Pith. sign in

REVIEW 4 major objections 7 minor 65 references

An illustration of formal moduli problems with differential graded Lie algebras

T0 review · 4 major / 7 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Over a characteristic-zero field, formal moduli problems are equivalent to differential graded Lie algebras, and the paper lays out this correspondence for non-experts.

desk verdict Honest, useful sketch of the Lurie–Pridham equivalence, but as-is it has a load-bearing definitional gap and enough typos that I would not hand it to a student. read the letter →

arxiv 2506.12977 v1 pith:TBZYQ2YW submitted 2025-06-15 math.AG

Ethan Eugene Wynner This is my paper · ORCID
classification math.AG MSC 14D1517B5518N60
keywords formalmoduliproblemsdifferentialgradedLiealgebrasdeformationtheoryderivedalgebraicgeometryKoszuldualityMaurer-Cartanequation∞-categoriesChevalley-Eilenbergcohomology
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper is an expository set of notes, not original research, whose goal is to make the central correspondence of derived deformation theory accessible: over a field of characteristic zero, formal moduli problems — functors on small commutative algebras that classify deformations — are equivalent, as ∞-categories, to differential graded Lie algebras. The author states the aim of being welcoming and insightful for the non-expert. The paper develops the dictionary: a dg Lie algebra $\mathfrak{g}_*$ controls a formal moduli problem through the Maurer-Cartan equation $dx = [x,x]$, and the Chevalley-Eilenberg cochain complex provides the bridge. A sympathetic reader would take the paper to establish the coPGI theorem by assembling the machinery of deformation theories, Koszul duality, and model categories, while delegating the two heaviest structural inputs to the cited sources.

What carries the argument

The load-bearing object is the Koszul duality functor $D : (\mathrm{CAlg}^{\mathrm{aug}}_k)^{\mathrm{op}} \to \mathrm{Lie}_k$, defined as the right adjoint of the cohomological Chevalley-Eilenberg functor $C^* : \mathfrak{g}_* \mapsto C^*(\mathfrak{g}_*)$. It carries the deformation-theoretic content: checking the axioms of a weak deformation theory reduces to a proposition about cofibrant dg Lie algebras freely generated by finite-dimensional graded spaces in negative degrees, and the extra sifted-colimit condition is verified through the free-Lie and forgetful adjunction. The equivalence itself is implemented by the Maurer-Cartan functor $MC(R, \mathfrak{g}_*) = \mathrm{Map}_{\mathrm{Lie}_k}(D(R), \mathfrak{g}_*)$, whose objects are solutions of $dx = [x,x]$ in $\mathfrak{m}_R \otimes \mathfrak{g}_*$.

What would settle it

Inspect the simplest nontrivial example: take $\mathfrak{g}_* = k$ concentrated in degree 0 with zero bracket, and compute the Maurer-Cartan space $\Psi(\mathfrak{g}_*)(k \oplus k[n])$; the equivalence predicts this space has the homotopy type of $\mathrm{Map}_{\mathrm{Lie}_k}(D(k \oplus k[n]), k)$, and a mismatch in any homotopy group for any $n$ would disprove the theorem.

Watch

Extended reading notes

Core claim

The central claim is Theorem 4.2.4 (coPGI): for a field $k$ of characteristic zero, inverting quasi-isomorphisms in the model category of differential graded Lie algebras yields an ∞-category $\mathrm{Lie}_k$, and there is an equivalence of ∞-categories $\Psi : \mathrm{Lie}_k \to \mathrm{Moduli}_k$ with the ∞-category of formal moduli problems over $k$. Concretely, a dg Lie algebra $\mathfrak{g}_*$ is sent to the functor $R \mapsto \mathrm{Map}_{\mathrm{Lie}_k}(D(R), \mathfrak{g}_*)$, where $D$ is the Koszul duality functor; this mapping space is the space of Maurer-Cartan elements in $\mathfrak{m}_R \otimes \mathfrak{g}_*$. The proof shows that $D$ satisfies the axioms of a deformation theory, so the general reconstruction theorem (Theorem 1.4.6) applies and forces $\Psi$ to be fully faithful with essential image exactly the formal moduli problems.

Load-bearing premise

The paper's proof of the equivalence inherits two heavyweight theorems from the cited sources as black boxes; if either theorem is mis-stated or the higher-categorical framework it assumes differs from the original, the claimed equivalence is not established by this exposition.

Editorial extensions

If this is right

  • Every formal moduli problem over a characteristic-zero field is presented up to equivalence by a dg Lie algebra, so deformation problems can be studied as Lie-algebra cohomology.
  • The deformation functor of a dg Lie algebra $\mathfrak{g}_*$ is explicitly the Maurer-Cartan space $MC(\mathfrak{m}_R \otimes \mathfrak{g}_*)$, making the equation $dx=[x,x]$ the operative deformation equation.
  • The first-order tangent space at a point is recovered from $\mathrm{H}^0$ of the tangent complex, and obstruction classes for extending deformations live in $\mathrm{H}^2$; the Lie bracket governs all higher-order structure.
  • The Koszul duality functor $D$ is a deformation theory, so the Yoneda embedding from $\mathrm{Lie}_k$ lands fully faithfully in formal moduli problems; no information is lost in passing from Lie algebras to their formal moduli.
  • In the paper's motivating example of deformations of a smooth proper scheme, the dictionary identifies automorphism groups of first-order deformations with $\mathrm{H}^0(Z;T_Z)$, isomorphism classes with $\mathrm{H}^1(Z;T_Z)$, and obstructions with $\mathrm{H}^2(Z;T_Z)$.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A direct corollary the author leaves implicit: any formal moduli problem whose classical truncation is a scheme's formal neighborhood should be governed by the derived infinitesimal automorphism Lie algebra of that object, so the whole formal neighborhood is recovered from one Lie algebra.
  • A testable extension is to run the dictionary on the deformation problem of a smooth proper scheme from the paper's example: the predicted dg Lie algebra should have Chevalley-Eilenberg cohomology matching $H^*(Z;T_Z)$, with $\mathrm{H}^2$ governing extension to second order.
  • The characteristic-zero hypothesis is the fragile input; because symmetric powers behave well and divided powers do not, one would expect positive-characteristic formal moduli problems to require divided-power Lie algebras or another enhancement, a direction this paper does not explore.
Share X Bluesky LinkedIn Reddit HN

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 7 minor

Summary. This paper is an expository survey, explicitly not claiming original results, of the Lurie--Pridham equivalence between differential graded Lie algebras and formal moduli problems over a field of characteristic zero. It introduces deformation contexts and formal moduli problems, reviews the tangent complex and deformation theories, gives the definition of formal moduli problems for commutative algebras, surveys differential graded Lie algebras and their Chevalley--Eilenberg (co)homology, and concludes with the central theorem (Theorem 4.2.4, coPGI): an equivalence of infinity-categories Psi : Lie_k -> Moduli_k. The main technical theorems are quoted from Lurie's DAG X and Pridham's work, and the paper provides background appendices on model categories and infinity-categorical miscellany. Because the paper is an exposition, its correctness depends on whether the definitions it uses are stated accurately and sufficiently connected to the quoted theorems.

Significance. If the exposition were accurate, it would fill a useful niche: a concise, readable introduction to a technically demanding subject, with the main equivalence correctly attributed to Lurie and Pridham. The paper is honest about its expository nature, and it includes useful details such as the model structure on dg Lie algebras, the universal enveloping algebra, and the Chevalley--Eilenberg complexes. However, several load-bearing definitions are currently inaccurate or insufficiently connected, most importantly the mismatch between the two notions of formal moduli problem used in Definition 1.2.4 and Definition 2.1.6, and the nonstandard definition of excisive functors in Definition 1.3.3. These issues prevent the proof of Theorem 4.2.4, as written, from establishing the stated equivalence for the stated category Moduli_k.

major comments (4)
  1. [§2.1.6 and §4.2.4] Theorem 4.2.4 asserts an equivalence Psi : Lie_k -> Moduli_k, where Moduli_k is presumably the category of Definition 2.1.6, but the proof invokes Theorem 1.4.6, whose input category Moduli_Gamma is defined in Definition 1.2.4 for an arbitrary deformation context. The paper never proves, or cites a theorem proving, that for the deformation context (CAlg^aug_k, {Sigma^infinity k}), the small morphisms of Definition 1.2.3 coincide with the square-zero extensions used in Definition 2.1.6, nor that the two pullback axioms are equivalent. Without this identification, the essential-image argument in the proof of Theorem 4.2.4 does not establish the theorem as stated; the theorem could hold for one notion of Moduli_k and not the other. This identification should be stated and proved or explicitly quoted from a precise source.
  2. [§2.1.6] Condition 2 of Definition 2.1.6 is not a well-formed mathematical statement. The text says that a pullback diagram 'admits a unique factorization S ... for any object S in S and maps S -> X(R0), S -> X(R1)', which does not assert the required condition that the induced map X(R) -> X(R0) x_{X(R01)} X(R1) is an equivalence. It also conflates square-zero extensions with arbitrary surjections by writing 'i.e. surjections pi_* R_i -> pi_* R_01'. Since this is the central definition of the paper's main object of study, it must be rewritten precisely: one should specify which maps are required to be square-zero extensions, and state the pullback condition as an equivalence of the appropriate mapping space.
  3. [§1.3.3] Definition 1.3.3 defines an excisive functor as one sending pushout diagrams to pushout diagrams. The standard definition used in the theory of spectrum objects, and the one needed for Stab(D) to consist of spectrum objects, is that an excisive functor sends pushout squares to pullback squares (and a reduced excisive functor also sends the initial object to a terminal object). With the definition as printed, the composition in Corollary 1.4.4.2 need not define a spectrum object, and the tangent complex construction in Definition 1.3.5 inherits the error. This should be corrected and aligned with the cited source (Higher Algebra).
  4. [§3.1.7 and §4.1.1] Theorem 4.2.4 is stated as the coPGI equivalence, but the proof relies on Theorem 3.2.1, whose statement concerns a category 'Moduli' in Fun(CAlg^sm_C, S), and on Theorem 4.2.3. The notation Modulik is introduced in Theorem 4.1.1 but is never explicitly defined there. Moreover, the statement of Theorem 3.2.1 gives a localization universal property for theta, but the proof of Theorem 4.2.4 does not explain how this universal property, together with Theorem 4.2.3, yields the fully faithful embedding and essential image claim for the specific category Modulik. The relation among the categories called Moduli in Theorems 3.2.1, 4.1.1, and 4.2.4 should be made explicit, and the proof of Theorem 4.2.4 should be expanded to a degree that a reader can verify the equivalence is with the same definition used in the statement.
minor comments (7)
  1. [§3.1.1] The graded Jacobi identity has a sign typo: the third term should carry (-1)^{q ell}, not (1)^{q ell}.
  2. [§1.3.4] Construction 1.3.4 appears to contain a typo: 'For any map f : K' -> K' ' should presumably be 'K' -> K' with K and K' different objects, and the direction of the homotopy group map should be checked.
  3. [§3.1.7] The proof of Lemma 3.1.7 cites '[73]' for the Poincare-Birkhoff-Witt theorem, but the reference list contains no item 73; only 19 references are listed.
  4. [§3.1.7] The proof of Proposition 3.1.6 ends with 'It follows (see T.A.2.6.13)', which appears to be an internal reference without a target; it should be a precise citation to Higher Algebra or another source.
  5. [References] Reference 9 is listed as Higher Algebra but the URL points to the Higher Topos Theory PDF, and reference 10 for Spectral Algebraic Geometry points to a different text; these URLs should be corrected.
  6. [§1.4.5] In Definition 1.4.5, 'textbfSpc' is a LaTeX error, and the final sentence about viewing Spc as a subcategory of excisive functors is unclear and should be rephrased.
  7. [§2.1.6] The notation CAlg^sm_C is used without an explicit definition of 'small E-infinity algebra' in Section 2; the relation to Definition 1.2.3 should be stated even if the full identification is relegated to a remark.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper is an exposition that credits Lurie and Pridham for the central theorems, and its claims do not reduce to its own definitions or fits.

full rationale

The paper explicitly presents itself as exposition rather than original work, stating in the abstract: 'this paper should not be viewed as a presentation of original work, but rather a concise introduction to the subject in the form of a set of organized notes.' The central equivalence, Theorem 4.2.4, is derived by combining Theorem 1.4.6, attributed to Lurie, with Theorem 4.2.3, also attributed to Lurie. These are external citations to non-overlapping authors, not self-citations, and the paper does not claim to prove the underlying theorems independently. There is no data fitting, no fitted parameter renamed as a prediction, and no definition that covertly builds in the target equivalence: the 'deformation theory' hypotheses in Definition 1.4.5 are sufficient conditions supplied by Lurie's theorem, not the conclusion itself. The possible mismatch between Definition 1.2.4 and Definition 2.1.6 of formal moduli problems is a genuine gap or correctness concern, since the paper does not explicitly prove their coincidence for the deformation context (CAlgaug_k, {E}); however, this is an unproved identification of two independently stated notions, not a circular reduction. The paper's reliance on Lurie and Pridham is standard scholarly delegation for an expository article and does not make the derivation circular. Therefore the appropriate circularity score is 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no free parameters, new objects, or fitted quantities. It is an exposition of known results, and its assumptions are the standard theorems and categorical foundations of derived algebraic geometry, cited from Lurie and Pridham.

assumptions (5)
  • domain assumption Lurie's theorem that a deformation theory D: Γ^op → Ξ induces an equivalence Ψ: Ξ → Moduli_Γ.
    Invoked as Theorem 1.4.6 and used in §4.2.4 to conclude the equivalence. The paper does not prove this theorem.
  • domain assumption Pridham's theorem that there is a functor θ: N(Lie^dg_C) → Moduli with the universal property showing Lie^dg_C[W^-1] ≃ Moduli.
    Stated as Theorem 3.2.1 and used to motivate the connection. Proven in Pridham [15], not in this paper.
  • standard math PBW theorem: for a dg Lie algebra over a field of characteristic zero, Sym*(g*) is chain-isomorphic to U(g*) via symmetrization.
    Used in Lemma 3.1.7 to prove the enveloping algebra preserves quasi-isomorphisms.
  • domain assumption The category Lie^dg_k of dg Lie algebras over a field of characteristic zero carries a left proper combinatorial model structure with quasi-isomorphisms as weak equivalences.
    Stated as Proposition 3.1.6 and used to form the ∞-category Liek by inverting weak equivalences. The paper gives only a sketch with references to external results.
  • standard math The ∞-categorical background (presentable ∞-categories, stabilization Stab(Γ), the ∞-category of spaces S, sifted colimits, model categories) is taken as given.
    Used throughout; Appendix B summarizes but does not prove these foundations.

how reviews work

0 comments
Cite this review

Pith. "Pith review of An illustration of formal moduli problems with differential graded Lie algebras." pith.science (2026). https://pith.science/paper/TBZYQ2YW

@misc{pith2026250612977,
  author       = {Pith},
  title        = {Pith review of: An illustration of formal moduli problems with differential graded Lie algebras},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/TBZYQ2YW}},
  note         = {Machine review of arXiv:2506.12977}
}
read the original abstract

This article provides an exposition to the topic of formal moduli problems, emphasizing its connections with differential graded Lie algebras, and mainly following from Jacob Lurie's DAG X: Formal Moduli Problems. As such, this paper should not be viewed as a presentation of original work, but rather a concise introduction to the subject in the form of a set of organized notes. I hoped to make this paper feel welcoming and insightful for the non-expert enjoyer of derived algebraic geometry, like myself. Enjoy!

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

65 extracted references · 65 canonical work pages

  1. [1]

    The space X(∗) is contractible

    work page

  2. [2]

    7 Let Moduli Γ denote the full subcategory of Fun(Γ sm, S) spanned by formal moduli problems

    If A′ B′ A B ϕ is a pullback diagram and ϕ : B′ − →B is small, then the image X(A′) X(B′) X(A) X(B) X(ϕ) is a pullback diagram in S. 7 Let Moduli Γ denote the full subcategory of Fun(Γ sm, S) spanned by formal moduli problems. We call Moduli Γ the ∞-category of formal moduli problems. 1.2.5 Example Let (Γ, {Eα}α∈T ) be a deformation context,A ∈ Γ an objec...

    work page

  3. [3]

    For any map f : K ′ − →K ′ of finite pointed spaces inducing a surjection on homotopy groups π0K − →π0K ′, the induced map Eα(K) − →Eα(K ′) is small in Γ

    work page

  4. [4]

    1.3.5 Definition Let Γsm − →S be a formal moduli problem

    For all K ∈ Sf in ∗ , Eα(K) ∈ Γ is small. 1.3.5 Definition Let Γsm − →S be a formal moduli problem. For every α, we view the composition Y (Eα) = Sf in ∗ Eα − − →Γsm Y − →S as an object of Spc, and we call Y (Eα) the tangent complex to Y at α. 1.3.6 Remark We can further identify the tangent space Y (Ω∞Eα) with the 0th rung of the tangent complex Y (Eα). ...

    work page

  5. [5]

    D admits a left adjoint D′ : Ξ − →Γop

    work page

  6. [6]

    (b) The initial object ∅ ∈ Ξ0

    Ξ has a full subcategory Ξ 0 such that (a) For each K ∈ Ξ0, the unit map K 7− →D(D′(K)) is an equivalence. (b) The initial object ∅ ∈ Ξ0. Hence ∅ ≃ D(D′(∅)) ≃ D(∗). (c) For all α, n≥ 1, there exists an object Kα,n ∈ Ξ and an equivalence Ω∞−nEα ≃ D′Kα,n, determining a map vα,n : Kα,n ≃ D(D′(Kα,n)) ≃ D(Ω∞−nEα) − →D(∗) ≃ ∅. (d) For every pushout Kα,n K ∅ K ′...

    work page

  7. [7]

    D carries terminal objects in Γ to initial objects in Ξ

    work page

  8. [8]

    The unit map D′(D(A)) is an eqiuivalence in Γ

    Let A = D′(K) ∈ Γ, K∈ Ξ. The unit map D′(D(A)) is an eqiuivalence in Γ

    work page

Show all 65 references
  1. [9]

    If A ∈ Γ is small, then D(A) ∈ Ξ0, and A − →D′(D(A)) is an equivalence in Γ

  2. [10]

    10 1.4.4 A pair of corollaries

    If σ = A′ B′ A B ϕ is a pullback diagram where A, B, ϕare small, then D(σ) is a pushout in Ξ . 10 1.4.4 A pair of corollaries

  3. [11]

    For every K ∈ Ξ, the composition Γsm ⊂ Γ D − →Ξop y(K) − − − →S is a formal moduli problem, which determines a functor Ψ : Ξ − →ModuliΓ ⊂ Fun(Γop, S)

    Let y : Ξ − →Fun(Ξ, S) be the Yoneda embedding. For every K ∈ Ξ, the composition Γsm ⊂ Γ D − →Ξop y(K) − − − →S is a formal moduli problem, which determines a functor Ψ : Ξ − →ModuliΓ ⊂ Fun(Γop, S)

  4. [12]

    For every α, K∈ Ξ, the composition Sf in ∗ Eα − − →Γ D − →Ξop y(K) − − − →S is strongly excisive, and can be indeitifed with a spectrum object eα(K) ∈ Spc

    Let D : Γop − →Ξ be a weak deformation theory. For every α, K∈ Ξ, the composition Sf in ∗ Eα − − →Γ D − →Ξop y(K) − − − →S is strongly excisive, and can be indeitifed with a spectrum object eα(K) ∈ Spc. This deter- mines a functor eα : Ξ − →Spc. 1.4.5 Definition Finally we get...

  5. [13]

    Moreover, every first order deformationZ has an automorphism group which is naturally isomorphic to H 0(Z; TZ), where TZ is the tangent bundle of Z

    The image under X ∧ of the ring of dual numbers, X ∧(C[t]/(t2), is the groupoid of first order deformations of the varietyZ. Moreover, every first order deformationZ has an automorphism group which is naturally isomorphic to H 0(Z; TZ), where TZ is the tangent bundle of Z

  6. [14]

    The collection of isomorphism classes of first order deformations of Z are naturally identified with the first cohomology H 1(Z; TZ)

  7. [15]

    The first two of these properties are nice and friendly, and can be exposited without too much extra machinery

    Every first order deformation η1 of Z can be assigned a class θ ∈ H 2(Z; TZ) which vanishes if and only if η1 extends to a second order deformation η2 ∈ X ∧(C[t]/(t3)). The first two of these properties are nice and friendly, and can be exposited without too much extra machine...

  8. [16]

    The space X(C) is contractible

  9. [17]

    Note here that Ri − →R01 are square zero (their kernel is order 2 nilpotent) extensions of R, i.e

    For every pullback diagram R R0 R1 R01 in CAlgsm C for which the underlying maps π0R0 − →π0R01 and π0R1 − →π0R01 are surjective, the diagram X(R) X(R0) X(R1) X(R01) admits a unique factorization S X(R) X(R0) X(R1) X(R01) for any object S ∈ S and maps S − →X(R0), S− →X(R1). Not...

  10. [18]

    a fibration if each induced map Vn − →Wn is surjective

  11. [19]

    a cofibration if each induced map Vn − →Wn is injective

  12. [20]

    3.1.6 Proposition Let k be a field of characteristic zero

    a weak equivalence if it is a quasi-isomorphism. 3.1.6 Proposition Let k be a field of characteristic zero. The category Liedg k has the structure of a left proper combinatorial model category. 3.1.7 Lemma in aid of proposition 3.1.6 Let f : g∗ − →g′ ∗ be a morphism of differe...

  13. [21]

    f is a quasi-isomorphism

  14. [22]

    Proof of lemma

    The induced map U (g∗) − →U (g′ ∗) is a quasi isomorphism of differential graded algebras. Proof of lemma. For every n ∈ Z, let ψ : g⊗n ∗ − →U (g∗) denote the multiplication map. For any permutation σ ∈ {1, 2, . . . , n}, let ϕσ be the induced automorphism of g⊗n ∗ . Then the ...

  15. [23]

    W is perfect (see appendix B) (this follows from Lurie [9])

  16. [24]

    We will now sketch the proof of 2

    If f : g∗ − →g′ ∗ is a quasi-isomorphism over k, and x ∈ gn−1 is a cycle which classifies the map Free(∂E (n)∗) − →gk, then the induced map g∗ a Free(∂E (n)∗) Free(E(n)∗) − →g′ ∗ a Free(∂E (n)∗) Free(E(n)∗) is a quasi-isomorphism. We will now sketch the proof of 2. Let F : U (...

  17. [25]

    Elements of Cn(g)n are of the form x + ϵy, where x ∈ gn, y∈ gn−1

    for all n ∈ Z, we define the vector space Cn(g)∗ by gn ⊕ gn−1. Elements of Cn(g)n are of the form x + ϵy, where x ∈ gn, y∈ gn−1

  18. [26]

    The differential satisfies d(x + ϵy) = dx + y − ϵdy

  19. [27]

    3.3.2 The homological Chevalley-Eilenberg complex Let g∗ be a differential graded Lie algebra over a field k

    The bracket is given by [ x + ϵy, x′ + ϵy′] = [x, x′] + ϵ([y, x′] + (−1)p[x, y′]) for x ∈ gp. 3.3.2 The homological Chevalley-Eilenberg complex Let g∗ be a differential graded Lie algebra over a field k. The zero map g∗ − →0 sneakily induces a map of differential graded algebr...

  20. [28]

    There exists a graded vector space V∗ ⊂ g∗ such that for each integer n, dim Vn < ∞

  21. [29]

    For all n ≥ 0, Vn is trivial

  22. [30]

    Then C satisfies axiom 3

    V∗ freely generates g∗ as a graded Lie algebra. Then C satisfies axiom 3. The proof of this relies on the following lemma, whose proof can be found in Lurie [7] §2. 4.2.2 Lemma in aid of Proposition Let g∗ be a differential graded Lie algebra over k, and assume that for each n...

    1967

  23. [31]

    If any pair of the three maps f, g,and g ◦ f are weak equivalences, then so is the third

    Let f, gbe morphisms in C such that g ◦ f is definable. If any pair of the three maps f, g,and g ◦ f are weak equivalences, then so is the third

  24. [32]

    If f is a retract of g and g is a weak equivalence, fibration, or cofibration, then so is f. 27

  25. [33]

    We mandate that acyclic cofibrations satisfy left lifting with respect to fibrations, and cofibrations satisfy left lifting with respect to acyclic fibrations

    A map f is called a trivial or acyclic (co)fibration if it is a (co)fibration and a weak equiva- lence. We mandate that acyclic cofibrations satisfy left lifting with respect to fibrations, and cofibrations satisfy left lifting with respect to acyclic fibrations. A model categ...

  26. [34]

    Equivalently, C has a fully faithful right adjoint localization C ,→ Psh(S), where PSh(S) is the category of presheaves on S

    a set S of small objects such that every object in C is a colimit over objects in S. Equivalently, C has a fully faithful right adjoint localization C ,→ Psh(S), where PSh(S) is the category of presheaves on S

  27. [35]

    Note also that a model category is left proper if for every diagram A X B X ∪ B f i h where i is a cofibration and f is a weak equivalence, the map h is also a weak equivalence

    a set of cofibrations and a set of acyclic cofibrations which ”generate” all (acyclic) cofibrations in C. Note also that a model category is left proper if for every diagram A X B X ∪ B f i h where i is a cofibration and f is a weak equivalence, the map h is also a weak equiva...

  28. [36]

    F preserves (trivial) cofibrations

  29. [37]

    G preserves (trivial) fibrations

  30. [38]

    F preserves cofibrations and G preserves fibrations

  31. [39]

    If any of these are satisfied, we say that ( F, G) deetermines a Quillen adjunction on the categories C and D

    F preserves trivial cofibrations and G preserves trivial fibrations. If any of these are satisfied, we say that ( F, G) deetermines a Quillen adjunction on the categories C and D. Suppose that C DF G is a Quillen adjunction. We also say that F is a left Quillen functor and G i...

  32. [40]

    LF : hC − →hD is an equivalence of categories

  33. [41]

    RG : hD − →hC is an equivalence of categories

  34. [42]

    For any cofibrant object C ∈ C, and any fibrant object D ∈ D, a map C − →G(D) is a weak equivalence in C if and only if the adjoint F (C) − →D is a weak equivalence in D. Proof. 1. ⇐ ⇒2. is immediate since RG and LF are adjoint. Both are equivalent to the statement that u : id...

  35. [43]

    Every isomorphism is an element of W

  36. [44]

    For any pair of composable morphisms f and g, if any pair of the three maps f, g, g◦ f is in W, then the third is as well

  37. [45]

    Let X = lim − → {Xα}, Y = lim − → {Yα}

    Let {fα : Xα − →Yα} be a collection of morphisms indexed by a filtered poset. Let X = lim − → {Xα}, Y = lim − → {Yα}. Let f : X − →Y be the induced map. If each fα ∈ W, then f ∈ W. This is equivalent to the statement that W is stable under filtered colimits

  38. [46]

    6.1.6 The nerve of a small category Let J be a small category

    There exists a subset W0 ⊂ W such that every f ∈ W is a filtered colimit of morphisms in W0. 6.1.6 The nerve of a small category Let J be a small category. The nerve N (J) of J is the simplicial set whose 0-simplices are objects of J, 1-simplices are morphisms in J, 2-simplice...

  39. [47]

    Calaque and Grivaux, Formal moduli problems and formal derived stacks, https://arxiv.org/pdf/1802.09556

    arXiv

  40. [48]

    nLab, ”Cofibrantly generated model category”, https://ncatlab.org/nlab/show/cofibrantly+generated+model+category

  41. [49]

    Benjamin, Inc., New York, 1969

    Fulton, W Algebraic Curves, W.A. Benjamin, Inc., New York, 1969. https://dept.math.lsa.umich.edu/ wful- ton/CurveBook.pdf

    1969

  42. [50]

    Greenless, J.P.C and MAy, J.P Equivariant stable homotopy theory https://www.math.uchicago.edu/ may/PAPERS/Newthird.pdf

  43. [51]

    Hartshorne, R Deformation Theory, Springer Graduate Texts in Mathematics

  44. [52]

    Model categories, https://people.math.rochester.edu/faculty/doug/otherpapers/hovey- model-cats.pdf

    Hovey, M. Model categories, https://people.math.rochester.edu/faculty/doug/otherpapers/hovey- model-cats.pdf

  45. [53]

    Derived Algebraic Geometry X: Formal Moduli Problems, https://people.math.harvard.edu/ lurie/papers/DAG- X.pdf

    Lurie, J. Derived Algebraic Geometry X: Formal Moduli Problems, https://people.math.harvard.edu/ lurie/papers/DAG- X.pdf

  46. [54]

    Higher Topos Theory, https://www.math.ias.edu/ lurie/papers/HTT.pdf

    Lurie, J. Higher Topos Theory, https://www.math.ias.edu/ lurie/papers/HTT.pdf

  47. [55]

    Higher Algebra, https://www.math.ias.edu/ lurie/papers/HTT.pdf

    Lurie, J. Higher Algebra, https://www.math.ias.edu/ lurie/papers/HTT.pdf

  48. [56]

    Spectral Algebraic Geometryhttps://www.math.uchicago.edu/ may/PAPERS/kmbooklatex.pdf

    Lurie, J. Spectral Algebraic Geometryhttps://www.math.uchicago.edu/ may/PAPERS/kmbooklatex.pdf

  49. [57]

    Derived Algebrai Geometry III: Commutative Algebrahttps://math.mit.edu/ lurie/papers/DAG- III.pdf

    Lurie, J. Derived Algebrai Geometry III: Commutative Algebrahttps://math.mit.edu/ lurie/papers/DAG- III.pdf

  50. [58]

    Derived Algebraic Geometry IV: Deformation Theoryhttps://math.mit.edu/ lurie/papers/DAG- IV.pdf

    Lurie, J. Derived Algebraic Geometry IV: Deformation Theoryhttps://math.mit.edu/ lurie/papers/DAG- IV.pdf

  51. [59]

    and Verity, D.Infinite Category Theory from Scratch, https://math.jhu.edu/ eriehl/scratch.pdf

    Riehl, E. and Verity, D.Infinite Category Theory from Scratch, https://math.jhu.edu/ eriehl/scratch.pdf

  52. [60]

    Matsumura, H Commutative Ring Theory , Cambridge University Press, 1986

    1986

  53. [61]

    Pridham, J Unifying derived deformation theories , Adv. Math. 224 (2010), no.3, 772-826

    2010

  54. [62]

    Derived Algebraic Geometry VI:E[k]-algebras,https://people.math.harvard.edu/ lurie/papers/DAG- VI.pdf

    Lurie, J. Derived Algebraic Geometry VI:E[k]-algebras,https://people.math.harvard.edu/ lurie/papers/DAG- VI.pdf

  55. [63]

    nLab, ”E-infinity algebra,” https://ncatlab.org/nlab/show/E-infinity+algebra

  56. [64]

    and May, J.P.Operads, Algebras, Modules, and Motives https://www.math.uchicago.edu/ may/PAPERS/kmbooklatex.pdf

    Kriz, I. and May, J.P.Operads, Algebras, Modules, and Motives https://www.math.uchicago.edu/ may/PAPERS/kmbooklatex.pdf

  57. [65]

    Soergel, W Koszul Duality and applications in representation theory https://people.mpim- bonn.mpg.de/geordie/Soergel.pdf 33

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.