Syntax and Semantics in Representable Functors

Syntax is something that prescribes how we are allowed to speak - it establishes the rules and constraints of our discourse.

Semantics is what our speech means - it gives content and interpretation to our syntactic expressions.

Let $F:\mathcal{C}\to \mathbf{Set}$$F:\mathcal C\to\mathbf{Set}$ be a representable functor, with $F\cong {\mathrm{Hom}}_{\mathcal{C}}(T,-)$$F\cong\mathrm{Hom}_{\mathcal C}(T,-)$.

We should view $T$$T$ as a kind of syntax and $x\in F(X)$$x\in F(X)$ as a kind of semantics or model of $T$$T$ in $X$$X$.

Consider the following examples:

For $\mathbf{Haus}$$\mathbf{Haus}$, consider the convergent sequence functor $\mathrm{Haus}(\mathbb{N}\cup \{\mathrm{\infty }\},-)$$\mathrm{Haus}(\mathbb N\cup\{\infty\},-)$, we should view ${\mathrm{id}}_{\mathbb{N}\cup \{\mathrm{\infty }\}}$$\mathrm{id}_{\mathbb N\cup\{\infty\}}$ as the universal convergent sequence.

A model of $(\mathbb{N}\cup \{\mathrm{\infty }\},{\mathrm{id}}_{\mathbb{N}\cup \{\mathrm{\infty }\}})$$(\mathbb N\cup\{\infty\},\mathrm{id}_{\mathbb N\cup\{\infty\}})$ in a space $X$$X$ is a convergent sequence in $X$$X$.

For $\mathbf{Top}$$\mathbf{Top}$, consider $U(X)=\{f\in {\mathrm{Hom}}_{\mathrm{Top}}(\mathbb{R},X):f(x+T)=f(x)\}$$U(X)=\{f\in\mathrm{Hom}_{\mathrm{Top}}(\mathbb R,X):f(x+T)=f(x)\}$. Then we have $U\cong \mathbf{Top}(\mathbb{R}/\mathbb{Z},-)$$U\cong\mathbf{Top}(\mathbb R/\mathbb Z,-)$. We could view ${\mathrm{id}}_{\mathbb{R}/\mathbb{Z}}$$\mathrm{id}_{\mathbb R/\mathbb Z}$ as the universal periodic function with $T=1$$T=1$.

For $\mathbf{CRing}$$\mathbf{CRing}$, $\mathbf{CRing}(\mathbb{Z}[X,Y]/(X^2+Y^2-1),-)$$\mathbf{CRing}(\mathbb Z[X,Y]/(X^2+Y^2-1),-)$ is the solution set functor of $X^2+Y^2-1$$X^2+Y^2-1$, and we should view ${\mathrm{id}}_{\mathbb{Z}[X,Y]/(X^2+Y^2-1)},X⟼X,Y⟼Y$$\mathrm{id}_{\mathbb Z[X,Y]/(X^2+Y^2-1)}, X\longmapsto X,Y\longmapsto Y$ as the universal solution of $X^2+Y^2-1$$X^2+Y^2-1$.

For an well powered Topos $\mathcal{E}$$\mathcal E$, let $\mathrm{\Omega }$$\Omega$ be the subobject classifier, we have $\mathrm{Sub}(-)\cong {\mathrm{Hom}}_{\mathcal{E}}(-,\mathrm{\Omega })$$\mathrm{Sub}(-)\cong\mathrm{Hom}_{\mathcal E}(-,\Omega)$.

We should view $\mathrm{\top }:1\to \mathrm{\Omega }$$\mathrm{\top}:1\to\Omega$, or its characteristic function ${\mathrm{id}}_{\mathrm{\Omega }}$$\mathrm{id}_{\Omega}$ as the universal subobject. Since every subobject is given by pullback of $\mathrm{\top }:1\to \mathrm{\Omega }$$\top:1\to\Omega$.

To formalize this idea, consider the category of elements $\begin{array}{c}{\int }_{\mathcal{C}}F\end{array}$$\int_{\mathcal C} F\\$, where $F\cong {\mathrm{Hom}}_{\mathcal{C}}(T,-)$$F\cong \mathrm{Hom}_{\mathcal C}(T,-)$.

The object $(T,{\mathrm{id}}_T)$$(T,\mathrm{id}_T)$ is the syntax, a model or a semantics of this syntax on $X$$X$ is a morphism

$$\boxed{{\displaystyle x:(T,{\mathrm{id}}_T)\to (X,x)}}$$

The identity is the universal model/semantics.

For example, in ${\displaystyle {\int }_{\mathbf{Haus}}\mathrm{Haus}(\mathbb{N}\cup \{\mathrm{\infty }\},-)}$$\displaystyle \int_{\mathbf{Haus}} \mathrm{Haus}(\mathbb N\cup\{\infty\},-)$, the identity is $(n{)}_{n=0}^{\mathrm{\infty }}$$(n)_{n=0}^{\infty}$. A convergent sequence on $X$$X$ is a continuous function

$$\boxed{{\displaystyle x:\mathbb{N}\cup \{\mathrm{\infty }\}\to X}}$$

It will correspond to

$$(X,x)$$

which is a semantics/model of $(\mathbb{N}\cup \{\mathrm{\infty }\},{\mathrm{id}}_{\mathbb{N}\cup \{\mathrm{\infty }\}})$$(\mathbb N\cup\{\infty\},\mathrm{id}_{\mathbb N\cup\{\infty\}})$ in $X$$X$.

Now let us consider the projection functor

$$\boxed{{\displaystyle \pi :{\int }_{\mathcal{C}}F⟶\mathcal{C},(X,x)⟼X}}$$

Then the fiber functor is representable

$$\boxed{{\displaystyle {\pi }^{-1}(X)\cong {\mathrm{Hom}}_{\mathcal{C}}(T,X)}}$$

Remark. It looks like the fiber functor in Grothendieck Galois theory, huh.

The fiber ${\pi }^{-1}(X)$$\pi^{-1}(X)$ is the collection of all the semantics/models of $X$$X$.

Now let us consider $\mathbf{CAT}$$\mathbf{CAT}$, which is the category of categories.

Fix a (small) category $\mathcal{D}$$\mathcal D$, and consider $\mathrm{Func}(\mathcal{D},-):\mathbf{CAT}\to \mathbf{CAT}$$\mathrm{Func}(\mathcal D,-):\mathbf{CAT}\to \mathbf{CAT}$. Sometimes we consider some structure preserving functor, and we consider the subcategory of $\mathbf{CAT}$$\mathbf{CAT}$.

We should view $\mathcal{D}$$\mathcal D$ as the syntax, and $\mathrm{Func}(\mathcal{D},\mathcal{C})$$\mathrm{Func}(\mathcal D,\mathcal C)$ as a kind of semantics/model on $\mathcal{C}$$\mathcal C$.

Again, we should consider the category of elements with respect to $\mathrm{Func}(\mathcal{D},-)$$\mathrm{Func}(\mathcal D,-)$.

The object is

$$\boxed{{\displaystyle (\mathcal{C},c),\text{ where }c\text{ is a functor from }\mathcal{D}\text{ to }\mathcal{C}\text{.}}}$$

Here $(\mathcal{D},{\mathrm{id}}_{\mathcal{D}})$$(\mathcal D,\mathrm{id}_\mathcal D)$ is the syntax, and $(\mathcal{C},c)$$(\mathcal C,c)$ is a model in $\mathcal{C}$$\mathcal C$, and $\mathrm{Func}(\mathcal{D},\mathcal{C})$$\mathrm{Func}(\mathcal D,\mathcal C)$ is the category of models in $\mathcal{C}$$\mathcal C$.

Example.

Consider the category $B(G)$$B(G)$, where $G$$G$ is a group, and $B(G)$$B(G)$ is a one-point category. Then

$$\boxed{{\displaystyle (B(G),{\mathrm{id}}_{B(G)})\text{ is the syntax of the group action, and }(\mathbf{Set},F)\text{ is a model in }\mathbf{Set},\text{ i.e., a }G\text{-set}}}$$

Consider the category

$${\mathcal{D}}^{\mathrm{op}}=E\stackrel{s}{\underset{t}{\rightrightarrows }}V$$

Then the syntax of quiver (edge, vertex, source and target) is given by

$$\boxed{{\displaystyle ({\mathcal{D}}^{\mathrm{op}},{\mathrm{id}}_{{\mathcal{D}}^{\mathrm{op}}}),\text{ and }(\mathbf{Set},F)\text{ is a model in }\mathbf{Set}}}$$

The semantics in $\mathbf{Set}$$\mathbf{Set}$ form a topos.

Let $R$$R$ be a ring and $B(R)$$B(R)$ be the one-point category. Let ${\mathcal{A}}^{+}$$\mathcal{A}^+$ be the category of additive categories (morphism is additive functor).

Then consider $\mathrm{Add}(B(R),-):{\mathcal{A}}^{+}\to {\mathcal{A}}^{+}$$\mathrm{Add}(B(R),-):\mathcal A^+\to \mathcal A^+$

Then

$$\boxed{{\displaystyle (B(R),{\mathrm{id}}_{B(R)})\text{ is the syntax of the ring action, and }(\mathbf{Ab},F)\text{ is a model in }\mathbf{Ab},\text{ i.e., an }R\text{-module}}}$$

Let ${\mathbb{T}}_{\mathbb{G}}$$\mathbb {T_G}$ be the Lawvere theory of groups, and consider the category of cartesian monoidal categories (the morphism is product-preserving functor), then

$$\boxed{{\displaystyle ({\mathbb{T}}_{\mathbb{G}},{\mathrm{id}}_{{\mathbb{T}}_{\mathbb{G}}}),\text{ is the syntax of group theory, and }(\mathbf{Set},F)\text{ is a model in }\mathbf{Set}}}$$