Definition 7.36.1. Let $f, g : \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \to \mathop{\mathit{Sh}}\nolimits (\mathcal{D})$ be two morphisms of topoi. A 2-morphism from $f$ to $g$ is given by a transformation of functors $t : f_* \to g_*$.
7.36 2-morphisms of topoi
This is a brief section concerning the notion of a $2$-morphism of topoi.
Pictorially we sometimes represent $t$ as follows:
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \rrtwocell ^ f_ g{t} & & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) } \]
Note that since $f^{-1}$ is adjoint to $f_*$ and $g^{-1}$ is adjoint to $g_*$ we see that $t$ induces also a transformation of functors $t : g^{-1} \to f^{-1}$ (usually denoted by the same symbol) uniquely characterized by the condition that the diagram
\[ \xymatrix{ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(\mathcal{G}, f_*\mathcal{F}) \ar[d]_{t \circ -} \ar@{=}[r] & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(f^{-1}\mathcal{G}, \mathcal{F}) \ar[d]^{- \circ t} \\ \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{D})}(\mathcal{G}, g_*\mathcal{F}) \ar@{=}[r] & \mathop{\mathrm{Mor}}\nolimits _{\mathop{\mathit{Sh}}\nolimits (\mathcal{C})}(g^{-1}\mathcal{G}, \mathcal{F}) } \]
commutes. Because of set theoretic difficulties (see Remark 7.15.4) we do not obtain a 2-category of topoi. But we can still define horizontal and vertical composition and show that the axioms of a strict 2-category listed in Categories, Section 4.29 hold. Namely, vertical composition of 2-morphisms is clear (just compose transformations of functors), composition of 1-morphisms has been defined in Definition 7.15.1, and horizontal composition of
\[ \xymatrix{ \mathop{\mathit{Sh}}\nolimits (\mathcal{C}) \rtwocell ^ f_ g{t} & \mathop{\mathit{Sh}}\nolimits (\mathcal{D}) \rtwocell ^{f'}_{g'}{s} & \mathop{\mathit{Sh}}\nolimits (\mathcal{E}) } \]
is defined by the transformation of functors $s \star t$ introduced in Categories, Definition 4.28.1. Explicitly, $s \star t$ is given by
\[ \xymatrix{ f'_*f_*\mathcal{F} \ar[r]^{f'_*t} & f'_*g_*\mathcal{F} \ar[r]^ s & g'_*g_*\mathcal{F} } \quad \text{or}\quad \xymatrix{ f'_*f_*\mathcal{F} \ar[r]^ s & g'_*f_*\mathcal{F} \ar[r]^{g'_*t} & g'_*g_*\mathcal{F} } \]
(these maps are equal). Since these definitions agree with the ones in Categories, Section 4.28 it follows from Categories, Lemma 4.28.2 that the axioms of a strict 2-category hold with these definitions.
Post a comment
Your email address will not be published. Required fields are marked.
In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).
All contributions are licensed under the GNU Free Documentation License.
Comments (2)
Comment #5980 by Carlos Simpson on
Comment #6154 by Johan on