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Stream: practice: terminology & notation

Topic: Differentiating discrete fibrations from left actions

Nathanael Arkor (Apr 26 2025 at 12:28):

A [[discrete fibration]] is a functor satisfying a property. However, every discrete fibration over a category $B$ is specified by simpler data: namely a set $|E|$, a function $p : |E| \to |B|$, and for each morphism $b \to p(e)$ an element of $|E|$ satisfying a unitality and associativity law.

I would like terminology for this data that differentiates it from a discrete fibration (I am working in a setting where the analogues of these two notions are different enough to warrant different names). However, I've been struggling to find good terminology in the literature.

• Mac Lane calls them "left-$B$-objects" or "internal diagrams".
• Borceux calls them "internal base-valued functors".

I would like terminology that doesn't involve "internal", and ideally captures the intuition of a "lifting" operation or fibres in some way. Is there any such terminology in the literature already, or does anyone have any suggestions?

Adittya Chaudhuri (Apr 26 2025 at 13:52):

In the context of Lie groupoids, I am aware of some terminology for fibrations with splitting cleavage described via left actions:

For example,

• In the context of the left action of a Lie groupoid $\mathbb{X}:=[X_1 \rightrightarrows X_0]$ on a manifold $E$ equipped with an right action of a Lie group $G$, they are sometimes called a principal $G$-bundle over the Lie groupoid $\mathbb{X}$ (Definition 2.2 in Chern-Weil map for principal bundles over groupoids).

• In the context of tangent bundle over a Lie groupoid they are sometimes called Ehresmann connection on Lie groupoids (as in the section 2.4 of Representations up to homotopy and Bott’s spectral sequence for Lie groupoids).

• In the context of representation theory of Lie groupoids, they are sometimes called representations of Lie groupoids (See section 2.1 of VB-GROUPOIDS AND REPRESENTATION THEORY OF LIE GROUPOIDS).

However, it is also possible that I might have misunderstood your question.

Amar Hadzihasanovic (Apr 26 2025 at 14:47):

What about calling it a $B$-graded set?

Mike Shulman (Apr 26 2025 at 14:49):

My instinct would be "left $B$-module".

James Deikun (Apr 26 2025 at 19:12):

Mine would be "bundled presheaf".