Examples of Topos and Essential Geometric Morphism

Topos and Geometric Morphism.Example of Topos and Geometric MorphismG-Set and QuiverA universal way to find essential geometric morphism between $\mathbf{Set}^{\mathcal C^{\mathrm{op}}}$ and $\mathbf{Set}$Reconstrcut the adjoint between $\mathbf{G-Set}$ and $\mathbf{Set}$.Another essential geometric morphism between $\mathbf{Quiver}$ and $\mathbf{Set}$.

Topos and Geometric Morphism.

An elementary topos is a kind of "good" category, the categorical property looks like $\mathbf{Set}$$\mathbf{Set}$.

i.e.

A quick formal definition is that an elementary topos is a category which

1. has finite limits,

2. is cartesian closed, and

3. has a subobject classifier.

This also implies that it has finite colimits.

A Geometirc Morphism between two elementary topos is a pair of adjoint functors $(f^{\ast },f_{\ast })$$(f^*,f_*)$ such that $f^{\ast }$$f^*$ preserve finite limit (hence $f_{\ast }$$f_*$) preserve finite colimit. If $f^{\ast }$$f^*$ admits a left adjoint $f_{!}$$f_!$ as well, then we say $(f_{!},f^{\ast },f_{\ast })$$(f_!,f^*,f_*)$ is a triple of essential geometric morphism.

Example of Topos and Geometric Morphism

Let $\mathcal{C}$$\mathcal C$ be a small category, then ${\mathbf{Set}}^{{\mathcal{C}}^{\mathrm{op}}}$$\mathbf{Set}^{\mathcal C^{\mathrm{op}}}$ is a topos. You compute each limit and exponential, subobject classifier ''pointwise''.

G-Set and Quiver

Recall that category of left $G-\mathrm{Set}$$G-\mathrm{Set}$ is the functor category $[B(G):\mathrm{Set}]$$[B(G):\mathrm{Set}]$.

If we consider ${\mathbf{Set}}^{B(G{)}^{\mathrm{op}}}$$\mathrm{\mathbf{Set}}^{B(G)^{\mathrm{op}}}$, then we obtain the category of right $G$$G$-sets (i.e.right group actions). This is a topos as well.

Category of Quiver is a functor category as well.

Consider

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

Then

$${\mathrm{Set}}^{{\mathcal{D}}^{\mathrm{op}}}=\mathrm{Graph}$$

Since a functor $F:{\mathcal{D}}^{\mathrm{op}}\to \mathrm{Set}$$F:\mathcal D^{\mathrm{op}}\to\mathrm{Set}$ is just $F(s),F(t):F(E)\to F(V)$$F(s),F(t):F(E)\to F(V)$. Here $F(E)$$F(E)$ is the edge set, $F(V)$$F(V)$ is the vertex set. $F(s)(x)$$F(s)(x)$ tells you the sorce of $x$$x$ and $F(t)(x)$$F(t)(x)$ is the target of $x$$x$.

The natural transformation is just the quiver homomorphism.

$$\begin{array}{ccc}F(E)& \stackrel{F(s),F(t)}{\to }& F(V)\\ {\alpha }_X\downarrow & & \downarrow {\alpha }_Y& \\ G(E)& \stackrel{G(s),G(t)}{\to }& G(V)\end{array}$$

There is an essential geometirc morphism between $\mathbf{Set}$$\mathbf{Set}$ and $\mathrm{Garph}$$\mathrm{Garph}$. (Click the link.)

Remark. This definition is very natural.

Try to draw the quiver of ${\mathrm{Hom}}_{{\mathcal{D}}^{\mathrm{op}}}(E,-),{\mathrm{Hom}}_{{\mathcal{D}}^{\mathrm{op}}}(V,-)$$\mathrm{Hom}_{\mathcal D^{\mathrm{op}}}(E,-),\mathrm{Hom}_{\mathcal D^{\mathrm{op}}}(V,-)$.

Now let us consider the essential geometric morphism between $G$$G$-Set and $\mathbf{Set}$$\mathbf{Set}$.

 

The $\mathrm{\Delta }$$\Delta$ admits a right adjoint as well, namely, $(-{)}^G$$(-)^G$.

For a $G$$G$-Set $X$$X$, $X^G:=\{x\in X:g\cdot x=x\}$$X^G:=\{x\in X:g\cdot x=x\}$. For a morphism $f:X\to Y,g\cdot f(x)=f(g\cdot x)=f(x)$$f:X\to Y,g\cdot f(x)=f(g\cdot x)=f(x)$. Hence we can induce a morphism $f^G:X^G\to Y^G$$f^G:X^G\to Y^G$.

Easy to see that

$${\mathrm{Hom}}_{G-\mathrm{Set}}(\mathrm{\Delta }(A),B)\cong {\mathrm{Hom}}_{\mathbf{Set}}(A,B^G)$$

Hence we have the essential geometric morphism

$$(\mathcal{O},\mathrm{\Delta },(-{)}^G)$$

A universal way to find essential geometric morphism between ${\mathbf{Set}}^{{\mathcal{C}}^{\mathrm{op}}}$$\mathbf{Set}^{\mathcal C^{\mathrm{op}}}$ and $\mathbf{Set}$$\mathbf{Set}$

We already now that limit is a right adjoint functor of $\mathrm{\Delta }$$\Delta$(Click here). By duality, we have colimt is the left adjoint of $\mathrm{\Delta }$$\Delta$.

In particular, for a small category $\mathcal{C}$$\mathcal C$, we have

$$\mathrm{colim}⊣\mathrm{\Delta }⊣lim$$

Where $\mathrm{\Delta }:\mathbf{Set}\to {\mathbf{Set}}^{{\mathcal{C}}^{\mathbf{op}}},(\mathrm{co})lim:{\mathbf{Set}}^{{\mathcal{C}}^{\mathrm{op}}}\to \mathbf{Set}$$\Delta:\mathbf{Set}\to\mathbf{Set^{\mathcal C^{\mathrm{}op}}},(\mathrm{co})\lim:\mathbf{Set}^{\mathcal C^{\mathrm{op}}}\to\mathbf{Set}$.

Reconstrcut the adjoint between $\mathbf{G}\mathbf{-}\mathbf{Set}$$\mathbf{G-Set}$ and $\mathbf{Set}$$\mathbf{Set}$.

We know that $\mathbf{G}\mathbf{-}\mathbf{Set}$$\mathbf{G-Set}$ is just ${\mathbf{Set}}^{\mathbf{B}\mathbf{(}\mathbf{G})}$$\mathbf{Set^{B(G)}}$. (Click here) Hence we have

$$\mathrm{colim}⊣\mathrm{\Delta }⊣lim$$

But what is a (co)limit of a $G$$G$-Set $X:B(G)\to \mathrm{Set}$$X:B(G)\to\mathrm{Set}$?

It is very easy to see that the colimit of $X$$X$ is just $X/G$$X/G$ and the limit is just $X^G$$X^G$.

Another essential geometric morphism between $\mathbf{Quiver}$$\mathbf{Quiver}$ and $\mathbf{Set}$$\mathbf{Set}$.

Recall that we have

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

Then the limit and colimit of $F$$F$ are just equalizer and coequalizer of $F(s),F(t)$$F(s),F(t)$.

Then

$$limF=\mathrm{Eq}(F(s),F(t))=\{e\in F(E):F(s)(e)=F(t)\}$$

i.e. those self loop.

$$\mathrm{colim}F=\mathrm{Coeq}(F(s),F(t))={\pi }_0(F)$$

That is, the path connected component of a quiver $F$$F$.

For a set $X$$X$, $\mathrm{\Delta }(X)$$\Delta(X)$ is

$$\mathrm{\Delta }(X)(E)=\mathrm{\Delta }(X)(V)=X,\mathrm{\Delta }(X)(s)=\mathrm{\Delta }(t)={\mathrm{id}}_X$$

Thus it is a discrete graph with all the self loop.