Category Theory Preliminaries II

Definition 431.1(Functor).

Let $\text{C},\text{D}$ be categories. A covariant functor

$$\mathcal{F}:\text{C}\to \text{D}$$

is an assignment of an object $\mathcal{F}(A)\in \text{Obj}(\text{D})$ for every $A\in \text{Obj}(\text{C})$ and of a function

$${\text{Hom}}_{\text{C}}(A,B)\to {\text{Hom}}_{\text{D}}(\mathcal{F}(A),\mathcal{F}(B))$$

for every pair of objects $A,B$ in $\text{C}$. This function of also denoted $\mathcal{F}$ and must preserve identities and compositions. That is,

$$\mathcal{F}(1_A)=1_{\mathcal{F}(A)}$$

for all objects $A$ of $\text{C}$, and

$$\mathcal{F}(\beta \circ \alpha )=\mathcal{F}(\beta )\circ \mathcal{F}(\alpha )$$

for all objects $A,B,C$ of $\text{C}$, and for all $\alpha \in {\text{Hom}}_{\text{C}}(A,B)$ and $\beta \in {\text{Hom}}_{\text{C}}(B,C)$.

A contravariant functor $\text{C}\to \text{D}$ is a covariant functor ${\text{C}}^{op}\to \text{D}$.

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Definition 431.2(Limit).

Let $\mathcal{F}:\text{I}\to \text{C}$ be a covariant functor. The limit of $\mathcal{F}$ (if it exists) is an object $L$ of $\text{C}$, endowed with morphisms ${\lambda }_I:L\to \mathcal{F}(I)$ for all objects $I$ of $\text{I}$, satisfying the following properties.

1. If $\alpha :I\to J$ is a morphism in $\text{I}$, then ${\lambda }_J=\mathcal{F}(\alpha )\circ {\lambda }_I$.
2. $L$ is final with respect to this property.

Definition 431.3(Definition).

Let $\text{C},\text{D}$ be categories, and let $\mathcal{F},\mathcal{G}$ be (say, covariant) functors $\text{C}\to \text{D}$. A natural transformation $\mathcal{F}⇝\mathcal{G}$ is the datum of a morphism ${\nu }_X:\mathcal{F}(X)\to \mathcal{G}(X)$ in $\text{D}$ for every object $X$ in $\text{C}$, such that $\forall \alpha :X\to Y$ in $\text{C}$ the diagram

commutes. A natural isomorphism is a natural transformation $\nu$ such that ${\nu }_X$ is an isomorphism for every $X$.

Definition 431.4(Adjoint).

Let $\text{C},\text{D}$ be categories, and let $\mathcal{F}:\text{C}\to \text{D}$, $\mathcal{G}:\text{C}\to \text{D}$ be functors. We say that $\mathcal{F}$ and $\mathcal{G}$ are adjoint (and we say that $\mathcal{G}$ is right-adjoint to $\mathcal{F}$ and $\mathcal{F}$ is left-adjoint to $\mathcal{G}$) if there are natural isomorphisms

$${\text{Hom}}_{\text{C}}(X,\mathcal{G}(Y))\stackrel{\sim }{\to }{\text{Hom}}_{\text{D}}(\mathcal{F}(X),Y)$$

for all objects $X$ of $\text{C}$ and $Y$ of $\text{D}$.

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Important

Left adjoints are applied to the left slot; likewise for right adjoints.

Proposition 431.5(Proposition).

Left adjoint functors preserve colimits; dually, right adjoint functors preserve limits.

Proof.

To show that left adjoint functors preserve colimits, it suffices to show that they preserve coproducts and pushouts. Check this.□

Definition 431.6(Exact functors).

A functor is exact if it preserves exactness, that is, it sends exact sequences to exact sequences.