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Stream: learning: questions

Topic: isococomma squares and pseudoepic morphisms

Emily (Apr 02 2024 at 23:42):

In Carboni–Johnson–Street–Verity's _Modulated Bicategories_ one finds the following:

image.png

Here "pseudopullback" actually means "isocomma object".

I wanted to confirm two basic/elementary questions I have:

1) Are pseudoepic morphisms the ones such that the analogous diagram as the above one is an "isococomma" square?
2) Is "opcomma object" the same thing as a "cocomma object"? The former is described e.g. here or in Street's Elementary Cosmoi I paper, while the later (specialised to the 2-category of categories) is described here

Matteo Capucci (he/him) (Apr 03 2024 at 07:15):

Uh never heard of opcommas. Do you think you can explain the difference between them and cocommas?

Nathanael Arkor (Apr 03 2024 at 07:57):

Emily said:

2) Is "opcomma object" the same thing as a "cocomma object"? The former is described e.g. here or in Street's Elementary Cosmoi I paper, while the later (specialised to the 2-category of categories) is described here

The terms "cocomma" and "opcomma" and "cospan" and "opspan" are used interchangeably in the 2-category literature, as far as I can tell. There is an unfortunate tension between the convention for duality in a 1-category, in which an $X$ in $\mathscr C^{\text{op}}$ is called a "co-$X$" in $\mathscr C$, and the convention for duality in a 2-category, in which there are two notions of dual, "op" and "co". It would make most sense to use the "op-" prefix for "op" and the "co-" prefix for "co", and this does recover the duality between monads and comonads, for instance. However, this choice conflicts with the terminology with 1-categories, meaning that one would call the categorification of a coproduct in a 1-category an "op-product" in a 2-category. (Arguably it is the convention for 1-categories that is bad, but it's not viable to change this now.) So different authors have chosen different conventions for 2-categories. Usually categorifications of 1-dimensional (co)limits use the "co-" prefix, whilst truly 2-dimensional (co)limits often use the "op-" prefix. But it really depends on the author, and there may be instances in the literature where neither of these conventions have been followed.

Mike Shulman (Apr 03 2024 at 14:50):

My impression is that "cocomma" is much more common in recent papers. If I saw "opcomma" I might expect it to mean a comma object in $\mathcal{C}^{\mathrm{co}}$, just as an opfibration is a fibration in $\mathcal{C}^{\mathrm{co}}$.

Emily (Apr 03 2024 at 16:46):

Perfect! Thank you so much, Nathanael and Mike :)

Mike Shulman (Apr 03 2024 at 18:51):

(Of course a comma object in $\mathcal{C}^{\mathrm{co}}$ is just a comma object in $\mathcal{C}$ of the transposed cospan.)