This is a thread to collect methods to produce symbols signifying higher cells, especially with current nLab software.
Another arrow-related thread is this, though not for higher cells.
]]>This is just to suggest another “test problem”, concerning a question implicity asked in the thread
namely how to professionally/routinely/usually handle the fact that according to the usual conventions, one may arbitrarily modify diagrammatically-given axioms by introducing identity arrows into them, and then the resulting diagram, while of course dismissable by “nothing happened, you did the category-theoretic equivalent of a Reidemeister move”, can “trigger” different category-theoretic concepts/ideas: I could point to at least one literature reference which treats the axioms for
by displaying the parallel pair of functors
$\array{ & & & & & \\ & & \overset{cod}{\longrightarrow} & & & \\ \mathsf{D}_1 & & & & & \mathsf{D}_0 \\ & & \overset{dom}{\longrightarrow}& & & \\ & & & & & }$as the span of functors
$\array{ & && & & \\ & & & & & \\ \mathsf{D}_0 & & & \overset{\text{----------------------}}{\text{------------------}} && & \mathsf{D}_0 \\ & \overset{cod}{\nwarrow} & & \overset{dom}{\nearrow} & & \\ & & \mathsf{D}_1 & & & }$which then may “trigger” the concept of pushouts-in-CAT, and another literature reference which arbitrarily”blows-up” the parallel pair of functors into a cospan of functors
$\array{ & && & & \\ & & & & & \\ \mathsf{D}_1 & & & \overset{\text{----------------------}}{\text{------------------}} && & \mathsf{D}_1 \\ & \overset{cod}{\searrow} & & \overset{dom}{\swarrow} & & \\ & & \mathsf{D}_0 & & & }$which may “trigger” the concept of pullbacks-in-CAT.
One of the references actually uses pullbacks-in-CAT in the sequel .
I do not expect there to be much to be said, in particular since pushouts and pullbacks are dual concepts (which in turn is reflected in how small the “edit-distance” from the above code for the above span into the above code of the above cospan is) so that here in a sense even less happens than in the case of the example of monoidal bicategories axiomatized with, alternatively, a triangular or a quadrangular $\mu$iddle unitor. Except for some evident things, such as that a reasonable definition of Mod() should judge
Mod(axiomatization of double-categories with a parallel pair)
“equal” to
Mod(axiomatization of double-categories with a span)
“equal” to
Mod(axiomatization of double-categories with a cospan)
and therefore must somehow always simultaneously be “aware” of both sides of any two dual cat-concepts.
Again, there probably is not much to say here, I am just floating this particularly “trivial” and symmetric and self-dual example. But who knows?
]]>Consider sets and functions. A relation between two sets can then be expressed as a subset of a Cartesian product, in other words, we can define it and describe it using functions.
Viceversa, can we describe functions using relations as the “fundamental” arrows? That is, can we define and describe functions without using other functions, only relations?
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