Revisiting colimits in $\mathbf{Cat}$ and homotopy category

Varinderjit Mann

arxiv: 2603.07773 · v2 · submitted 2026-03-08 · 🧮 math.CT

Revisiting colimits in Cat and homotopy category

Varinderjit Mann This is my paper

Pith reviewed 2026-05-15 14:29 UTC · model grok-4.3

classification 🧮 math.CT

keywords colimitsCathomotopy categorynerve functorsimplicial setsweighted colimitsreflective embeddinglimits

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The pith

The homotopy category functor from simplicial sets to categories exists exactly when certain weighted colimits exist in the category of categories.

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

The paper establishes an equivalence between the homotopy category functor h from simplicial sets to categories and a specific class of weighted colimits in the category of categories. By constructing those colimits explicitly from properties of simplicial sets and the nerve functor, it shows that the nerve embedding of categories into simplicial sets is reflective. This reflectiveness immediately yields that the category of categories has all small limits and colimits. The same explicit method is then used to give new constructions for coequalizers and localizations of categories.

Core claim

We first demonstrate an equivalence between the existence of the homotopy category functor h : sSet → Cat and the existence of a specific class of weighted colimits in Cat. We then construct these weighted colimits explicitly by using certain properties of simplicial sets and the nerve functor. Consequentially, the embedding N : Cat ↪ sSet is reflective, and can be used to infer the (co)completeness of Cat. Finally, we use this approach to reformulate the construction of coequalizers and localizations in Cat.

What carries the argument

The equivalence relating the homotopy category functor h to a class of weighted colimits in Cat, constructed explicitly via the nerve functor.

If this is right

Cat has all small colimits.
Cat has all small limits.
Coequalizers in Cat arise directly as the weighted colimits tied to the homotopy functor.
Localizations of categories admit reformulations as instances of these weighted colimits.

Where Pith is reading between the lines

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

The construction may let one compute colimits in Cat by first forming the corresponding simplicial object and then applying the homotopy functor.
Reflectiveness of the nerve could transfer other categorical properties between Cat and sSet without separate proofs.
One could verify the method by explicitly building the pushout of two functors in Cat and checking it matches the standard definition.

Load-bearing premise

That the standard properties of simplicial sets and the nerve functor suffice to construct the required weighted colimits explicitly without additional data or choices.

What would settle it

An explicit weighted colimit in Cat that cannot be recovered from the homotopy category functor, or failure to obtain a known colimit such as the coequalizer of two parallel functors by the described nerve-based construction.

read the original abstract

In this paper, we justify and make precise an elementary approach that establishes the existence of (co)limits in $\mathbf{Cat}$. This approach, while conceptually evident, has not been made fully explicit or systematically described in the literature. We first demonstrate an equivalence between the existence of the homotopy category functor $h : \mathbf{sSet} \rightarrow \mathbf{Cat}$ and the existence of a specific class of weighted colimits in $\mathbf{Cat}$. We then construct these weighted colimits explicitly by using certain properties of simplicial sets and the nerve functor. Consequentially, the embedding $N : \mathbf{Cat} \hookrightarrow \mathbf{sSet}$ is reflective, and can be used to infer the (co)completeness of $\mathbf{Cat}$. Finally, we use this approach to reformulate the construction of coequalizers and localizations in $\mathbf{Cat}$.

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

The paper spells out an explicit construction of weighted colimits in Cat via the nerve and simplicial sets to show reflectivity and completeness, but the canonicity needs checking.

read the letter

The paper's main contribution is spelling out an explicit construction for certain weighted colimits in the category of categories using the nerve functor from Cat to simplicial sets and standard properties of simplicial sets. It links this to the homotopy category functor h from sSet to Cat, shows an equivalence between their existence, and then uses the reflectivity of the nerve embedding to conclude that Cat is complete and cocomplete. It also applies this to give reformulations of coequalizers and localizations in Cat. What the paper does well is taking a conceptually straightforward but often hand-wavy argument and making the steps more systematic and elementary. For someone working in category theory, having a clear recipe for these colimits can help when dealing with constructions that involve localizations or quotients. The potential issue is whether this construction is truly explicit and free of choices. The stress test points out that realizing weighted colimits often involves quotienting by 2-simplex homotopies, and the nerve doesn't canonically select representatives. If the paper provides a formula that avoids any auxiliary choices and is strictly functorial, then it works fine. But if it relies on picking representatives or additional data, that would undermine the claim of explicitness without extra assumptions. The argument seems to rest on standard facts about simplicial sets, so no obvious circularity there. This is the kind of paper that would interest category theorists who care about foundational constructions and want to see them laid out plainly. It is not revolutionary, but the explicitness could be valuable for teaching or for organizing related results. I would recommend sending it to peer review so that experts can verify the details of the construction and see if the canonicity holds up.

Referee Report

2 major / 2 minor

Summary. The paper claims an equivalence between the existence of the homotopy category functor h: sSet → Cat and the existence of a specific class of weighted colimits in Cat. It constructs these colimits explicitly from standard properties of simplicial sets and the nerve functor N: Cat ↪ sSet, thereby proving that N is reflective and that Cat is (co)complete. The approach is then used to reformulate the construction of coequalizers and localizations in Cat.

Significance. If the explicit construction succeeds without non-canonical choices, the paper would provide a clear, elementary justification for the standard fact that Cat is cocomplete, linking it directly to the nerve and homotopy category functors. This could serve as a useful reference for explicit colimit computations in Cat and strengthen the connection between simplicial sets and categories.

major comments (2)

The central construction of the weighted colimits (described after the equivalence in the main body): the paper must exhibit a strictly functorial formula that produces the colimit object in Cat without auxiliary choices of representatives for homotopy classes of 1-simplices induced by 2-simplices. The standard nerve does not canonically select such representatives, so any quotienting step risks depending on non-canonical data unless a canonical selection is proven.
The claimed equivalence between the existence of h and the weighted colimits (stated in the first main result): the direction 'existence of colimits implies existence of h' is plausible via the nerve, but the converse requires an explicit construction of h from the colimits that is shown to be functorial and to satisfy the universal property of the homotopy category; this step is load-bearing for the reflectivity claim and needs a detailed verification.

minor comments (2)

Notation for the weighted colimits should be introduced with a precise definition (e.g., as a coequalizer in Cat or via a localization) before the explicit construction is given.
The reformulation of coequalizers and localizations at the end would benefit from a short comparison table or diagram showing how the new description differs from the classical one.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thorough review and insightful comments, which help clarify the presentation of our explicit constructions. We address each major comment below and will revise the manuscript to incorporate the requested details and verifications.

read point-by-point responses

Referee: The central construction of the weighted colimits (described after the equivalence in the main body): the paper must exhibit a strictly functorial formula that produces the colimit object in Cat without auxiliary choices of representatives for homotopy classes of 1-simplices induced by 2-simplices. The standard nerve does not canonically select such representatives, so any quotienting step risks depending on non-canonical data unless a canonical selection is proven.

Authors: We agree that strict functoriality without auxiliary choices is essential. Our construction defines the weighted colimit in Cat by taking objects as the 0-simplices of the relevant simplicial set and morphisms as the quotient of 1-simplices by the equivalence relation generated by the 2-simplices (via the face and degeneracy maps). This relation is defined directly from the simplicial identities without selecting representatives for homotopy classes. We prove functoriality by showing that any morphism of diagrams induces a unique morphism of colimits via the universal property, using only the naturality of the nerve and the simplicial structure. We will add an explicit lemma in the revised manuscript verifying that the resulting functor is strictly functorial in both the diagram and the weight, with no non-canonical data involved. revision: yes
Referee: The claimed equivalence between the existence of h and the weighted colimits (stated in the first main result): the direction 'existence of colimits implies existence of h' is plausible via the nerve, but the converse requires an explicit construction of h from the colimits that is shown to be functorial and to satisfy the universal property of the homotopy category; this step is load-bearing for the reflectivity claim and needs a detailed verification.

Authors: We acknowledge that the converse direction requires a fully detailed verification to support the reflectivity of N. In the proof, we construct h(X) explicitly as the weighted colimit of the diagram given by the nerve of the terminal category weighted by X; functoriality follows because a map f: X → Y induces a map of weights, hence a unique map of colimits by the universal property. The universal property of h is verified by showing that maps from h(X) to any category C correspond exactly to maps from X to N(C) via the colimit universal property and the adjunction. We will expand this into a dedicated subsection with step-by-step checks of functoriality, naturality, and the universal property in the revised version. revision: yes

Circularity Check

0 steps flagged

No circularity: explicit construction from standard sSet and nerve properties

full rationale

The paper establishes an equivalence between the homotopy category functor h and a class of weighted colimits in Cat, then constructs those colimits directly from standard properties of simplicial sets and the nerve functor N. This is used to show reflectivity of N and hence (co)completeness of Cat. No quoted step reduces a prediction or central claim to a fitted parameter, self-definition, or load-bearing self-citation; the derivation chain remains independent of the target results and relies on external, non-circular inputs from simplicial set theory.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard properties of simplicial sets, the nerve functor, and the definition of weighted colimits; no free parameters, invented entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)

standard math Standard properties of simplicial sets and the nerve functor suffice to construct the required weighted colimits
Invoked to build the colimits explicitly from the homotopy category functor equivalence.

pith-pipeline@v0.9.0 · 5445 in / 1235 out tokens · 35804 ms · 2026-05-15T14:29:08.509982+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

12 extracted references · 12 canonical work pages

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M. A. Bednarczyk, A. M. Borzyszkowski, and W. Pawlowski. Generalized Congruences - Epimorphisms in Cat.Theory and Applications of Categories, 5(11):266–280, 1999. URL http://eudml.org/doc/120226. Transmitted by Michael Barr. p. 1

work page 1999

[2] [2]

Betti, A

R. Betti, A. Carboni, R. Street, and R. Walters. Variation through enrichment.Journal of Pure and Applied Algebra, 29(2):109–127, Aug. 1983. ISSN 0022-4049. doi: 10.1016/0022-4049(83) 90100-7. p. 1

work page doi:10.1016/0022-4049(83 1983

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Borceux.Handbook of Categorical Algebra, volume 1 : Basic Category Theory

F. Borceux.Handbook of Categorical Algebra, volume 1 : Basic Category Theory. Cambridge University Press, Aug. 1994. ISBN 9780511525858. doi: 10.1017/cbo9780511525858. p. 1

work page doi:10.1017/cbo9780511525858 1994

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Borceux.Handbook of Categorical Algebra, volume 2 : Categories and Structures

F. Borceux.Handbook of Categorical Algebra, volume 2 : Categories and Structures. Cambridge University Press, Nov. 1994. ISBN 9780511525865. doi: 10.1017/cbo9780511525865. pp. 11, 12

work page doi:10.1017/cbo9780511525865 1994

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P. G. Goerss and J. F. Jardine.Simplicial Homotopy Theory. Birkhäuser Basel, 2009. ISBN 9783034601894. doi: 10.1007/978-3-0346-0189-4. p. 22

work page doi:10.1007/978-3-0346-0189-4 2009

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Johnson and D

N. Johnson and D. Yau.2-Dimensional Categories. Oxford University Press USA - OSO, 2021. ISBN 9780192645678. p. 4

work page 2021

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ReprintsinTheoryandApplicationsof Categories, 2005

G.M.Kelly.BasicConceptsofEnrichedCategoryTheory. ReprintsinTheoryandApplicationsof Categories, 2005. URLhttp://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf. Reprint of the 1982 Cambridge University Press edition. pp. 10, 11

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Loregian,(Co)end Calculus, Cambridge University Press, 1 ed

F. Loregian.(Co)end calculus. Number 468 in London Mathematical Society lecture note series. Cambridge University Press, Cambridge, 2021. ISBN 9781108778657. doi: https: //doi.org/10.1017/9781108778657. p. 11

work page doi:10.1017/9781108778657 2021

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Lurie.Higher topos theory

J. Lurie.Higher topos theory. Number No. 170 in Annals of mathematics studies. Princeton University Press, Princeton, N.J, 2009. ISBN 1400830559. p. 18

work page 2009

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Riehl.Category theory in context

E. Riehl.Category theory in context. Dover Publications, Inc., 2016. ISBN 048680903X. pp. 1, 2, 3, 20, 27

work page 2016

[11] [11]

H. Wolff. V-cat and V-graph.Journal of Pure and Applied Algebra, 4(2):123–135, Apr. 1974. ISSN 0022-4049. doi: 10.1016/0022-4049(74)90018-8. p. 1

work page doi:10.1016/0022-4049(74)90018-8 1974

[12] [12]

doi: 10.1007/978-3-030-61203-0

D.Yau.InvolutiveCategoryTheory.SpringerInternationalPublishing,2020.ISBN9783030612030. doi: 10.1007/978-3-030-61203-0. p. 1 35

work page doi:10.1007/978-3-030-61203-0 2020