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Stream: theory: category theory

Topic: a kind of two-dimensional limits

Matteo Capucci (he/him) (May 29 2025 at 15:03):

So... [[strict 2-limits]], or rather Cat-limits, are strict representing objects of strict cones, i.e. satisfy

$$K(X,\lim^W F) \cong [D, {\bf Cat}]_s(J, K(X, F-))$$

where $F:J \to K$ is a diagram, $W:J \to {\bf Cat}$ is a weight, $K$ is a 2-category and ${\bf Cat}$ is the 1-category of (small) categories.
Thus the first strict refers to $\cong$ (as opposed to $\simeq$) while the second refers to $[D, {\bf Cat}]$ being the 2-category whose 1-cells are 2-natural transformations (and thus giving rise to strictly commutative cones).
OTOH, [[pseudolimits]] are strict representing objects of pseudocones, i.e. satisfy

$$\mathbb{K}(X,\operatorname{pslim}^W F) \cong [D, \mathbb{\bf Cat}]_p(J, \mathbb{K}(X, F-))$$

I wonder if there is a complementary notion where we keep the equivalence but have strict pseudocones, i.e.

$$\mathbb{K}(X,L) \simeq [D, \mathbb{\bf Cat}]_s(J, \mathbb{K}(X, F-))$$

Matteo Capucci (he/him) (May 29 2025 at 15:03):

?

Nathanael Arkor (May 29 2025 at 17:03):

The terminology for two-dimensional limits is not particularly consistent, but a reasonable choice is to call these strict bilimits (and more generally use the terminology strict/pseudo/lax/colax 2-limit and strict/pseudo/lax/colax bilimit to distinguish between the various universal properties). @Sori Lee has written about these here, for instance.

Matteo Capucci (he/him) (May 30 2025 at 06:22):

Invaluable as always @Nathanael Arkor!