat{X}(Z) and XX , YY , ZCZ\in \mathcal{C} . It is clear that φ^ϕ^=φϕundefined\hat{\varphi}\circ \hat{\phi}=\widehat{\varphi\circ\phi} . Conversely, suppose fHomC^(X^,Y^)f\in \operatorname{Hom}_{\hat{\mathcal{C}}} (\hat{X},\hat{Y}) , define map i:HomC^(X^,Y^)HomC(X,Y)i:\operatorname{Hom}_{\hat{\mathcal{C}}} (\hat{X},\hat{Y})\to \operatorname{Hom}_\mathcal{C} (X,Y) by i:ff(X)(idX), i:f\mapsto f(X)(\operatorname{id}_X), where idXHomC(X,X)=X^(X)\operatorname{id}_X\in \operatorname{Hom}_\mathcal{C}(X,X)=\hat{X}(X) and f(idX)Y^(X)f(\operatorname{id}_X)\in \hat{Y}(X) . Then i(φ^)=φ^(X)(idX)=φidX=φ. i(\hat{\varphi})=\hat{\varphi}(X)(\operatorname{id}_X)=\varphi\circ \operatorname{id}_X=\varphi. On the other hand, we should show i(f)undefined=f\widehat{i(f)}=f when fHomC^(X^,Y^)f\in \operatorname{Hom}_{\hat{\mathcal{C}}} (\hat{X},\hat{Y}) , and it's equivalent to show i(f)undefined(Z)=f(Z)\widehat{i(f)}(Z)=f(Z) for all ZCZ\in\mathcal{C} . Now suppose a morphism φ:ZX\varphi:Z\to X , i(f)φ=i(f)undefined(Z)(φ)=f(Z)(φ). i(f)\circ\varphi=\widehat{i(f)}(Z)(\varphi)=f(Z)(\varphi). Using the commutativity of the diagram, \[\require{AMScd} \begin{CD} \hat{X}(X) @>f(X)>> \hat{Y}(X)\\ @V{\hat{X}(\varphi)}VV @VV{\hat{Y}(\varphi)}V \\ \hat{X}(Z) @>f(Z)>> \hat{Y}(Z) \end{CD} \] then f(Z)X^(φ)(idX)=Y^(φ)f(X)(idX)=Y^(φ)(i(f))=i(f)φ f(Z)\circ\hat{X}(\varphi)(\operatorname{id}_X)=\hat{Y}(\varphi)\circ f(X)(\operatorname{id}_X)=\hat{Y}(\varphi)(i(f))=i(f)\circ \varphi and f(Z)X^(φ)(idX)=f(Z)(idXφ)=f(Z)(φ). f(Z)\circ\hat{X}(\varphi)(\operatorname{id}_X)=f(Z)(\operatorname{id}_X\circ\varphi)=f(Z)(\varphi). Thus ii is an isomorphism that i:HomC^(X^,Y^)HomC(X,Y)i:\operatorname{Hom}_{\hat{\mathcal{C}}} (\hat{X},\hat{Y})\cong \operatorname{Hom}_\mathcal{C} (X,Y) .

If XX represents the functor FF , HomC^(Y^,F)HomC^(Y^,X^)HomC(Y,X)\operatorname{Hom}_{\hat{\mathcal{C}}}(\hat{Y},F)\cong \operatorname{Hom}_{\hat{\mathcal{C}}}(\hat{Y},\hat{X})\cong\operatorname{Hom}_\mathcal{C}(Y,X) . If YY also represents FF , then there exists an isomorphism φ:Y^F\varphi:\hat{Y}\cong F , then i(φ)i(\varphi) is the according isomorphism between XX and YY . Thus, the representing object of a representable functor is defined uniquely up to a isomorphism.

Definition 2: Suppose XX , YCY\in\mathcal{C} , the direct product X×YX\times Y is (upto an isomorphism) the object ZZ representing the functor (if such functor is representable, or if such ZZ exists) WX^(W)×Y^(W), W\mapsto \hat{X}(W)\times\hat{Y}(W), where X^(W)×Y^(W)\hat{X}(W)\times\hat{Y}(W) is the direct product of Set which has been constructed directly.

Definition 3: Suppose XX , YY , SSetS\in\textit{Set} , f:XSf:X\to S and g:YSg:Y\to S , the pullback of ff and gg or the fibre product of XX and YY is X×SY={(x,y)X×Y:f(x)=g(y)}. X\times_S Y=\{(x,y)\in X\times Y:f(x)=g(y)\}. In a general category C\mathcal{C} , X×SYX\times_SY is defined as the object ZZ representing the functor WX^(W)×S^(W)Y^(W). W\mapsto \hat{X}(W)\times_{\hat{S}(W)}\hat{Y}(W). When f=gf=g are constant morphisms, X×SYX×YX\times_S Y\cong X\times Y . When ff and gg are embedding, X×SYXYX\times_S Y\cong X\cap Y .

The universal property can be directly vertified. Since ZZ represents the functor WX^(W)×Y^(W)W\mapsto \hat{X}(W)\times\hat{Y}(W) , then Z^(W)=X^(W)×Y^(W)\hat{Z}(W)=\hat{X}(W)\times\hat{Y}(W) . Let W=ZW=Z , the image of idZZ^(Z)\operatorname{id}_Z\in \hat{Z}(Z) is that idZ=(πX,πY)\operatorname{id}_Z=(\pi_X,\pi_Y) , where πXY^(Z)\pi_X\in \hat{Y}(Z) and πYY^(Z)\pi_Y\in \hat{Y}(Z) . This can be writen as XundefinedπXZundefinedπYYX\xleftarrow{\pi_X} Z \xrightarrow{\pi_Y}Y

Now suppose pX:WXp_X:W\to X and pY:WYp_Y:W\to Y , then there exist η=(pX,pY)X^(W)×Y^(W)=Z^(W)\eta=(p_X,p_Y)\in \hat{X}(W)\times\hat{Y}(W)= \hat{Z}(W) . The last thing we should to show is that pX=πXηp_X=\pi_X\circ \eta and pY=πYηp_Y=\pi_Y\circ \eta . Indeed, for all zZz\in Z , (πX,πY)((pX,pY)(z))=idZ((pX,pY)(z))=(pX,pY)(z). (\pi_X,\pi_Y)((p_X,p_Y)(z))=\operatorname{id}_Z((p_X,p_Y)(z))=(p_X,p_Y)(z). The uniqueness of X×YX\times Y up to an isomorphism can be drived from the above theorem or universal property directly.

Because of some psychological reasons, we usually use a commutative diagram to visualize the universal property described above: \[ \begin{xy} \xymatrix{ &A\ar[dl]\ar[dr]\ar@{-->}[d]&\\ X&X\times Y\ar[l]\ar[r]&Y\\ } \end{xy} \] Similarly, the universal property of fibre product can be written as a commutative diagram: \[ \begin{xy} \xymatrix{ U \ar@/_/[ddr]_y \ar@{.>}[dr]|{\langle x,y \rangle} \ar@/^/[drr]^x \\ & X \times_Z Y \ar[d]^q \ar[r]_p & X \ar[d]_f \\ & Y \ar[r]^g & Z } \end{xy} \]

We fix a category J\mathcal{J} , and define the diagonal functor Δ:CFunct(J,C)\Delta:\mathcal{C}\to \textit{Funct}(\mathcal{J},\mathcal{C}) as follow:

  • On objects: ΔX\Delta X is the set of constant functors with the value XX . In other words, ΔX(j)=X\Delta X(j)=X for all jJj\in\mathcal{J} , ΔX(φ)=idX\Delta X(\varphi)=\operatorname{id}_X for all morphism φ\varphi in J\mathcal{J} .
  • On morphisms: Suppose ψ:XY\psi:X\to Y is a morphism in C\mathcal{C} , Δψ:ΔXΔY\Delta\psi:\Delta X\to \Delta Y is defined as follows: Δψ(j):X=ΔX(j)Y=ΔY(j)\Delta\psi(j):X=\Delta X(j)\to Y=\Delta Y(j) for all jJj\in \mathcal{J} .

It is clear that Δ(φψ)=ΔφΔψ\Delta(\varphi\circ\psi)=\Delta\varphi\circ\Delta\psi , so Δ\Delta is indeed a functor.

Definition 4: Suppose F:JCF:\mathcal{J}\to \mathcal{C} is a functor, the (projective or inverse) limit of FF in the category C\mathcal{C} according to J\mathcal{J} is an object XCX\in\mathcal{C} representing the functor YHomFunct(J,C)(ΔY,F):CSet. Y\mapsto \operatorname{Hom}_{\textit{Funct}(\mathcal{J},\mathcal{C})}(\Delta Y,F):\mathcal{C}^\circ\to \textit{Set}. The limit of FF is denoted by X=limundefinedFX=\underleftarrow{\lim} F .

Buwai Lee

Buwai Lee

交换图都不会画的魔法师