Span composition using fake pullbacks

Ross Street

arxiv: 1907.02695 · v1 · pith:VMTXHTENnew · submitted 2019-07-05 · 🧮 math.CT

Span composition using fake pullbacks

Ross Street This is my paper

Pith reviewed 2026-05-25 02:01 UTC · model grok-4.3

classification 🧮 math.CT

keywords spansfake pullbacksEM-spansPuppe-exact categoriesrelationscompositionPROP for monoids

0 comments

The pith

Spans of EM-spans can be composed using fake pullbacks in categories without traditional pullbacks.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper shows how to build a category of spans in categories that lack ordinary pullbacks. It does so by restructuring an existing proof for relations in Puppe-exact categories, observing that these relations are spans of EM-spans. Because EM-spans admit fake pullbacks, their spans can be composed, and this works in a setting more general than Puppe-exact categories, including the PROP for monoids.

Core claim

Relations are spans of EM-spans, and EM-spans admit fake pullbacks so that spans of EM-spans compose. This allows the construction of a category of spans in some categories C which do not have pullbacks in the traditional sense.

What carries the argument

Fake pullbacks of EM-spans, which serve as the mechanism for composing spans of EM-spans.

If this is right

The category of spans exists in the PROP for monoids.
Associativity of composition of relations holds via this restructured proof in Puppe-exact categories.
The construction applies to a broader class of categories than Puppe-exact ones.
Spans can be formed without requiring the category to have all pullbacks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This method may extend to other algebraic categories or PROPs where pullbacks are absent.
It could provide a framework for studying relations in more general homological settings.
Similar fake pullback techniques might apply to other composition problems in category theory.

Load-bearing premise

That EM-spans admit fake pullbacks in the given category as needed for the composition to be well-defined and associative.

What would settle it

Constructing or identifying a category where EM-spans do not admit fake pullbacks, which would prevent the composition from being associative.

read the original abstract

The construction of a category of spans can be made in some categories $\CC$ which do not have pullbacks in the traditional sense. The PROP for monoids is a good example of such a $\CC$. The 2012 book concerning homological algebra by Marco Grandis gives the proof of associativity of relations in a Puppe-exact category based on a 1967 paper of M.\v{S}. Calenko. The proof here is a restructuring of that proof in the spirit of the first sentence of this Abstract. We observe that these relations are spans of EM-spans and that EM-spans admit fake pullbacks so that spans of EM-spans compose. Our setting is more general than Puppe-exact categories.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

This note rephrases the Calenko-Grandis argument so spans compose via fake pullbacks on EM-spans in categories without ordinary pullbacks.

read the letter

The punchline is that this short note lets you build a category of spans in some categories that lack pullbacks, by switching to fake pullbacks on EM-spans. It applies to the PROP for monoids and gives a setting wider than Puppe-exact categories. What is new is the specific restructuring: relations are spans of EM-spans, and those admit fake pullbacks so the composition works. It takes the Calenko 1967 argument as given in Grandis 2012 and rewrites it around this idea instead of ordinary pullbacks. The paper does well at keeping the argument short and focused on the change in setup. The citation pattern is solid, pointing back to the original sources without claiming more than a rephrasing. The soft spot is the load-bearing assumption that EM-spans do admit fake pullbacks in the target categories and that this yields associative composition. The note invokes this to get the result, so the reader has to verify that step in each new category. If that holds, the rest follows. No circularity or invented entities here. This paper is for category theorists who need span or relation constructions outside the usual exact categories. A reader working on homological algebra or algebraic theories in general categories would get value from seeing how the proof adapts. It deserves a serious referee because the claim is concrete and the change in perspective is useful even if incremental. Recommendation: yes, send it out for review.

Referee Report

2 major / 2 minor

Summary. The manuscript re-structures the Calenko (1967)/Grandis (2012) proof of associativity for relation composition in Puppe-exact categories. It observes that relations are spans of EM-spans, that EM-spans admit fake pullbacks, and that this permits well-defined associative span composition in categories lacking ordinary pullbacks (e.g., the PROP for monoids), yielding a setting strictly more general than Puppe-exact categories.

Significance. If the fake-pullback construction is verified, the result supplies a conceptual route to span categories in algebraic and diagrammatic settings where pullbacks fail to exist, extending the reach of relation-based homological algebra beyond the Puppe-exact case while preserving the original proof architecture.

major comments (2)

[Abstract] The load-bearing step is the assertion that EM-spans admit fake pullbacks yielding associative composition (Abstract and the re-structured argument). No explicit construction or verification is supplied for the PROP-for-monoids example; without it the claimed generality over Puppe-exact categories remains unconfirmed.
[Section 3 (re-structuring of the proof)] The re-structuring invokes the existence of fake pullbacks on EM-spans to replace ordinary pullbacks in the Calenko/Grandis argument. The manuscript must exhibit the concrete diagrams or universal property that guarantees these fake pullbacks exist and compose associatively when ordinary pullbacks do not.

minor comments (2)

The citation to Grandis 2012 should include full bibliographic details (title, series, publisher) rather than the year alone.
Notation for EM-spans and fake pullbacks should be introduced with a short diagram or universal-property statement on first use to aid readers unfamiliar with the 1967/2012 sources.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and for identifying points where the manuscript would benefit from additional explicit detail. We address each major comment below and will revise the manuscript to incorporate the requested constructions and diagrams.

read point-by-point responses

Referee: [Abstract] The load-bearing step is the assertion that EM-spans admit fake pullbacks yielding associative composition (Abstract and the re-structured argument). No explicit construction or verification is supplied for the PROP-for-monoids example; without it the claimed generality over Puppe-exact categories remains unconfirmed.

Authors: We agree that the manuscript states the PROP for monoids as an illustrative example without supplying a concrete verification of the fake-pullback construction in that specific category. The general argument for EM-spans is developed abstractly, but to substantiate the claim of strict generality we will add an explicit construction of the relevant fake pullbacks together with a direct check of associativity for that example. revision: yes
Referee: [Section 3 (re-structuring of the proof)] The re-structuring invokes the existence of fake pullbacks on EM-spans to replace ordinary pullbacks in the Calenko/Grandis argument. The manuscript must exhibit the concrete diagrams or universal property that guarantees these fake pullbacks exist and compose associatively when ordinary pullbacks do not.

Authors: Section 3 replaces ordinary pullbacks by the universal property of fake pullbacks in the re-structured argument. We accept that the current text would be clearer with explicit diagrams. The revision will include diagrams that display the fake-pullback squares for EM-spans and a short verification that the resulting composition remains associative by the same diagram-chasing steps used in the original Calenko/Grandis proof. revision: yes

Circularity Check

0 steps flagged

No circularity: restructured proof relies on external citations and definitional observations without self-reduction.

full rationale

The paper explicitly restructures the Calenko/Grandis proof of associativity for relations in Puppe-exact categories, observing that relations are spans of EM-spans which admit fake pullbacks. This observation is presented as a direct consequence of the definitions in the given category C (e.g., PROP for monoids), not derived from fitted parameters, self-citations that bear the load, or ansatzes smuggled from prior author work. The central generality claim (composition via fake pullbacks in categories lacking ordinary pullbacks) is supported by reference to 1967 and 2012 external literature rather than reducing to the paper's own inputs by construction. No equations or steps equate a claimed result to its own definition or fit.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The construction rests on the domain assumption that EM-spans admit fake pullbacks; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption EM-spans admit fake pullbacks enabling associative composition of spans of EM-spans
Invoked as the mechanism allowing the category of spans to be constructed without traditional pullbacks.

pith-pipeline@v0.9.0 · 5634 in / 1015 out tokens · 19527 ms · 2026-05-25T02:01:28.055542+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We observe that these relations are spans of EM-spans and that EM-spans admit fake pullbacks so that spans of EM-spans compose. Our setting is more general than Puppe-exact categories.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The fake pullback of a cospan ... is constructed as follows. Factorize r ≃ x ∘ a and s ≃ y ∘ b ... take the bipullback ...

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
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