Remark 3.2.4.1.2 (01D7): Analogies Between Set Theory and Category Theory: Powersets and Categories of Presheaves

3.2.4 Tables of Analogies Between Set Theory and Category Theory

Here we record some analogies between notions in set theory and category theory. The analogies relating to presheaves relate equally well to copresheaves, as the opposite $X^{\mathrm{op}}$ of a set $X$ is just $X$ again.

Remark 3.2.4.1.1 ▶ Basic Analogies Between Set Theory and Category Theory

The basic analogies between set theory and category theory are summarised in the following table:

Set Theory	Category Theory
Enrichment in $\{ \mathsf{true},\mathsf{false}\} $	Enrichment in $\mathsf{Sets}$
Set $X$	Category $\mathcal{C}$
Element $x\in X$	Object $X\in \operatorname {\mathrm{Obj}}\webleft (\mathcal{C}\webright )$
Function $f\colon X\to Y$	Functor $F\colon \mathcal{C}\to \mathcal{D}$
Function $X\to \{ \mathsf{true},\mathsf{false}\} $	Copresheaf $\mathcal{C}\to \mathsf{Sets}$
Function $X\to \{ \mathsf{true},\mathsf{false}\} $	Presheaf $\mathcal{C}^{\mathsf{op}}\to \mathsf{Sets}$

Remark 3.2.4.1.2 ▶ Analogies Between Set Theory and Category Theory: Powersets and Categories of Presheaves

The category of presheaves $\mathsf{PSh}\webleft (\mathcal{C}\webright )$ and the category of copresheaves $\mathsf{CoPSh}\webleft (\mathcal{C}\webright )$ on a category $\mathcal{C}$ are the 1-categorical counterparts to the powerset $\mathcal{P}\webleft (X\webright )$ of subsets of a set $X$. The further analogies built upon this are summarised in the following table:

Set Theory

Category Theory

Powerset $\mathcal{P}\webleft (X\webright )$

Presheaf category $\mathsf{PSh}\webleft (\mathcal{C}\webright )$

Characteristic function

$\chi _{\left\{ x\right\} }\colon X\to \{ \mathsf{t},\mathsf{f}\} $

Representable presheaf

$h_{X}\colon \mathcal{C}^{\mathsf{op}}\hookrightarrow \mathsf{Sets}$

Characteristic embedding

$\chi _{\webleft (-\webright )}\colon X\hookrightarrow \mathcal{P}\webleft (X\webright )$

Yoneda embedding

${\text{よ}}\colon \mathcal{C}^{\mathsf{op}}\hookrightarrow \mathsf{PSh}\webleft (\mathcal{C}\webright )$

Characteristic relation

$\chi _{X}\webleft (-_{1},-_{2}\webright )\colon X\times X\to \{ \mathsf{t},\mathsf{f}\} $

Hom profunctor

$\operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (-_{1},-_{2}\webright )\colon \mathcal{C}^{\mathsf{op}}\times \mathcal{C}\to \mathsf{Sets}$

The Yoneda lemma for sets

$\operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{U}\webright )=\chi _{U}\webleft (x\webright )$

The Yoneda lemma for categories

$\operatorname {\mathrm{Nat}}\webleft (h_{X},\mathcal{F}\webright )\cong \mathcal{F}\webleft (X\webright )$

The characteristic

embedding is fully faithful,

$\operatorname {\mathrm{Hom}}_{\mathcal{P}\webleft (X\webright )}\webleft (\chi _{x},\chi _{y}\webright )=\chi _{X}\webleft (x,y\webright )$

The Yoneda

embedding is fully faithful,

$\operatorname {\mathrm{Nat}}\webleft (h_{X},h_{Y}\webright )\cong \operatorname {\mathrm{Hom}}_{\mathcal{C}}\webleft (X,Y\webright )$

Subsets are unions

of their elements

$\displaystyle U=\bigcup _{x\in U}\left\{ x\right\} $

$\displaystyle \chi _{U}=\operatorname*{\operatorname {\mathrm{colim}}}_{\chi _{x}\in \mathcal{P}\webleft (U\webright )}\webleft (\chi _{x}\webright )$

Presheaves are colimits

of representables,

$\displaystyle \mathcal{F}\cong \operatorname*{\operatorname {\mathrm{colim}}}_{h_{X}\in \int _{\mathcal{C}}\mathcal{F}}\webleft (h_{X}\webright )$

Remark 3.2.4.1.3 ▶ Analogies Between Set Theory and Category Theory: Categories of Elements

We summarise the analogies between un/straightening in set theory and category theory in the following table:

Set Theory

Category Theory

Assignment

$U\mapsto \chi _{U}$

Assignment

$\mathcal{F}\mapsto \textstyle \int _{\mathcal{C}}\mathcal{F}$

Un/straightening isomorphism

$\mathcal{P}\webleft (X\webright )\cong \mathsf{Sets}\webleft (X,\{ \mathsf{t},\mathsf{f}\} \webright )$

Un/straightening equivalence

$\mathsf{PSh}\webleft (\mathcal{C}\webright )\mathrel {\smash {\overset {\scriptscriptstyle \text{eq.}}\cong }}\mathsf{DFib}\webleft (\mathcal{C}\webright )$

Remark 3.2.4.1.4 ▶ Analogies Between Set Theory and Category Theory: Functions Between Powersets and Functors Between Presheaf Categories

We summarise the analogies between functions $\mathcal{P}\webleft (A\webright )\to \mathcal{P}\webleft (B\webright )$ and functors $\mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$ in the following table:

Set Theory

Category Theory

Direct image function

$f_{*}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright )$

Direct image functor

$F_{*}\colon \mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$

Inverse image function

$f^{-1}\colon \mathcal{P}\webleft (Y\webright )\to \mathcal{P}\webleft (X\webright )$

Inverse image functor

$F^{*}\colon \mathsf{PSh}\webleft (\mathcal{D}\webright )\to \mathsf{PSh}\webleft (\mathcal{C}\webright )$

Direct image with

compact support function

$f_{!}\colon \mathcal{P}\webleft (X\webright )\to \mathcal{P}\webleft (Y\webright )$

Direct image with

compact support functor

$F_{!}\colon \mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$

Remark 3.2.4.1.5 ▶ Analogies Between Set Theory and Category Theory: Relations and Profunctors

We summarise the analogies between functions relations and profunctors in the following table:

Set Theory	Category Theory
Relation $R\colon X\times Y\to \{ \mathsf{t},\mathsf{f}\} $	Profunctor $\mathfrak {p}\colon \mathcal{D}^{\mathsf{op}}\times \mathcal{C}\to \mathsf{Sets}$
Relation $R\colon X\to \mathcal{P}\webleft (Y\webright )$	Profunctor $\mathfrak {p}\colon \mathcal{C}\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$
Relation as a cocontinuous morphism of posets $R\colon \webleft (\mathcal{P}\webleft (X\webright ),\subset \webright )\to \webleft (\mathcal{P}\webleft (Y\webright ),\subset \webright )$	Profunctor as a colimit-preserving functor $\mathfrak {p}\colon \mathsf{PSh}\webleft (\mathcal{C}\webright )\to \mathsf{PSh}\webleft (\mathcal{D}\webright )$

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