This post answers a question that had come up some time ago with Arthur Apter, and more recently with Philipp Schlicht and Arthur Apter.

Definition. A cardinal 𝜅 is uniformly  <𝜃-supercompact if there is an embedding 𝑗 :𝑉 →𝑀 having critical point 𝜅, with 𝑗⁡(𝜅) >𝜃 and 𝑀<𝜃 ⊂𝑀.

(Note:  This is typically stronger than merely asserting that 𝜅 is 𝛾-supercompact for every 𝛾 <𝜃, a property which is commonly denoted <𝜃-supercompact, so I use the adjective “uniformly” to highlight the distinction.)

Two easy observations are in order.  First, if 𝜃 is singular, then 𝜅 is uniformly <𝜃-supercompact if and only if 𝜅 is 𝜃-supercompact, since the embedding 𝑗 :𝑉 →𝑀 will have 𝑗⁢′′⁢𝜆 ∈𝑀 for every 𝜆 <𝜃, and we may assemble 𝑗⁢′′⁢𝜃 from this inside 𝑀, using a sequence of length cof⁡(𝜃). Second, in the successor case, 𝜅 is uniformly <𝜆+-supercompact if and only if 𝜅 is 𝜆-supercompact, since if 𝑗 :𝑉 →𝑀 has 𝑀𝜆 ⊂𝑀, then it also has 𝑀<𝜆+ ⊂𝑀. So we are mainly interested in the concept of uniform <𝜃-supercompactness when 𝜃 is weakly inaccessible.

Definition. Let us say of a cardinal 𝜅 that ⟨𝜇𝜆 ∣𝜆 <𝜃⟩ is a coherent 𝜃-system of normal fine measures, if each 𝜇𝜆 is a normal fine measure on 𝑃𝜅⁢𝜆, which cohere in the sense that if 𝜆 <𝛿 <𝜃, then 𝜇𝜆 ≤𝑅⁢𝐾𝜇𝛿, and more specifically 𝑋 ∈𝜇𝜆 if and only if {𝜎 ∈𝑃𝜅⁢𝛿 ∣𝜎 ∩𝜆 ∈𝑋} ∈𝜇𝛿.  In other words, 𝜇𝜆 =𝑓 ∗𝜇𝛿, where 𝑓 :𝑃𝜅⁢𝛿 →𝑃𝜅⁢𝜆 is the function that chops off at 𝜆, so that 𝑓 :𝜎 ↦𝜎 ∩𝜆.

Theorem.  The following are equivalent, for any regular cardinals 𝜅 ≤𝜃.

1. The cardinal 𝜅 is uniformly <𝜃-supercompact.

2. There is a coherent 𝜃-system of normal fine measures for 𝜅.

Proof. The forward implication is easy, since if 𝑗 :𝑉 →𝑀 has 𝑀<𝜃 ⊂𝑀, then we may let 𝜇𝜆 be the normal fine measure on 𝑃𝜅⁢𝜆 generated by 𝑗⁢′′⁢𝜆 as a seed, so that 𝑋 ∈𝜇𝜆 ⟺ 𝑗⁡′′⁢𝜆 ∈𝑗⁡(𝑋).  Since the seeds 𝑗⁢′′⁢𝜆 cohere as initial segments, it follows that 𝜇𝜆 ≤𝑅⁢𝐾𝜇𝛿 in the desired manner whenever 𝜆 <𝛿 <𝜃.

Conversely, fix a coherent system ⟨𝜇𝜆 ∣𝜆 <𝜃⟩ of normal fine measures. Let 𝑗𝜆 :𝑉 →𝑀𝜆 be the ultrapower by 𝜇𝜆. Every element of 𝑀𝜆 has the form 𝑗𝜆⁡(𝑓)⁢(𝑗⁡′′⁢𝜆).  Because of coherence, we have an elementary embedding 𝑘𝜆,𝛿 :𝑀𝜆 →𝑀𝛿 defined by 𝑘𝜆,𝛿:𝑗𝜆⁡(𝑓)⁢(𝑗⁡′′⁢𝜆)↦𝑗𝛿⁡(𝑓)⁢(𝑗⁡′′⁢𝜆). It is not difficult to check that these embeddings altogether form a commutative diagram, and so we may let 𝑗 :𝑉 →𝑀 be the direct limit of the system, with corresponding embeddings 𝑘𝜆,𝜃 :𝑀𝜆 →𝑀.  The critical point of 𝑘𝜆,𝛿 and hence also 𝑘𝜆,𝜃 is larger than 𝜆.  This embedding has critical point 𝜅, and I claim that 𝑀<𝜃 ⊂𝑀. To see this, suppose that 𝑧𝛼 ∈𝑀 for each 𝛼 <𝛽 where 𝛽 <𝜃.  So 𝑧𝛼 =𝑘𝜆𝛼,𝜃⁡(𝑧∗𝛼) for some 𝑧∗𝛼 ∈𝑀𝜆𝛼. Since 𝜃 is regular, we may find 𝜆 <𝜃 with 𝜆𝛼 ≤𝜆 for all 𝛼 <𝛽 and also 𝛽 ≤𝜆, and so without loss we may assume 𝜆𝛼 =𝜆 for all 𝛼 <𝛽. Since 𝑀𝜆 is closed under 𝜆-sequences, it follows that ⃗𝑧∗ =⟨𝑧∗𝛼 ∣𝛼 <𝛽⟩ ∈𝑀𝜆.  Applying 𝑘𝜆,𝜃 to ⃗𝑧∗ gives precisely the desired sequence ⃗𝑧 =⟨𝑧𝛼 ∣𝛼 <𝛽⟩ inside 𝑀, showing this instance of 𝑀<𝜃 ⊂𝑀. QED

The theorem does not extend to singular 𝜃.

Theorem.  If 𝜅 is 𝜃-supercompact for a singular strong limit cardinal 𝜃 above 𝜅, then there is a transitive inner model in which 𝜅 has a coherent system ⟨𝜇𝜆 ∣𝜆 <𝜃⟩  of normal fine measures, but 𝜅 is not uniformly <𝜃-supercompact.

Thus, the equivalence of the first theorem does not hold generally for singular 𝜃.

Proof.  Suppose that 𝜅 is 𝜃-supercompact, where 𝜃 is a singular strong limit cardinal. Let 𝑗 :𝑉 →𝑀 be a witnessing embedding, for which 𝜅 is not 𝜃-supercompact in 𝑀 (use a Mitchell-minimal measure).  Since 𝜃 is singular, this means by the observation after the definition above that 𝜅 is not uniformly <𝜃-supercompact in 𝑀. But meanwhile, 𝜅 does have a coherent system of normal fine ultrafilters in 𝑀, since the measures defined by 𝑋 ∈𝜇𝜆 ⟺ 𝑗⁡′′⁢𝜆 ∈𝑗⁡(𝑋) form a coherent system just as in the theorem, and the sequence ⟨𝜇𝜆 ∣𝜆 <𝜃⟩ is in 𝑀 by 𝜃-closure. QED

The point is that in the singular case, the argument shows only that the direct limit is <cof⁡(𝜃)-closed, which is not the same as <𝜃-closed when 𝜃 is singular.

The example of singular 𝜃 also shows that 𝜅 can be <𝜃-supercompact without being uniformly <𝜃-supercompact, since the latter would imply full 𝜃-supercompactness, when 𝜃 is singular, but the former does not. The same kind of reasoning separates uniform from non-uniform <𝜃-supercompactness, even when 𝜃 is regular.

Theorem. If 𝜅 is uniformly <𝜃-supercompact for an inaccessible cardinal 𝜃, then there is a transitive inner model in which 𝜅 is <𝜃-supercompact, but not uniformly <𝜃-supercompact.

Proof. Suppose that 𝜅 is uniformly <𝜃-supercompact, witnessed by embedding 𝑗 :𝑉 →𝑀, with 𝑀<𝜃 ⊂𝑀, and furthermore assume that 𝑗⁡(𝜅) is as small as possible among all such embeddings. It follows that there can be no coherent 𝜃-system of normal fine measures for 𝜅 inside 𝑀, for if there were, the direct limit of the associated embedding would send 𝜅 below 𝑗⁡(𝜅), which from the perspective of 𝑀 is a measurable cardinal far above 𝜅 and 𝜃. But meanwhile, 𝜅 is 𝛽-supercompact in 𝑀 for every 𝛽 <𝜃. Thus, 𝜅 is <𝜃-supercompact in 𝑀, but not uniformly <𝜃-supercompact, and so the notions do not coincide. QED

Meanwhile, if 𝜃 is weakly compact, then the two notions do coincide. That is, if 𝜅 is <𝜃-supercompact (not necessarily uniformly), and 𝜃 is weakly compact, then in fact 𝜅 is uniformly <𝜃-supercompact, since one may consider a model 𝑀 of size 𝜃 with 𝜃 ∈𝑀 and 𝑉𝜃 ⊂𝑀, and apply a weak compactness embedding 𝑗 :𝑀 →𝑁. The point is that in 𝑁, we get that 𝜅 is actually 𝜃-supercompact in 𝑁, which provides a uniform sequence of measures below 𝜃.