\(
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\)
Fix the variance and universal properties of the categorical constructions used by the algebraic and lattice chapters. Core and groupoid completion are defined in Definition 2.10.
Definition 3.1 (Category of elements) Let \(F\colon C^{\mathrm{op}}\to\mathbf{Set}\) be a presheaf. Its category of elements \(\int_C F\) has objects \((c,x)\) with \(x\in F(c)\). A morphism \((c,x)\to(d,y)\) is a morphism \(f\colon c\to d\) satisfying \(F(f)(y)=x\). The projection \[
p_F\colon\int_C F\longrightarrow C,\qquad(c,x)\longmapsto c,
\] is a discrete fibration (nLab authors 2026b; Riehl 2016, sec. 2.4). For a covariant functor \(C\to\mathbf{Set}\) the corresponding projection is a discrete opfibration.
Grothendieck construction
For a pseudofunctor \(F\colon C^{\mathrm{op}}\to\mathbf{Cat}\), the Grothendieck construction \(\int_C F\to C\) is a fibration. The category of elements is its \(\mathbf{Set}\)-valued special case (nLab authors 2026d). A general forgetful functor need not arise from a category of elements; later chapters specify each forgetful functor directly.
Definition 3.2 (Subobjects, full and replete subcategories) A subobject of \(Y\in\mathcal C\) is a monomorphism \(m:X\to Y\) in \(\mathcal C\); the datum is the arrow \(m\), not merely the domain \(X\) (Lurie 2018, Tag 04VD). Equivalently, \(m\) is a subterminal object of the slice \(\mathcal C_{/Y}\), and if the relevant pullback exists, \(m\) is monic exactly when the diagonal \(\Delta_m:X\to X\times_Y X\) is an equivalence.
For ordinary \(\mathbf{Cat}\), a functor \(F:\mathcal A\to\mathcal B\) is monic if and only if it is injective on objects and faithful; hence subobjects in \(\mathbf{Cat}\) are ordinary subcategories, up to isomorphism over \(\mathcal B\). In \(\operatorname{Cat}_{\infty,\infty}\), a monomorphism must also contain every equivalence between objects in its image, so not every ordinary subcategory remains a subobject after passing to \(\operatorname{Cat}_{\infty,\infty}\).
A full subcategory \(\mathcal D\subseteq\mathcal C\) is the special case where \(m\) is fully faithful: a collection of objects of \(\mathcal C\) with mapping spaces inherited from \(\mathcal C\), \[
\operatorname{Map}_{\mathcal D}(x,y)\simeq\operatorname{Map}_{\mathcal C}(x,y).
\] It is replete if every object of \(\mathcal C\) equivalent to an object of \(\mathcal D\) also belongs to \(\mathcal D\). An isomorphism-invariant property of objects of \(\mathcal C\) defines a replete full subcategory (The Stacks project authors 2025, Tag 001D; nLab authors 2026g).
Definition 3.3 (Concrete categories) A concrete category is a category \(C\) equipped with a faithful functor \(U\colon C\to\mathbf{Set}\). The set \(U(X)\) is the underlying set of \(X\). Any further underlying-set construction is stated as a factorization through this functor.
For a concrete category, a morphism whose underlying function is injective is monic, and one whose underlying function is surjective is epic; each converse is a separate claim about \(U\). When \(U\) is corepresented by an object \(P\) as in Definition 15.1, \(U(f)=\operatorname{Hom}_C(P,f)\) and \(f\) is monic exactly when \(U(f)\) is injective: one direction applies the definition of a monomorphism to \(P\), the other uses faithfulness. The corresponding claim for epimorphisms fails: the inclusion \(\mathbb Z\hookrightarrow\mathbb Q\) is monic and epic in \(\mathbf{Ring}\), and it is neither surjective nor invertible (Riehl 2016, Example 1.2.10). Invertibility is asked of the morphism, by exhibiting a two-sided inverse in \(C\).
Comma categories and universal arrows
Definition 3.4 (Comma categories) Let \(T\colon\mathcal E\to\mathcal C\) and \(S\colon\mathcal D\to\mathcal C\) be functors. The comma category \((T\downarrow S)\) has as objects the triples \(\langle e,d,f\rangle\) with \(e\in\mathcal E\), \(d\in\mathcal D\), and \(f\colon Te\to Sd\) in \(\mathcal C\), and as morphisms \(\langle e,d,f\rangle\to\langle e',d',f'\rangle\) the pairs \(\langle k\colon e\to e',\,h\colon d\to d'\rangle\) with \[
f'\circ Tk=Sh\circ f.
\] The two projections send \(\langle e,d,f\rangle\) to \(e\) and to \(d\) (Mac Lane 1998, sec. II.6).
Taking \(T\) or \(S\) to be a functor \(\mathbf 1\to\mathcal C\), that is an object of \(\mathcal C\), gives the coslice \((b\downarrow S)\) and the slice \((T\downarrow a)\); taking both to be \(\operatorname{id}_{\mathcal C}\) gives the arrow category \((\mathcal C\downarrow\mathcal C)\), whose objects are the morphisms of \(\mathcal C\); taking both to be objects \(a,b\) gives the discrete category \(\operatorname{Hom}_{\mathcal C}(b,a)\).
For an ordinary category \(C\), the arrow category is also \([[1],C]\), and evaluation at the source and target gives a functor to \(C\times C\). Local hom-objects and walking-arrow categories in \(\mathbf{Cat}_{\infty,\infty}\) are defined in Definition 2.5 and Definition 2.16.
Definition 3.5 (Universal arrows) Let \(S\colon\mathcal D\to\mathcal C\) and \(c\in\mathcal C\). A morphism \(u\colon c\to Sr\) is universal from \(c\) to \(S\) when, for every \(d\in\mathcal D\) and every \(f\colon c\to Sd\), there is a unique \(g\colon r\to d\) with \(Sg\circ u=f\). Equivalently, \(\langle r,u\rangle\) is an initial object of the comma category \((c\downarrow S)\), so it is unique up to a unique isomorphism of \((c\downarrow S)\) and \(r\) is unique up to isomorphism in \(\mathcal D\) (Mac Lane 1998, sec. III.1). Dually, a morphism \(v\colon Sr\to c\) is couniversal when \(\langle r,v\rangle\) is terminal in \((S\downarrow c)\).
Free constructions are of this form: the insertion of a graph into the underlying graph of the free category on it is universal from the graph to the forgetful functor \(U\colon\mathbf{Cat}\to\mathbf{Grph}\) (Definition 5.7). The corepresenting objects listed in Definition 15.1 are the values of the corresponding free constructions on a one-element set, and the universal arrow is the choice of that element.
Additive and abelian categories
Definition 3.6 (Preadditive and additive categories) A preadditive category, or \(\mathbf{Ab}\)-category, is a category \(\mathcal A\) in which each \(\operatorname{Hom}_{\mathcal A}(b,c)\) is an abelian group and composition is bilinear. Its zero elements are the zero morphisms \(0\colon b\to c\), and a composite with a zero morphism is zero.
In a preadditive category an object \(z\) is initial exactly when it is terminal, exactly when \(1_z=0\), and exactly when \(\operatorname{Hom}_{\mathcal A}(z,z)\) is the zero group; such a \(z\) is a zero object (Mac Lane 1998, sec. VIII.2, Proposition 1).
A biproduct diagram for \(a,b\in\mathcal A\) consists of an object \(c\) and morphisms \[
i_1\colon a\to c,\quad i_2\colon b\to c,\quad p_1\colon c\to a,\quad p_2\colon c\to b
\] satisfying \[
p_1i_1=1_a,\qquad p_2i_2=1_b,\qquad i_1p_1+i_2p_2=1_c.
\] Two objects of \(\mathcal A\) have a product exactly when they have a biproduct, exactly when they have a coproduct; the biproduct object with \(p_1,p_2\) is their product and with \(i_1,i_2\) their coproduct, and it is written \(a\oplus b\) (Mac Lane 1998, sec. VIII.2, Theorem 2).
An additive category is a preadditive category with a zero object and a biproduct for each pair of objects (Mac Lane 1998, sec. VIII.2).
Definition 3.7 (Kernels and cokernels) In a category with a zero object, the kernel of \(f\colon a\to b\) is the equalizer of the pair \(f,0\colon a\to b\): a morphism \(k\colon s\to a\) with \(fk=0\) through which every \(h\) with \(fh=0\) factors uniquely. The cokernel is the coequalizer of the same pair (Mac Lane 1998, sec. VIII.1).
Definition 3.8 (Abelian categories) An abelian category is an additive category in which every morphism has a kernel and a cokernel and, for every morphism \(f\), the canonical map \[
\operatorname{Coim}(f)\longrightarrow\operatorname{Im}(f)
\] is an isomorphism (The Stacks project authors 2025, Tag 0109). In an abelian category, a morphism is monic exactly when its kernel is zero and epic exactly when its cokernel is zero.
An equivalent axiomatization asks for a zero object, binary biproducts, kernels and cokernels of every morphism, and that every monomorphism be a kernel and every epimorphism a cokernel (Mac Lane 1998, sec. VIII.3). Kernels and cokernels then supply the remaining finite limits and colimits: the equalizer of \(f,g\) is the kernel of \(f-g\), and biproducts are the finite products, so an abelian category is finitely complete and finitely cocomplete (Mac Lane 1998, sec. VIII.3).
Theorem 3.9 (The canonical factorization) In an abelian category every morphism \(f\) factors as \(f=me\) with \(m\) monic and \(e\) epic, where \[
m=\ker(\operatorname{coker}f),
\qquad
e=\operatorname{coker}(\ker f),
\] and the factorization is functorial: a commutative square from \(f\) to \(f'\) induces a unique morphism between the two factorizations commuting with both halves. The object \(\operatorname{Im}(f)\) is the domain of \(m\) and \(\operatorname{Coim}(f)\) the codomain of \(e\) (Mac Lane 1998, sec. VIII.3, Proposition 1).
Theorem 3.10 (Diagram lemmas) Let \(\mathcal A\) be abelian. For a morphism \(\langle f,g,h\rangle\) of short exact sequences: if \(f\) and \(h\) are monic then \(g\) is monic, and if \(f\) and \(h\) are epic then \(g\) is epic (short five lemma); there is a connecting morphism \(\delta\colon\ker h\to\operatorname{coker}f\) making \[
\ker f\to\ker g\to\ker h
\xrightarrow{\;\delta\;}
\operatorname{coker}f\to\operatorname{coker}g\to\operatorname{coker}h
\] exact (snake lemma). In a commutative diagram with exact rows and five vertical morphisms, if the four outer ones are isomorphisms then so is the middle one (five lemma) (Mac Lane 1998, sec. VIII.4, Lemmas 1, 4, and 5).
Exactness is a property of a composable pair: \(a\xrightarrow{f}b\xrightarrow{g}c\) with \(gf=0\) is exact at \(b\) when the monomorphism \(\operatorname{Im}(f)\to b\) and the kernel of \(g\) represent the same subobject of \(b\) (Definition 15.2).
Endomorphisms and automorphisms
Definition 3.11 (The endomorphism monoid and the automorphism group) For an object \(X\) of a category \(\mathcal C\), composition makes \(\operatorname{Hom}_{\mathcal C}(X,X)\) a monoid with unit \(\operatorname{id}_X\). Its group of invertible elements is \(\operatorname{Aut}_{\mathcal C}(X)\), the automorphism group of \(X\), which is \(\operatorname{Iso}_{\mathcal C}(X,X)\) in the notation of Section 7.1. When \(\mathcal C\) is preadditive, \(\operatorname{Hom}_{\mathcal C}(X,X)\) is a ring under composition and the group addition, and \(\operatorname{Aut}_{\mathcal C}(X)\) is its group of units.
The arity-\(1\) term of the endomorphism operad of Section 4.5 is this monoid; the higher arities record the operations \(X^{\times n}\to X\) and are not composition.
Pseudo-pullbacks
Given functors \(F\colon A\to C\) and \(G\colon B\to C\), their pseudo-pullback has objects \((a,b,\alpha)\) with \(\alpha\colon F(a)\xrightarrow{\sim}G(b)\) and the evident compatible morphisms. It is the 2-categorical pullback up to equivalence (nLab authors 2026a).
If one leg is an isofibration, its strict pullback presents the pseudo-pullback up to equivalence. A replete full inclusion is an isofibration (nLab authors 2026e). Thus strict pullbacks suffice for replete full subcategories, while forgetful functors from chosen structures generally require pseudo-pullbacks.
Connected components and fibers
The functor \(\pi_0\colon\mathcal S\to\mathbf{Set}\) sends a space to its set of components. It does not preserve arbitrary homotopy pullbacks. For a fiber sequence \(F\to E\to B\), the associated exact sequence of pointed sets contains \[
\pi_1(B,b)\longrightarrow\pi_0(F)\longrightarrow\pi_0(E)
\longrightarrow\pi_0(B).
\] Consequently, the fiber of \(\pi_0(E)\to\pi_0(B)\) need not equal \(\pi_0(F)\) (May 1999). A set-valued invariant formed from this sequence must specify whether it uses the fiber after \(\pi_0\) or the components of the homotopy fiber.
Definition 3.12 (Generalized elements) A generalized element of \(X\in C\) with domain \(T\) is a morphism \(T\to X\). For a presheaf \(F\colon C^{\mathrm{op}}\to\mathbf{Set}\), an element \(x\in F(T)\) is an object \((T,x)\) of \(\int_C F\).
Definition 3.13 (Representable presheaves) The presheaf \(F\) is representable if there is an object \(X\) and a natural isomorphism \(F\cong\operatorname{Hom}_C(-,X)\). Under this isomorphism, \(\operatorname{id}_X\) corresponds to the universal element. The dual convention applies to corepresentable covariant functors.
Theorem 3.14 (The Yoneda lemma) Let \(C\) be locally small, \(F\colon C\to\mathbf{Set}\) a functor, and \(c\in C\). Evaluation of a natural transformation at \(\operatorname{id}_c\) is a bijection \[
\operatorname{Nat}\bigl(\operatorname{Hom}_C(c,-),\,F\bigr)
\;\cong\;
F(c),
\qquad
\alpha\longmapsto\alpha_c(\operatorname{id}_c),
\] natural in \(c\) and in \(F\) (Riehl 2016, Theorem 2.2.4; Mac Lane 1998, sec. III.2). Consequently the Yoneda embeddings \[
C\hookrightarrow\mathbf{Set}^{C^{\mathrm{op}}},
\qquad
C^{\mathrm{op}}\hookrightarrow\mathbf{Set}^{C}
\] are fully faithful: natural transformations between represented functors correspond to morphisms between the representing objects (Riehl 2016, Corollary 2.2.8).
A representation is therefore determined by its universal element, and two representations of the same functor are related by a unique isomorphism compatible with the universal elements.
Definition 3.15 (The $\int_C F$ convention)
For every presheaf \(F\colon C^{\mathrm{op}}\to\mathbf{Set}\), use the category of elements and discrete-fibration convention of Definition 3.1. Thus the projection \(\int_C F\to C\) is a discrete fibration; no additional opposite category is taken after forming \(\int_C F\).
Definition 3.16 (Smooth manifolds as locally ringed spaces) Let \(\mathsf{LRS}_{\mathbb{R}}\) denote the category of locally \(\mathbb{R}\)-ringed spaces: topological spaces equipped with a sheaf of \(\mathbb{R}\)-algebras whose stalks are local rings (nLab authors 2026f). The standard local model of dimension \(n\) is \[
M_n := (\mathbb{R}^n,\, C^\infty_{\mathbb{R}^n}),
\] the Euclidean space with its sheaf of smooth real-valued functions.
A smooth atlas for an object \(X \in \mathsf{LRS}_{\mathbb{R}}\) is a morphism \(p\colon U \to X\) such that:
\(U\) is isomorphic to a coproduct of local models: \(U \cong \coprod_{i \in I} M_{n_i}\).
For every component inclusion \(\iota_i\colon M_{n_i} \hookrightarrow U\), the composite \(p \circ \iota_i\colon M_{n_i} \to X\) is an open immersion.
\(p\) is an effective epimorphism: it is the coequalizer of its kernel pair \[ U \times_X U \;\rightrightarrows\; U \;\xrightarrow{p}\; X.
\]
A smooth manifold is an object \(X \in \mathsf{LRS}_{\mathbb{R}}\) that admits a smooth atlas (nLab authors 2026c, 2026h).
The data encode the classical construction as follows. The object \(U\) is the disjoint union of charts. The fibered product \(U \times_X U\) represents the pairwise intersections of charts in \(X\). The two projections \(\pi_1, \pi_2\colon U \times_X U \to U\) encode the transition functions. The coequalizer condition forces \(X\) to be the colimit obtained from charts glued along their transitions: the classical manifold, recovered without point-set intersections.
Definition 3.17 (Schemes as locally ringed spaces) The smooth manifold definition of Definition 3.16 transfers to algebraic geometry by substituting the ambient category and local models.
Let \(\mathsf{LRS}\) denote the category of all locally ringed spaces. The local models are affine schemes: for a commutative ring \(A\), write \[
M_A := (\operatorname{Spec}(A),\, \mathcal{O}_{\operatorname{Spec}(A)}).
\] A Zariski atlas for an object \(X \in \mathsf{LRS}\) is a morphism \(p\colon U \to X\) such that:
\(U\) is isomorphic to a coproduct of affine schemes: \(U \cong \coprod_{i \in I} M_{A_i}\).
For every component inclusion \(\iota_i\colon M_{A_i} \hookrightarrow U\), the composite \(p \circ \iota_i\colon M_{A_i} \to X\) is an open immersion.
\(p\) is an effective epimorphism: it is the coequalizer of its kernel pair \[ U \times_X U \;\rightrightarrows\; U \;\xrightarrow{p}\; X.
\]
A scheme is an object \(X \in \mathsf{LRS}\) that admits a Zariski atlas.
The coequalizer is essential in algebraic geometry because the intersection of two affine open subschemes need not be affine. The fibered product \(U \times_X U\) encodes the scheme-theoretic intersections of the affine covers without requiring the intersections themselves to be local models.
Definition 3.18 (Generalization by Grothendieck topology) The formulation of Definition 3.16 and Definition 3.17 admits a uniform generalization. Fix a category \(\mathcal{C}\) of locally ringed spaces, a class \(\mcl\) of local models, and a class \(\mca\) of admissible morphisms. A \(\mca\)-atlas for \(X \in \mathcal{C}\) is an effective epimorphism \(p\colon \coprod_{i \in I} L_i \to X\) with each \(L_i \in \mcl\) and each component an \(\mca\)-morphism. Different choices of \(\mca\) yield different Grothendieck topologies and geometric objects:
| Open immersions |
Zariski |
Schemes |
| Étale morphisms |
Étale |
Algebraic spaces, Deligne–Mumford stacks |
| Faithfully flat morphisms of finite presentation |
fppf |
Artin stacks |
| Faithfully flat quasi-compact morphisms |
fpqc |
fpqc sheaves |
The smooth manifold case uses \(\mcl = \{M_n\}_{n \geq 0}\) and \(\mca =\) open immersions in \(\mathsf{LRS}_{\mathbb{R}}\).
Dawes, Matthew. 2022.
Orbits in Lattices.
https://doi.org/10.48550/arXiv.2205.10601.
Lurie, Jacob. 2018.
“Kerodon.” https://kerodon.net.
Mac Lane, Saunders. 1998. Categories for the Working Mathematician. Second edition. Graduate Texts in Mathematics 5. Springer Science+Business Media, LLC.
May, J. Peter. 1999. A Concise Course in Algebraic Topology. Chicago Lectures in Mathematics. University of Chicago Press.
Riehl, Emily. 2016. Category Theory in Context. Vol. 1. Aurora: Dover Modern Math Originals. Dover Publications.
Sikirić, Mathieu Dutour, and Klaus Hulek. 2023.
Moduli of Polarized Enriques Surfaces – Computational Aspects.
https://doi.org/10.48550/arXiv.2302.01679.
The Stacks project authors. 2025.
The Stacks Project.
https://stacks.math.columbia.edu/tags.