\((\infty,1)\)-Categories: Foundations¶

Job: fix models of \((\infty,1)\)-categories (quasicategories, complete Segal spaces), state limits/colimits and presentability as used in HTT/HA and derived geometry.

Core¶

Motivations¶

Phenomenon	\(1\)-categorical failure	\(\infty\)-fix
Quasi-isomorphisms	inverted in \(D(\mathcal{A})\) lose coherent cones	stable \(\infty\)-category
Homotopy limits	derived functors ad hoc	limits in \(\mathcal{S}\) or \(\mathcal{C}\)
Stacks / descent	\(2\)-limits partial	sheaves of spaces / \(\infty\)-topoi
\(A_\infty\)-algebras	coherence diagrams infinite	algebras over \(\infty\)-operads

An \((\infty,1)\)-category has objects, mapping spaces (not just sets), associative composition up to coherent homotopy, and all \(k\)-morphisms for \(k\ge 2\) invertible. The homotopy hypothesis identifies \(\infty\)-groupoids with spaces (Kan complexes).

Quasicategories (Joyal–Lurie)¶

A quasicategory (weak Kan complex, inner Kan complex) is a simplicial set \(\mathcal{C}\) such that every inner horn \(\Lambda^n_i\hookrightarrow\Delta^n\) (\(0<i<n\)) admits a filler \(\Delta^n\to\mathcal{C}\).

Simplicial set	Role
\(\mathcal{C}_0\)	objects
\(\mathcal{C}_1\)	\(1\)-morphisms
\(\mathcal{C}_n\)	coherent \(n\)-simplices of composable arrows
Horn fillers	composition (\(n=2\)) and associativity/coherence (\(n\ge 3\))

Examples. Kan complexes \(=\) quasicategories that are \(\infty\)-groupoids. Nerve \(N(\mathbf{C})\) of an ordinary category is a quasicategory. DG-nerve of a DG-category; simplicial nerve of a fibrant simplicial category.

Mapping spaces. For \(x,y\in\mathcal{C}_0\), \(\mathrm{Map}_{\mathcal{C}}(x,y)\) is a Kan complex (Joyal), well-defined up to homotopy; \(h\mathcal{C}\) has \(\mathrm{Hom}= \pi_0\mathrm{Map}\).

Functors. Map of simplicial sets \(\mathcal{C}\to\mathcal{D}\); \(\mathrm{Fun}(\mathcal{C},\mathcal{D})=\mathrm{Fun}_{\mathbf{sSet}}(\mathcal{C},\mathcal{D})\) is again a quasicategory if \(\mathcal{D}\) is.

Joyal model structure. Cofibrations \(=\) monomorphisms; fibrant objects \(=\) quasicategories; weak equivalences \(=\) categorical equivalences (induce equivalences on mapping spaces and essentially surjective on \(h\mathcal{C}\)). Quillen equivalent to other models of \((\infty,1)\)-categories.

Complete Segal spaces (Rezk)¶

A Segal space is a simplicial space \(X_\bullet\) (bisimplicial set) such that $$ X_n\xrightarrow{\sim} X_1\times_{X_0}\cdots\times_{X_0}X_1 $$ (Segal maps are weak homotopy equivalences). Completeness: the map \(X_0\to X_{\mathrm{hoequiv}}\) (subspace of homotopy equivalences in \(X_1\)) is a weak equivalence — objects are determined by their identity arrows up to homotopy.

Feature	Quasicategories	Complete Segal spaces
Underlying data	single simplicial set	simplicial space
Mapping spaces	cut out by joins / slices	\(X_1\) with source/target to \(X_0\)
Strictness of composition	weak (fillers)	Segal condition weakens pullbacks
Comparison	Joyal–Tierney / Lurie	Quillen equivalent to quasicats

When to prefer CSS. Homotopy-theoretic constructions (localization, Rezk completion) and some \((\infty,n)\) extensions (Θ-spaces, \(n\)-fold complete Segal spaces) are natural in the CSS world. Research papers in derived AG typically use quasicategories (Lurie).

Limits and colimits¶

A limit of a diagram \(p:K\to\mathcal{C}\) is a terminal object of the \(\infty\)-category of cones \(\mathcal{C}_{/p}\); dually colimits are initial objects of cocones. Equivalent: \(\mathrm{Map}_{\mathcal{C}}(X,\lim F)\simeq\lim \mathrm{Map}_{\mathcal{C}}(X,F(-))\) in \(\mathcal{S}\).

\(1\)-categorical notion	\(\infty\)-upgrade
Pullback	homotopy pullback (space-valued mapping)
Equalizer	derived equalizer
Adjunction	unit/counit + triangle identities up to homotopy; Hom-space adjunction
Kan extension	pointwise via colimits when presentable

Theorem (HTT). \(\mathcal{C}\) admits all small colimits iff it admits small coproducts and geometric realizations (or pushouts + coproducts, etc., under standard generators). Same dual for limits.

Presentable \(\infty\)-categories (pointer)¶

\(\mathcal{C}\) is presentable if it is accessible (generated under \(\kappa\)-filtered colimits by a small set of \(\kappa\)-compact objects for some regular \(\kappa\)) and admits all small colimits. Equivalently: accessible localization of a presheaf \(\infty\)-category \(\mathcal{P}(\mathcal{C}_0)=\mathrm{Fun}(\mathcal{C}_0^{\mathrm{op}},\mathcal{S})\).

Fact	Content
Adjoint functor theorem	\(F:\mathcal{C}\to\mathcal{D}\) between presentables preserves limits \(\Rightarrow\) left adjoint exists under accessibility hyp.
\(\mathrm{Pr}^L\)	presentable \(\infty\)-cats + left adjoints (colimit-preserving functors)
Tensor product	\(\mathcal{C}\otimes\mathcal{D}\) in \(\mathrm{Pr}^L\); underlies higher algebra of presentable stable \(\infty\)-cats
Stable	finite limits exist, suspension equivalence; homotopy cat triangulated

Examples. \(\mathcal{S}\); \(\mathrm{Sp}\) (spectra); \(\mathrm{Mod}_R\) for \(\mathbb{E}_\infty\)-ring \(R\); \(\mathrm{QCoh}(X)\) for derived/spectral schemes; sheaves of spaces on a site.

Straightening / unstraightening (one line)¶

Left fibrations over \(\mathcal{C}\) \(\leftrightarrow\) functors \(\mathcal{C}\to\mathcal{S}\) (HTT Ch.\ 2); Cartesian fibrations \(\leftrightarrow\) functors to \(\mathrm{Cat}_\infty\). This is the \(\infty\)-categorical Grothendieck construction — indispensable for families of categories and \(\infty\)-topoi.

See it¶

Dold–Kan and the stable nerve. The \(\infty\)-category of chain complexes \(\mathrm{Ch}_{\ge 0}(R)\) (connective) is equivalent to \(\mathrm{Mod}_{HR}^{\mathrm{cn}}\) in spectra; the classical derived category is \(h\) of the stable \(\infty\)-category of modules. Computing \(\mathrm{Tor}\) as a colimit of bar constructions is literally a geometric realization in an \(\infty\)-category — explaining why simplicial resolutions compute derived functors.

Homotopy limits of spaces. \(\mathrm{holim}\) of a cosimplicial space is the limit in the quasicategory \(\mathcal{S}\); Tot of a Bousfield–Kan tower is a special case. Spectral sequences of homological algebra / spectral sequences arise from filtrations of such limits.

Check¶

Show that a Kan complex is a quasicategory, and that \(N(\mathbf{C})\) has unique inner horn fillers.

Kan: fillers for all horns, hence for inner horns \(\Rightarrow\) quasicategory. Nerve: an \(n\)-simplex is a string of \(n\) composable arrows; an inner horn specifies all but one composite, uniquely determined by composition in \(\mathbf{C}\) — unique filler. (Outer horns need not fill: that would force all arrows invertible.)

Why does the existence of all small colimits + accessibility imply a free cocompletion property?

Presentable \(\mathcal{C}\simeq\) accessible localization of \(\mathcal{P}(\mathcal{C}_0)\); \(\mathcal{P}(\mathcal{C}_0)\) is the free cocompletion of small \(\mathcal{C}_0\) under small colimits (HTT). Localizations kill additional colimit relations — same logic as ordinary locally presentable categories (Gabriel–Ulmer).

Pitfalls¶

• Horn direction: inner vs outer; confusing Kan with Joyal fibrancy is the standard first error.
• Equivalence vs isomorphism: objects equivalent if related by a morphism invertible in \(h\mathcal{C}\) (homotopy equivalence in mapping spaces), not strictly equal.
• Homotopy category loses limits: \(h(\lim F)\not\simeq\lim (hF)\) in general — compute limits \(\infty\)-categorically before passing to \(h\).
• Size: universe issues same as in ordinary category theory; presentability is the practical fix.

Refs¶

• Lurie, Higher Topos Theory
• Joyal, quasi-category notes; Kerodon (Lurie)
• Rezk, A model for the homotopy theory of homotopy theory
• Riehl–Verity, Elements of \(\infty\)-Category Theory (model-independent)

Next¶

Higher categories index · \(\infty\)-operads · Category theory · Derived AG · Homological algebra · Topos theory · Geometric Langlands — DG / presentable \(\infty\)-categories on both sides of the categorical equivalence