3/10/2023

General Nonsense 1- Yoneda Lemmma

This i A fibration is a functor $F:\mathcal{E}\rightarrow \mathcal{B}$ such that $\forall u: B' \rightarrow FE$ in $\mathcal{B}$ there is a unique $t: E'\rightarrow E$ such that $Ft = u$ and moreover $Fs=uu' \implies s= tt'$ with $Ft' = s'$. A morphism in $t \in \mathcal{E}$ is in the fiber over $B \in \mathcal{B}$ of a fibration $F:\mathcal{E}\rightarrow\mathcal{B}$ denoted by $t \in F^{-1}\left(B\right)$ if $Ft = id_B$. A fibration is discrete if the fiber over any object is a set. We shall denote $\mbox{DiscFib}\left(\mathcal{B}\right)$ for the category of discrete fibrations over $B$. Define a presheaf to be a contravariant set valued functor over a locally small category and denote $\mathcal{B}^{\wedge}$ for the category of presheaves and their natural transformations. In this post we recollect some generalities conerning the topos of presheaves, begining by showing that $\mbox{DiscFib}\left(\mathcal{B}\right) \simeq \mathcal{B}^{\wedge}$. There is a discrete fibration $\rho_X$ associated to any presheaf $X \in \mathcal{B}^{\wedge}$ given by a forgetful functor $\left(x,B\right)\mapsto B: \mathcal{B}_X \rightarrow \mathcal{B}$, where $\mathcal{B}_X$ is the category of elements of $X: \mathcal{B}^{o}\rightarrow Set$ with $u \in \mathcal{B}_X\left(\left(x',B'\right),\left(x,B\right)\right)$ if $u \in \mathcal{B}\left(B',B\right)$ and $Xux=x'$. A section $x \in XB$ of a presheaf $X$ gives a presheaf morphism $x:B \rightarrow X$ whose component at $B' \in \mathcal{B}$ is given by $u \mapsto Xux:\mathcal{B}: \mathcal{B}\left(B',B\right)\rightarrow XB'$. Here, $B \in \mathcal{B}^{\wedge}$ is an abuse of notation for a presheaf representable by $B \in \mathcal{B}$, that is -in anticipation of the uses of Yoneda lemma- we write $B$ instead of $\mathcal{B}\left(-,B\right):\mathcal{B}^{o}\rightarrow Set$ and $x:B \rightarrow X$, for the presheaf over $X$ determined by sending the generic element $id_B$ of $B$ to $x \in XB$. Moreover, there is a quasi-equivalance of big categories of the category of presheaves over $X$ denoted $\mathcal{B}^{\wedge}/X$ and the category $\left(\mathcal{B}_X\right)^{\wedge}$ of presheaves over the category of elements of $X$. Recall that the Yoned embedding $\mathscr{Y}:\mathcal{B}\rightarrow \mathcal{B}^{\wedge}$ sends $B\in \mathcal{B}$ to $\mathcal{B}\left(-,B\right)\in \mathcal{B}^{\wedge}$. But, besides the Yoneda embedding $\mathscr{Y}''\left(x,B\right)= \mathcal{B}_X\left(-,\left(x,B\right)\right):\left(\mathcal{B}_{X}\right)^{o}\rightarrow Set$ of the category of elements of $X$ into the category of presheaves over the category of elements of $X$, there is also $\mathscr{Y}':\mathcal{B}_X \rightarrow \mathcal{B}^{\wedge}/X$ where $\left(\phi,Z\right) \in \mathcal{B}^{\wedge}/X$ if $\phi \in \mathcal{B}^{\wedge}\left(Z,X\right)$ and $\mathscr{Y}'\left(x,B\right) = \left(x,B\right)$ where $x:B \rightarrow X$ is determined by $id_B \mapsto x$. Its easy to see $\mathcal{B}^{\wedge}/X \simeq \left(\mathcal{B}_X\right)^{\wedge}$ as any presheaf $F: \left(\mathcal{B}_X\right)^{\wedge}\rightarrow Set$ can be sent to $\pi: \lambda\left(F\right) \rightarrow X$ where $\lambda\left(F\right): \mathcal{B}^{o}\rightarrow Set$ sends $B \in \mathcal{B}$ to the set $\left\{\left(x,s\right)| x \in XB, s \in F\left(x,B\right)\right\}$ and $\pi_B\left(x,s\right):=x$ gives the component at $B \in \mathcal{B}$. On the other hand, any $\alpha: Z \rightarrow X$ gives a presheaf $\gamma\left(\alpha\right) \in \left(\mathcal{B}_X\right)^{\wedge}$ which maps $\left(x,B\right)$ to $\left\{z \in ZB|\alpha z = x\right\}$ and we have: $\gamma \circ \lambda \simeq id_{\left(\mathcal{B}_X\right)^{\wedge}}$ and $\lambda \circ \gamma \simeq id_{\mathcal{B}^{\wedge}/X}$. Seeing that $\rho_X: \mathcal{B}_{X} \rightarrow \mathcal{B}$ defined by $\left(x,B\right) \mapsto B$ associated to any presheaf $X:\mathcal{B}^{o}\rightarrow Set$ is a discrete fibration is the matter of unpacking $X$ being a presheaf. We show in detail how any discrete fibration $F: \mathcal{E}\rightarrow \mathcal{B}$ gives a presheaf, namely the presheaf $B \mapsto F^{-1}\left(B\right):\mathcal{B}\rightarrow Set$. To this end, suppose $r:E \rightarrow E_0$$ \in F^{-1}\left(B\right)$ is in the fiber over $B$ and $u:B' \rightarrow FE$ in the base category $\mathcal{B}$ are given. By definition, there must be $t:E'\rightarrow E$ with $Ft = u$ and moreover $Fs=uu' \implies s=tt'$ with $Ft'=u'$. In particular, letting $s=rt$ we see $Fs = FrFt = id_{B}u = u id_{B'}$ and therefore $rt = tt'$ for some unique $t' \in F^{-1}\left(B'\right)$. For a functor $\alpha:I \rightarrow \mathcal{B}$ we define the ind-limit of $\alpha$ to be the colimit of $\mathscr{Y} \circ \alpha: I \rightarrow \mathcal{B}^{\wedge}$ where $\mathscr{Y}:\mathcal{B}\rightarrow \mathcal{B}^{\wedge}$ is the Yoneda embedding $B \mapsto \mathcal{B}\left(-,B\right)$. The category $\mathcal{B}^{\wedge}$ of presheaves over $B$ has colimits (of small diagrams) as $Set$ has such colimits as we know: Let $\psi: J \rightarrow \left[\mathcal{C},\mathcal{D}\right]$ be a diagram in the functor category $\left[\mathcal{C},\mathcal{D}\right]$ and suppose $\mathcal{D}$ has $J$-shaped colimits. Then the colimit of $\psi$ exists and is given by $C \mapsto \varinjlim \beta_ C: \mathcal{C}\rightarrow \mathcal{D}$ where $\beta_C: J \rightarrow \mathcal{D}$ for $C \in \mathcal{C}$ is given by $j \mapsto \psi \left(j\right)\left(C\right)$. In other word, colimits (and similalry limits) are computed objectwise. So a presheaf $"\varinjlim \alpha ": \mathcal{B}^{o} \rightarrow Set$ called the ind-limit of $\alpha$ must exists when $I$ is small and $$\mathcal{B}^{\wedge}\left("\varinjlim" \alpha, Z\right)\simeq \varprojlim \mathcal{B}^{\wedge}\left(\mathscr{Y}\alpha,Z\right) \simeq I^{\wedge}\left(1, \mathcal{B}^{\wedge}\left(\mathscr{Y}\alpha,Z\right)\right)$$.