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1. • CommentRowNumber1.
   • CommentAuthorPeter Heinig
   • CommentTimeJul 10th 2017
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   Author: Peter Heinig
   Format: MarkdownItexThis is just to suggest another "test problem", concerning a question implicity asked in the thread [Higher model theory](https://nforum.ncatlab.org/post/63903/) 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 * * a vertically-strict, horizontally-weak double-category 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?

   This is just to suggest another “test problem”, concerning a question implicity asked in the thread

   Higher model theory

   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

   • • a vertically-strict, horizontally-weak double-category

   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?

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeJul 10th 2017
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   Author: Mike Shulman
   Format: MarkdownItex2-categorical algebra is very well-studied. 2-monads are one aspect of it, also there are 2-algebraic theories and so on. I'm not sure what you mean by "trigger", but if you just mean that drawing things in a different way may make the reader think of different ideas, then that has nothing to do with categorification.

   2-categorical algebra is very well-studied. 2-monads are one aspect of it, also there are 2-algebraic theories and so on. I’m not sure what you mean by “trigger”, but if you just mean that drawing things in a different way may make the reader think of different ideas, then that has nothing to do with categorification.