Smash Products for Non-cartesian Internal Prestacks
Pith reviewed 2026-05-24 17:22 UTC · model grok-4.3
The pith
An analogue of the smash product exists for prestacks internal to non-cartesian monoidal categories.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper defines a smash product construction for an internal prestack F from the opposite of a base to the category of internal categories in a non-cartesian monoidal category, producing a fibration whose base is the original base, and proves that taking fibers recovers the prestack and coinvariants do likewise. The construction simultaneously generalizes the Grothendieck construction for prestacks and smash products for B-module algebras over a bialgebra B.
What carries the argument
The smash product analogue for non-cartesian internal prestacks, which converts an internal prestack into a fibration over its base while preserving the original data via fibers or coinvariants.
If this is right
- The Grothendieck construction extends to internal prestacks in monoidal categories that lack cartesian products.
- Smash products for B-module algebras arise as the special case when the internal prestack comes from a bialgebra.
- Recovery of the prestack by fibers or coinvariants continues to hold after the generalized construction.
- Fibrations built this way inherit the internal structure from the starting prestack.
Where Pith is reading between the lines
- The same method could be tested in concrete non-cartesian examples such as modules over a ring equipped with tensor product.
- It may supply a route to internal versions of other fibration-related constructions that currently assume cartesian structure.
- Applications in Hopf algebra theory or quantum group settings become available once the internal prestack language is set up.
Load-bearing premise
A non-cartesian monoidal category must admit a well-behaved notion of internal prestack so that the smash product analogue can be defined while preserving the fibration and recovery properties.
What would settle it
An explicit non-cartesian monoidal category together with an internal prestack such that the constructed object is not a fibration or such that taking fibers or coinvariants fails to recover the original prestack.
read the original abstract
The smash product construction (or the Grothendieck construction) takes a functor (or prestack) $F \colon B^{op} \to \mathbf{Cat}$ and returns a fibration $p \colon A \to B$. In this paper, we develop an analogue of the smash product for prestacks internal to a non-cartesian monoidal category. Our construction simultaneously generalizes the Grothendieck construction for prestacks and smash products for $B$-module algebras over a bialgebra $B$. Further, taking fibers or coinvariants allows one to recover the original prestack.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper develops an analogue of the smash product (Grothendieck construction) for prestacks internal to a non-cartesian monoidal category C. The construction simultaneously generalizes the standard Grothendieck construction for ordinary prestacks and smash products for B-module algebras over a bialgebra B, producing a fibration p : A → B from which the original prestack is recovered by taking fibers or coinvariants.
Significance. If the construction is rigorously defined with appropriate hypotheses, the result would unify fibration-theoretic and algebraic smash-product techniques in a broader class of monoidal categories, potentially enabling new applications in settings such as modules over rings or other non-cartesian structures where cartesian assumptions fail.
major comments (3)
- [Abstract and §2] Abstract and §2 (definition of internal prestack): the central claim requires a well-behaved notion of internal prestack in a non-cartesian monoidal category C, yet no explicit hypotheses are stated on C (e.g., existence of coequalizers for coinvariants, closedness, or tensor compatibility with the internal category structure). This is load-bearing for the fibration property and the recovery statements via fibers/coinvariants.
- [§4] §4 (smash product construction): the generalization from the cartesian/Grothendieck case and from B-module algebras must be verified to preserve the fibration axiom without additional structure; the abstract provides no indication that the requisite coequalizers or universal properties are assumed or constructed.
- [§5] §5 (recovery via fibers or coinvariants): the claim that the original prestack is recovered must be shown to hold under the stated (or missing) conditions on C; without explicit verification that coinvariants exist and commute appropriately with the monoidal structure, the recovery property may fail in the non-cartesian setting.
minor comments (2)
- [Notation] Clarify notation distinguishing internal prestacks from ordinary prestacks throughout the text.
- [§2] Add a dedicated subsection listing all standing assumptions on the monoidal category C.
Simulated Author's Rebuttal
We thank the referee for the careful reading and for identifying points where the hypotheses on the base monoidal category and the verifications of the fibration and recovery properties need to be made fully explicit. We address each major comment below and will revise the manuscript to incorporate the requested clarifications and proofs.
read point-by-point responses
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Referee: [Abstract and §2] Abstract and §2 (definition of internal prestack): the central claim requires a well-behaved notion of internal prestack in a non-cartesian monoidal category C, yet no explicit hypotheses are stated on C (e.g., existence of coequalizers for coinvariants, closedness, or tensor compatibility with the internal category structure). This is load-bearing for the fibration property and the recovery statements via fibers/coinvariants.
Authors: We agree that the standing assumptions on C were not collected in one place. The construction uses coequalizers to form the coinvariants that appear in the smash product and in the recovery statements; these are assumed to exist and to be preserved by the tensor product in each variable. In the revised manuscript we will add an explicit paragraph at the start of §2 listing these hypotheses (existence of coequalizers, compatibility with the monoidal structure, and the internal-category axioms that are preserved). This makes the load-bearing conditions visible without altering the statements of the main theorems. revision: yes
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Referee: [§4] §4 (smash product construction): the generalization from the cartesian/Grothendieck case and from B-module algebras must be verified to preserve the fibration axiom without additional structure; the abstract provides no indication that the requisite coequalizers or universal properties are assumed or constructed.
Authors: The definition of the smash product in §4 is given by a coequalizer that is required to exist under the hypotheses we will state in §2. We will expand the argument in §4 to include an explicit verification that the resulting internal category satisfies the fibration axioms (i.e., that the appropriate pullback diagrams exist and that the universal property of the coequalizer yields the required cartesian lifts). The verification will be written so that it specializes directly to both the classical Grothendieck construction and the smash product of B-module algebras, thereby confirming that no extra structure beyond the listed hypotheses is needed. revision: yes
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Referee: [§5] §5 (recovery via fibers or coinvariants): the claim that the original prestack is recovered must be shown to hold under the stated (or missing) conditions on C; without explicit verification that coinvariants exist and commute appropriately with the monoidal structure, the recovery property may fail in the non-cartesian setting.
Authors: We accept that the recovery statements require a self-contained proof under the hypotheses on C. In the revised §5 we will insert two propositions: one showing that the fiber functor recovers the original prestack when C is cartesian, and one showing that the coinvariant functor recovers it when the coinvariants are taken with respect to the bialgebra action. Each proof will explicitly use the coequalizer compatibility assumptions added in §2. If any additional commutation condition is needed in the non-cartesian case, it will be stated as a further hypothesis rather than left implicit. revision: yes
Circularity Check
No circularity; new construction is independently defined
full rationale
The paper presents an original construction for smash products of internal prestacks in non-cartesian monoidal categories that generalizes two known cases (Grothendieck for prestacks and smash products for B-module algebras). No equations, definitions, or claims in the provided text reduce a result to its own inputs by construction, rename a fitted parameter as a prediction, or rely on load-bearing self-citations whose content is unverified. The central claim rests on a novel definition whose validity is independent of the target recovery properties, making the derivation self-contained.
discussion (0)
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