Algebraic Spaces, Stacks, and Derived Algebraic Geometry Foundations

Fix a base scheme $S$ (Schemes, Schemes and morphisms over a base). An fppf covering of an $S$-scheme $T$ is a family $\{T_i\to T{\}}_{i\in I}$ of morphisms of $S$-schemes (Morphisms of schemes) such that each $T_i\to T$ is flat (Flat morphism of schemes) and locally of finite presentation (Locally finite presentation morphisms), and the images of the underlying maps $\mathrm{\mid }{T}_i\mathrm{\mid }\to \mathrm{\mid }T\mathrm{\mid }$ cover $T$. The fppf site $(Sch\mathrm{/}S{)}_{fppf}$ is the category of $S$-schemes with the pretopology whose coverings of $T$ are the fppf coverings of $T$, with the identity refinements and the usual composition of coverings of an fppf topology.

The covering condition is a topological surjectivity condition on the index family together with the two morphism properties; the index set $I$ need not be finite. The images may overlap. The maps need not be open immersions. We work inside one fixed big fppf site of $S$-schemes; nothing below uses size questions beyond those conventions.

Three standard properties are used constantly and are recorded here with their proofs. First, a Zariski open cover is an fppf covering (Open immersions of schemes): an open immersion is flat and locally of finite presentation, and its underlying map is an open topological embedding, so the images of the members of an open cover of $T$ cover $T$. Second, fppf coverings are stable under base change: for a morphism $T^{\mathrm{\prime }}\to T$ the base-changed family $\{T_i{\times }_T{T}^{\mathrm{\prime }}\to T^{\mathrm{\prime }}\}$ is fppf, because flatness and local finite presentation are stable under base change and images of the base-changed maps still cover $T^{\mathrm{\prime }}$ (Fibre product of schemes). Third, fppf coverings are stable under composition: if $\{T_i\to T{\}}_{i\in I}$ is fppf and $\{T_{ij}\to T_i{\}}_{j\in J_i}$ is fppf for every $i$, then the composites $\{T_{ij}\to T\}$ form an fppf covering of $T$, because a composite of flat morphisms is flat, a composite of locally finitely presented morphisms is locally of finite presentation, and the images of the composites cover $T$ by the covering property of the two families together.

No choice principle is used to state the definition or its consequences: a covering is a single family of morphisms, and the stability assertions are element-wise. A covering may be indexed by an empty set only when the target $T$ is empty, in which case the covering condition is vacuous.

Fix a base scheme $S$ (Schemes and morphisms over a base). A groupoid in $S$-schemes is a tuple $(U,R,s,t,c,e,i)$ consisting of $S$-schemes $U$ and $R$ and morphisms of $S$-schemes $s,t\text{\hspace{0.17em}⁣}:R\to U$ (source and target), $c\text{\hspace{0.17em}⁣}:R{\times }_{s,U,t}R\to R$ (composition), $e\text{\hspace{0.17em}⁣}:U\to R$ (identity) and $i\text{\hspace{0.17em}⁣}:R\to R$ (inverse), subject to the usual identities of a small groupoid. Here $c$ is defined on composable pairs $(r_1,r_2)$ with $s(r_1)=t(r_2)$ and $c(r_1,r_2)$ is the composite "$r_2$ first, then $r_1$", with $s(c(r_1,r_2))=s(r_2)$ and $t(c(r_1,r_2))=t(r_1)$; the identities are $s\circ e=t\circ e={\mathrm{i}\mathrm{d}}_U,s\circ c=s\circ {\mathrm{p}\mathrm{r}}_2,t\circ c=t\circ {\mathrm{p}\mathrm{r}}_1,$ $c\circ (c\times {\mathrm{i}\mathrm{d}}_R)=c\circ ({\mathrm{i}\mathrm{d}}_R\times c),c\circ ({\mathrm{i}\mathrm{d}}_R,e\circ s)=c\circ (e\circ t,{\mathrm{i}\mathrm{d}}_R)={\mathrm{i}\mathrm{d}}_R,$ $s\circ i=t,t\circ i=s,c\circ ({\mathrm{i}\mathrm{d}}_R,i)=e\circ t,c\circ (i,{\mathrm{i}\mathrm{d}}_R)=e\circ s,$ where the fibre products and projections are those of Fibre product of schemes and all morphisms are morphisms of $S$-schemes (Morphisms of schemes); associativity is stated on the triple fibre product $R{\times }_{s,U,t}R{\times }_{s,U,t}R$, where $c\times {\mathrm{i}\mathrm{d}}_R$ and ${\mathrm{i}\mathrm{d}}_R\times c$ are formed using the source/target identifications. A groupoid in $S$-schemes is precisely a groupoid object in the category of $S$-schemes in the sense of these diagrams, and its functor of points on the category of $S$-schemes is a groupoid-valued functor.

With $j=(t,s)\text{\hspace{0.17em}⁣}:R\to U{\times }_{S}U$, the groupoid is a relation when $j$ is a monomorphism (Monomorphism and epimorphism by left and right cancellation); then $j$ presents $R$ as a subobject of $U{\times }_{S}U$, and the groupoid axioms exhibit a reflexive ($e$), symmetric ($i$) and transitive ($c$) set-theoretic relation on the points of $U$ in the sense of Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$. The groupoid is an equivalence relation on $U$ over $S$ when it is a relation, and an etale equivalence relation when in addition $s$ and $t$ are etale (Étale morphism of schemes).

Restriction is well defined as follows. Let $g\text{\hspace{0.17em}⁣}:U^{\mathrm{\prime }}\to U$ be a morphism of $S$-schemes and form $R^{\mathrm{\prime }}=R{\times }_{U{\times }_{S}U}(U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }}),$ the fibre product along $j$ and $g\times g$, with its two projections ${\mathrm{p}\mathrm{r}}_R$ and ${\mathrm{p}\mathrm{r}}_{U^{\mathrm{\prime }}\times U^{\mathrm{\prime }}}$; set $t^{\mathrm{\prime }}={\mathrm{p}\mathrm{r}}_1\circ {\mathrm{p}\mathrm{r}}_{U^{\mathrm{\prime }}\times U^{\mathrm{\prime }}}$ and $s^{\mathrm{\prime }}={\mathrm{p}\mathrm{r}}_2\circ {\mathrm{p}\mathrm{r}}_{U^{\mathrm{\prime }}\times U^{\mathrm{\prime }}}$. The map $e^{\mathrm{\prime }}$ sends $u^{\mathrm{\prime }}$ to $(e(g(u^{\mathrm{\prime }})),(u^{\mathrm{\prime }},u^{\mathrm{\prime }}))$, where $(u^{\mathrm{\prime }},u^{\mathrm{\prime }})$ is the diagonal $U^{\mathrm{\prime }}\to U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }}$; the map $i^{\mathrm{\prime }}$ sends $(r,(u_1^{\mathrm{\prime }},u_2^{\mathrm{\prime }}))$ to $(i(r),(u_2^{\mathrm{\prime }},u_1^{\mathrm{\prime }}))$; and $c^{\mathrm{\prime }}$ sends a composable pair $((r_1,(u_1^{\mathrm{\prime }},u_2^{\mathrm{\prime }})),(r_2,(u_2^{\mathrm{\prime }},u_3^{\mathrm{\prime }})))$ to $(c(r_1,r_2),(u_1^{\mathrm{\prime }},u_3^{\mathrm{\prime }}))$. All three are defined by the universal property of the relevant fibre products, and the groupoid identities for $(U^{\mathrm{\prime }},R^{\mathrm{\prime }},s^{\mathrm{\prime }},t^{\mathrm{\prime }},c^{\mathrm{\prime }},e^{\mathrm{\prime }},i^{\mathrm{\prime }})$ follow from those for $(U,R,s,t,c,e,i)$ after applying the universal property; this tuple is the restriction $R{\mathrm{\mid }}_{U^{\mathrm{\prime }}}$ of the groupoid along $g$. If $j$ is a monomorphism, then so is $j^{\mathrm{\prime }}=(t^{\mathrm{\prime }},s^{\mathrm{\prime }})$, because a monomorphism is stable under base change in any category with fibre products: given two morphisms into the fibre product with equal composites to $U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }}$ and to $R$, the universal property of the fibre product makes them equal. Hence restricting an equivalence relation along an arbitrary morphism of $S$-schemes yields an equivalence relation. Restriction of the etale property needs $g$ etale and is recorded separately in Restriction of an etale equivalence relation: its local flatness, finite-presentation and fibre arguments establish the required stability without a choice assumption.

Let $\mathcal{C}$ be a category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). A functor $p:\mathcal{S}\to \mathcal{C}$ (Covariant functor, identity functor, composite functor, and contravariant functor) is a category fibred in groupoids over $\mathcal{C}$ if every arrow $f:V\to U$ of $\mathcal{C}$ and every object $x$ of $\mathcal{S}$ over $U$ admit a cartesian arrow $\phi :y\to x$ over $f$, and every fibre category ${\mathcal{S}}_U$ is a groupoid (Isomorphism, groupoid, and connected category).

Cartesian means: for every $g:W\to V$, every object $z$ of $\mathcal{S}$ over $W$ and every arrow $\psi :z\to x$ with $p(\psi )=fg$, there is a unique arrow $\chi :z\to y$ over $g$ with $\phi \circ \chi =\psi$. The fibre ${\mathcal{S}}_U$ is the subcategory of objects over $U$ and morphisms over ${\mathrm{i}\mathrm{d}}_U$. The condition is equivalent to: every arrow of $\mathcal{S}$ is cartesian, and for every $f:V\to U$ and every object $x$ over $U$ there is an arrow $y\to x$ over $f$. Since an arrow over an identity is an isomorphism exactly when it is cartesian, this is a genuine condition on $p$ and not a matter of choosing arrows.

A 1-morphism $F:(p:\mathcal{S}\to \mathcal{C})\to (q:\mathcal{T}\to \mathcal{C})$ is a functor over $\mathcal{C}$, i.e. $qF=p$ (such a functor automatically preserves cartesian arrows). A 2-morphism is a natural transformation $F\Rightarrow G$ over $\mathcal{C}$, i.e. one whose components lie in the fibres (Natural transformation and its components). An equivalence is a 1-morphism admitting an inverse over the base up to natural isomorphisms over the base. It induces fully faithful, essentially surjective functors on all fibres. Conversely, fibrewise full faithfulness and essential surjectivity imply equivalence when choices of preimages and vertical isomorphisms are supplied for all target objects; for set-sized total categories these choices follow from AC (The Axiom of Choice). Indeed cartesian factorization turns fibrewise full faithfulness into full faithfulness on arrows over each base arrow. For each target object x choose y over the same base object and an isomorphism F(y) to x; full faithfulness lifts the conjugated target arrows uniquely to define the inverse functor and its two natural isomorphisms (Vistoli, Proposition 3.36 and Lemma 3.37). These objects, 1-morphisms and 2-morphisms form a strict 2-category: composition of 1-morphisms is strictly associative and the identity 1-morphisms act strictly; the coherence isomorphisms familiar from a pseudofunctor description appear only after choosing pullbacks.

The definition requires neither a cleavage nor a simultaneous choice of pullbacks. If one does choose, for every arrow $f$, a single cartesian lift of each object $x$, then the chosen lifts compose only up to the canonical isomorphism supplied by cartesian uniqueness, and this global choice may use AC; the fibred-in-groupoids definition requires no such choice. The equivalence criterion above has its separately stated choice hypothesis. In particular the empty category is fibred in groupoids over $\mathcal{C}$ vacuously, and if $\mathcal{S}$ is empty then the fibre categories are empty groupoids.

Let $\mathrm{\Delta }$ be the simplex category: its objects are the finite nonempty ordered sets $[n]=\{0<1<\cdots <n\}$ for $n\ge 0$, and its morphisms are the order-preserving maps. A simplicial object in a category $\mathcal{C}$ is a functor ${\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}\to \mathcal{C}$ (Category, object, morphism, domain, codomain, identity, composition, and hom-collection, Covariant functor, identity functor, composite functor, and contravariant functor). A simplicial set is a simplicial object in sets; a simplicial commutative ring is a simplicial object in commutative rings (Commutative ring), so that each $A_n$ is a commutative ring and the structure maps are ring homomorphisms (Ring homomorphism: additive, multiplicative, and required to send $1$ to $1$); a simplicial module over a simplicial ring $A_{\bullet }$ is a simplicial object $M_{\bullet }$ in abelian groups together with a compatible $A_{\bullet }$-action, and the resulting category of simplicial $A_{\bullet }$-modules is abelian.

Writing $d_i\text{\hspace{0.17em}⁣}:M_n\to M_{n-1}$ for the image of the injection $[n-1]\to [n]$ omitting $i$ (a face map) and $s_i\text{\hspace{0.17em}⁣}:M_n\to M_{n+1}$ for the image of the surjection $[n+1]\to [n]$ repeating $i$ (a degeneracy map), these maps satisfy the usual simplicial identities $d_i{d}_j=d_{j-1}{d}_i(i<j),s_i{s}_j=s_{j+1}{s}_i(i\le j),d_i{s}_j=\{\begin{array}{ll}{s}_{j-1}{d}_i,& i<j,\\ \mathrm{i}\mathrm{d},& i=j\text{ or }i=j+1,\\ s_j{d}_{i-1},& i>j+1.\end{array}$ Conversely, such a sequence of face and degeneracy maps determines the functor.

The Moore complex of a simplicial abelian group or module $M_{\bullet }$ is the chain complex concentrated in nonnegative degrees with $M_n$ in degree $n$ and differential ${\mathrm{\partial }}_n={\sum }_{i=0}^n(-1{)}^i{d}_i\text{\hspace{0.17em}⁣}:M_n\to M_{n-1}(n\ge 1),$ with ${\mathrm{\partial }}_0=0$; the $d_i$ are the face maps (Chain complex in an abelian category); the simplicial identities give ${\mathrm{\partial }}_{n-1}{\mathrm{\partial }}_n=0$. Its homology is written ${\pi }_n(M_{\bullet })=H_n(M_{\bullet })(n\ge 0)$ (Homology object of a chain complex). By the Dold-Kan normalization theorem this agrees with the classical homotopy-group definition $H_n(N{M}_{\bullet })$ via the normalized subcomplex, so the convention is canonical; no model-category machinery is introduced here.

For a simplicial commutative ring $A_{\bullet }$, each ${\pi }_n(A_{\bullet })$ is defined as above, and ${\pi }_0(A_{\bullet })=A_0\mathrm{/}(d_0-d_1)(A_1)$ is a commutative ring. The image is an ideal: for $a\in A_0$, multiplying a representative $b\in A_1$ by $s_{0}a$ gives $a(d_{0}b-d_{1}b)$. Every ${\pi }_n(A_{\bullet })$ is a ${\pi }_0(A_{\bullet })$-module. Indeed multiplication by the totally degenerate simplex of a vertex $a\in A_0$ is a chain map on the Moore complex: its faces are the corresponding totally degenerate simplex of $a$ in the preceding degree. If two vertices are the endpoints of $b\in A_1$, multiplication by the simplicial path defined by $b$ gives a homotopy between these chain maps (the alternating prism sum inserts the degeneracies of $b$). Consequently $(d_0-d_1)(b)$ acts as zero on homology, so the action factors through the displayed quotient. Additivity, associativity and the unit descend from levelwise ring multiplication.

A morphism of simplicial commutative rings is a natural transformation $A_{\bullet }\to B_{\bullet }$ (Natural transformation and its components). Such a morphism is a weak equivalence when it induces isomorphisms ${\pi }_n(A_{\bullet })\to {\pi }_n(B_{\bullet })$ for all $n\ge 0$. The category of derived rings is the localization of the category of simplicial commutative rings at the weak equivalences; constructions on derived rings are used below only through statements that are independent of the chosen replacement up to canonical isomorphism.

Assume the Axiom of Choice (AC) (The Axiom of Choice). Fix a map $A\to B$ of commutative unital rings (Commutative ring) and an infinite cardinal $\kappa$ at least the cardinalities of $A$ and $B$ and of every variable set occurring in the specified countable polynomial resolutions of $B$ over $A$ that are used below.

For an ordinal $\lambda \le \kappa$ let $A[\lambda ]$ denote the polynomial $A$-algebra on the variable set $\lambda$ (The polynomial ring $R[x_i:i\in I]$ as finitely supported coefficient families on monomials). A polynomial presentation of $B$ over $A$ (bounded by $\kappa$) is an $A$-algebra map $\pi \text{\hspace{0.17em}⁣}:A[\lambda ]\to B$ for some ordinal $\lambda \le \kappa$; since $A[\lambda ]$ is free on $\lambda$, such a $\pi$ is determined by the family $(\pi (x_{\alpha }){)}_{\alpha <\lambda }$ in $B$, so the collection of all bounded polynomial presentations is a set. Here “presentation” means a polynomial factorization of $A\to B$; the augmentation need not be surjective. This permits the coefficient-change functor for arbitrary ring squares, even when $B{\otimes }_A{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$ is not surjective.

Define ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$ to be the category whose objects are the bounded polynomial presentations $\pi \text{\hspace{0.17em}⁣}:A[\lambda ]\to B$ and whose morphisms $\pi \to {\pi }^{\mathrm{\prime }}$ are the $A$-algebra maps $\phi \text{\hspace{0.17em}⁣}:A[\lambda ]\to A[{\lambda }^{\mathrm{\prime }}]$ with ${\pi }^{\mathrm{\prime }}\circ \phi =\pi$, i.e. maps commuting with the augmentations. Every hom collection is a set (a morphism is determined by the images of the variables of its source, which are polynomials over $A$ in ${\lambda }^{\mathrm{\prime }}$ variables), so ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$ is a small category (Category, object, morphism, domain, codomain, identity, composition, and hom-collection). The bounded polynomial-factorization category is its opposite ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }:=({\mathcal{P}}_{B\mathrm{/}A}^{\kappa }{)}^{\mathrm{o}\mathrm{p}},$ and we write $P\to B$ for the object corresponding to a presentation. The ordinal representatives and the specified presentations are transported into this model degreewise; conjugating all face and degeneracy maps by the resulting algebra isomorphisms preserves the simplicial identities.

The cotangent diagram is the contravariant functor $F_{B\mathrm{/}A}\text{\hspace{0.17em}⁣}:{\mathcal{C}}_{B\mathrm{/}A}^{\kappa }\to B\text{-}\mathrm{M}\mathrm{o}\mathrm{d},F_{B\mathrm{/}A}(P\stackrel{\pi }{\to }B)={\mathrm{\Omega }}_{P\mathrm{/}A}{\otimes }_{P}B,$ where ${\mathrm{\Omega }}_{P\mathrm{/}A}$ is the module of Kähler differentials (Universal Kähler differential module) and the tensor product is taken along $\pi$ (The tensor product $M{\otimes }_{R}N$ from the additive group underlying the free $\mathbb{Z}$-module on $M\times N$, elementary tensors, and finite tensor sums); on a morphism $\phi \text{\hspace{0.17em}⁣}:P\to P^{\mathrm{\prime }}$ of presentations, functoriality of Kähler differentials gives ${\mathrm{\Omega }}_{P\mathrm{/}A}{\otimes }_P{P}^{\mathrm{\prime }}\to {\mathrm{\Omega }}_{P^{\mathrm{\prime }}\mathrm{/}A}$ and hence a $B$-linear map $F_{B\mathrm{/}A}(P)\to F_{B\mathrm{/}A}(P^{\mathrm{\prime }})$. The corresponding arrow in ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }$ goes from $P^{\mathrm{\prime }}$ to $P$, so $F_{B\mathrm{/}A}$ is contravariant (Covariant functor, identity functor, composite functor, and contravariant functor). Since $P$ is a polynomial $A$-algebra, ${\mathrm{\Omega }}_{P\mathrm{/}A}$ is a free $P$-module and $F_{B\mathrm{/}A}(P)$ is a free $B$-module.

A simplicial polynomial resolution $P_{\bullet }\text{\hspace{0.17em}⁣}:{\mathrm{\Delta }}^{\mathrm{o}\mathrm{p}}\to {\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$ (with each $P_n$ a polynomial presentation) yields by composition with the inclusion a simplicial object of ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }$, i.e. a cosimplicial object of ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }$.

Finally, a commutative square of ring maps $\begin{array}{ccc}A& ⟶& B\\ \downarrow & & \downarrow \\ A^{\mathrm{\prime }}& ⟶& B^{\mathrm{\prime }}\end{array}$ induces, after replacing $\kappa$ by a common bound for the two squares, a functor ${\mathcal{P}}_{B\mathrm{/}A}^{\kappa }\to {\mathcal{P}}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}^{\kappa }$ sending $\pi \text{\hspace{0.17em}⁣}:A[\lambda ]\to B$ to $P{\otimes }_A{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$, where the target is the composite $P{\otimes }_A{A}^{\mathrm{\prime }}\to B{\otimes }_A{A}^{\mathrm{\prime }}\to B^{\mathrm{\prime }}$, and therefore a functor ${\mathcal{C}}_{B\mathrm{/}A}^{\kappa }\to {\mathcal{C}}_{B^{\mathrm{\prime }}\mathrm{/}{A}^{\mathrm{\prime }}}^{\kappa }$; the variable set, hence the bound, is unchanged.

Use of AC. AC is used only to choose, once and for all, the transported models of the specified countable presentations inside the bounded category and to bound the union of the countably many specified variable sets by a single infinite cardinal $\kappa$; the subsequent definition of the diagram, the contravariance and the coefficient-change functor are choice-free. Since a polynomial ring on at most $\kappa$ variables over a ring of size at most $\kappa$ has size at most $\kappa$, the standard resolution fits at every stage and no universe axiom beyond the ambient set theory of Category, object, morphism, domain, codomain, identity, composition, and hom-collection is introduced.

Assume the Axiom of Choice (AC). For a small category $C$ (Covariant functor, identity functor, composite functor, and contravariant functor), a commutative ring $R$ and ${\mathcal{A}}_R(C)=\mathrm{Fun}(C^{\mathrm{o}\mathrm{p}},R\text{-}\mathrm{M}\mathrm{o}\mathrm{d})$, the category ${\mathcal{A}}_R(C)$ is abelian (Abelian category) with pointwise exactness. The diagrams $R_U=R[{\mathrm{Hom}}_C(-,U)]$ are projective (Projective object), and every diagram has a canonical epimorphism from a direct sum of them. Bounded-above projective replacements can therefore be supplied in this category (Bounded above complexes admit projective replacements), and $L{\mathrm{colim}}_{C^{\mathrm{o}\mathrm{p}}}$ exists there using the published supplied-replacement derived-functor interfaces (Projective complexes model the bounded above derived category, Existence of the bounded above left total derived functor). Moreover $\mathrm{colim}{R}_U=R$, and for a diagram $F$ in degree zero its derived colimit is computed by the bar complex $K_n(F)={⨁}_{U_n\to \cdots \to U_0}F(U_0)$ with alternating face differential (the face dropping $U_0$ uses the restriction of coefficients $F(U_0)\to F(U_1)$); this complex is constructed with the direct-sum total complex of a double complex (Direct sum total complex of a double complex).

A model category is a category $\mathcal{M}$ with all small limits and colimits (Category, object, morphism, domain, codomain, identity, composition, and hom-collection), together with three classes $\mathcal{W},\mathrm{F}\mathrm{i}\mathrm{b},\mathrm{C}\mathrm{o}\mathrm{f}$ of maps, each closed under retracts (that is, if a map is a retract of a map in the class, in the arrow category, then it lies in the class), subject to the following axioms.

1. $\mathcal{W}$ satisfies two-out-of-three: if two of $f$, $g$ and $g\circ f$ lie in $\mathcal{W}$, then so does the third.
2. Lifting. For a cofibration $i\text{\hspace{0.17em}⁣}:A\to B$, a fibration $p\text{\hspace{0.17em}⁣}:X\to Y$ and morphisms $a\text{\hspace{0.17em}⁣}:A\to X$, $b\text{\hspace{0.17em}⁣}:B\to Y$ with $p\circ a=b\circ i$, if $i$ or $p$ lies in $\mathcal{W}$, then there is a lift $\mathrm{\ell }\text{\hspace{0.17em}⁣}:B\to X$ with $\mathrm{\ell }\circ i=a$ and $p\circ \mathrm{\ell }=b$.
3. Factorization. Every map $f$ factors both as $f=p\circ i$ with $i$ in $\mathrm{C}\mathrm{o}\mathrm{f}$ and $p$ in $\mathrm{F}\mathrm{i}\mathrm{b}\cap \mathcal{W}$, and as $f=q\circ j$ with $j$ in $\mathrm{C}\mathrm{o}\mathrm{f}\cap \mathcal{W}$ and $q$ in $\mathrm{F}\mathrm{i}\mathrm{b}$.

A trivial fibration is a map in $\mathrm{F}\mathrm{i}\mathrm{b}\cap \mathcal{W}$ and a trivial cofibration is a map in $\mathrm{C}\mathrm{o}\mathrm{f}\cap \mathcal{W}$. An object $X$ is cofibrant when the structure map $\mathrm{\varnothing }\to X$ from an initial object lies in $\mathrm{C}\mathrm{o}\mathrm{f}$, and fibrant when the structure map $X\to \ast$ to a terminal object lies in $\mathrm{F}\mathrm{i}\mathrm{b}$; these properties are independent of the chosen initial and terminal objects, since any two are canonically isomorphic. A map in $\mathcal{W}$ is a weak equivalence.

A Quillen adjunction $L⊣R$ between model categories $\mathcal{M}$ and $\mathcal{N}$ is an adjunction (in the sense of The unit-counit, hom-set, unit-universal, and counit-universal encodings of an adjunction are equivalent) whose right adjoint $R$ sends fibrations to fibrations and trivial fibrations to trivial fibrations. It is equivalent to require that the left adjoint $L$ sends cofibrations to cofibrations and trivial cofibrations to trivial cofibrations: transposing each lifting square across the adjunction bijection identifies a lift on the left with a lift on the right, so the two conditions are exchanged by the adjunction. A Quillen equivalence is a Quillen adjunction such that for every cofibrant $X$ and fibrant $Y$ a map $LX\to Y$ lies in $\mathcal{W}$ exactly when its adjoint $X\to RY$ does.

When source and target carry simplicial mapping objects, an enriched Quillen adjunction is a Quillen adjunction together with natural isomorphisms $\mathrm{M}\mathrm{a}\mathrm{p}(LX,Y)\cong \mathrm{M}\mathrm{a}\mathrm{p}(X,RY)$ compatible with the simplicial operators and natural with respect to the enriched mapping objects. A simplicial Quillen adjunction between simplicial model categories is an enriched adjunction in this sense whose underlying adjunction is Quillen.

These definitions do not assert the existence of any model structure: being a model category is structure on a category, and a functor between model categories is not required to preserve anything. Two objects require care throughout: the initial and terminal objects. In the category of unital commutative $A$-algebras the initial object is $A$ and the terminal object is the zero ring, and the two must not be conflated when cofibrancy and fibrancy are read off from the structure maps.

Throughout, $S$ is a fixed base scheme and $(Sch\mathrm{/}S{)}_{fppf}$ is the fppf site of Fppf coverings and the fppf site.

A presheaf of sets on $(Sch\mathrm{/}S{)}_{fppf}$ is a contravariant functor from the category of $S$-schemes to the category of sets (Presheaves, covariantly and contravariantly representable functors, and representations, Covariant functor, identity functor, composite functor, and contravariant functor); the associated representable presheaf of a scheme is the contravariant functor it represents. It is an fppf sheaf when for every fppf covering $\{T_i\to T{\}}_{i\in I}$ the diagram $F(T)⟶{\prod }_{i}F(T_i)\rightrightarrows {\prod }_{i,j}F(T_i{\times }_T{T}_j)$ is an equalizer of sets (Fibre product of schemes), the two maps being the pullbacks along the two projections $T_i{\times }_T{T}_j\to T_i,T_j$. Equivalently, restriction identifies $F(T)$ with the set of families $(s_i{)}_{i\in I}$, $s_i\in F(T_i)$, whose two pullbacks to every $T_i{\times }_T{T}_j$ agree (Equivalence relation, equivalence class, and the quotient set $A\mathrm{/}\sim$ for the underlying set-theoretic relation); the two descriptions agree because an equalizer in sets consists of the elements on which the two maps coincide.

A morphism of presheaves is a natural transformation (Natural transformation and its components); the presheaves on $(Sch\mathrm{/}S{)}_{fppf}$ thus form a category. The sheafification of a presheaf $F$ is an fppf sheaf $F^{\mathrm{a}}$ together with a morphism $F\to F^{\mathrm{a}}$ such that every morphism $F\to G$ with $G$ an fppf sheaf factors uniquely through $F\to F^{\mathrm{a}}$. When it exists it is unique up to unique isomorphism, by the usual universal property; in this library its existence is established separately for the presheaves used below. A representable presheaf is an fppf sheaf (Scheme morphisms satisfy fppf descent), under the Axiom of Choice recorded there, since fppf descent for morphisms of schemes is effective.

All sheaves below are set-valued unless stated otherwise. The empty family is an fppf covering of the empty scheme, so for a sheaf the sheaf condition on that covering forces $F(\mathrm{\varnothing })$ to be a one-point set; this holds in particular for every representable presheaf, since ${\mathrm{Hom}}_S(\mathrm{\varnothing },X)$ is a one-point set for every $S$-scheme $X$, because the empty scheme is initial in the category of $S$-schemes.

Let $j\text{\hspace{0.17em}⁣}:R\to U{\times }_{S}U$ be an etale equivalence relation on $U$ over $S$ (Groupoids in schemes, relations and etale equivalence relations) and let $g\text{\hspace{0.17em}⁣}:U^{\mathrm{\prime }}\to U$ be a morphism of $S$-schemes. Form the restriction $R^{\mathrm{\prime }}=R{\mathrm{\mid }}_{U^{\mathrm{\prime }}}=R{\times }_{U{\times }_{S}U}(U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }})$ with source and target the standard projections (Fibre product of schemes). Then $j^{\mathrm{\prime }}\text{\hspace{0.17em}⁣}:R^{\mathrm{\prime }}\to U^{\mathrm{\prime }}{\times }_S{U}^{\mathrm{\prime }}$ is an equivalence relation on $U^{\mathrm{\prime }}$ over $S$; if $g$ is etale (Étale morphism of schemes), then $j^{\mathrm{\prime }}$ is an etale equivalence relation. When $g$ is etale, each restricted source or target is a composition of a base change of $g$ with a base change of $s$ or $t$, hence etale.