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Stream: theory: category theory

Topic: the walking monomorphism

Morgan Rogers (he/him) (Jun 21 2022 at 09:13):

Let $C$ be a category, and consider its "mono-arrow category" $C^{\hookrightarrow}$, whose objects are monomorphisms in $C$, and whose morphisms are pairs of arbitrary morphisms forming commuting squares. This is a full subcategory of the usual arrow category.

Unless I'm making an interesting mistake, one can characterize this category, when equipped with the evident domain and codomain functors to $C$, as the universal span over the cospan of identity functors of $C$ in which the square is filled by a monic natural transformation.

But how can one capture this construction as a 2-limit? The usual arrow category construction is obtained by weighting over the walking arrow, but there is no "walking monomorphism". Is there a neat trick for capturing a "universal monic natural tranformation" like this?

David Michael Roberts (Jun 21 2022 at 14:05):

Can you use the fact a morphism $f\colon x \to y$ is a mono iff the pullback of $x\to y \leftarrow x$ has cone $x \stackrel{=}{\leftarrow} x \stackrel{=}{\to} x$?

Morgan Rogers (he/him) (Jun 21 2022 at 14:21):

Maybe in Lex, but in Cat you can't force the preservation of pullbacks so if you use that as a weight I think it doesn't work :thinking:

Mike Shulman (Jun 21 2022 at 16:45):

I don't think this can be a 2-limit construction in Cat. If it were, it would presumably be functorial, but since an arbitrary functor may not preserve monos this construction is not functorial on Cat.

Morgan Rogers (he/him) (Jun 21 2022 at 20:58):

@Nathanael Arkor made a sensible-sounding suggestion about enriching over "categories with a distinguished class of arrows" so as to be able to keep track of them. It's still surprising to me that this is hard to lift to a 2-categorical construction :)

Mike Shulman (Jun 21 2022 at 21:08):

Yes, that could also work.

Martin Szyld (Jun 28 2022 at 19:24):

Hi there, if I'm not mistaken you can see this as an example of a type of 2-dimensional limit (originally due to Gray), for the diagram 2 --> Cat mapping the arrow to $id_C$. This limit is taken w.r.to a family of 2-cells in the 2-category where your diagram lives (so in this case the monic natural transformations in Cat), and it is just like a lax limit but you require the structural 2-cells of the cones to be in this fixed family (in this case there is just one structural 2-cell, filling your square which is also a triangle).

(I don't see this working for these limits however if you take the cospan you mention as your diagram because you would get two 2-cells in opposite directions).

Note that this is not so different to the idea of enriching over "categories with a distinguished class of arrows", since the arrows in the hom-categories are 2-cells :)

By Mike's answer, it seems like this is not naturally a 2-limit, this makes me like your example, so thanks for your question! :)