Useful TeX Diagrams

Feb 20 2023 · LastMod: Feb 20 2023

Diagrams are mainly generated with Quiver, maybe accompanied by some comments on the concepts.

Fibered category theory

Strong cartesianness

Definition(Strong cartesianness) . For categories $\mathcal{E},\mathcal{B}$ and a functor $p:\mathcal{E}\to\mathcal{B}$, a morphism $f\in\operatorname{Mor}(\mathcal{E})$ is called $p$-strongly cartesian, if for any $f'\in\operatorname{Mor}(\mathcal{E})$ s.t. $\operatorname{cod}(f)=\operatorname{cod}(f')$ and any morphism $\psi:p(\operatorname{dom}(f')) \to p(\operatorname{dom}(f))$ s.t. $p(f)\circ \psi = p(f')$, there is a unique $g:\operatorname{dom}(f')\to\operatorname{dom}(f)$ with $f \circ g = f'$ and $p(g)=\psi$.

 2	e && {e'} & {\mathcal{E}} \\
 3	& a \\
 4	{p(e)} && {p(e')} & {\mathcal{B}} \\
 5	& {p(a)}
 6	\arrow["{p(f)}"', color={rgb,255:red,217;green,104;blue,104}, from=3-1, to=4-2]
 7	\arrow["{p(f')}", color={rgb,255:red,117;green,117;blue,117}, from=3-3, to=4-2]
 8	\arrow["{\forall \psi}"{description}, color={rgb,255:red,107;green,107;blue,107}, curve={height=-6pt}, tail reversed, no head, from=3-1, to=3-3]
 9	\arrow["f"', color={rgb,255:red,163;green,41;blue,41}, from=1-1, to=2-2]
10	\arrow["{\forall f'}"', tail reversed, no head, from=2-2, to=1-3]
11	\arrow["{\exists ! g }", curve={height=-6pt}, dotted, tail reversed, no head, from=1-1, to=1-3]
12	\arrow["p"{description}, from=1-4, to=3-4]

When $\psi$ is restricted to be an identity, $f$ is called $p$-weakly cartesian. For a Grothendieck fibration (that is, $p$ s.t. composition of cartesian arrows are cartesian, and for all $a\in\mathcal{E}$ and all morphisms $\phi$ with codomain $p(a)$, there is a cartesian lifting of $\phi$), strong cartesianness can be replaced by weak cartesianness. A (contravariant) pseudofunctor sending $b\in\mathcal{B}$ and $1_b$ to their inverse images (which form a category) is the Grothendieck construction.