nLab algebraic pattern

Idea

An algebraic pattern is a blueprint for a notion of functors on a fixed category satisfying a Segal condition, suitable for formalizing homotopy-coherent algebra in the Cartesian setting.

Definitions

Definition

An algebraic pattern is an (∞,1)-category $\mathcal{O}$ together with the following data:

• a pair of wide subcategories $\mathcal{O}^{\mathrm{int}},\mathcal{O}^{\mathrm{act}} \subset \mathcal{O}$, whose morphisms are called inert and active morphisms, and

• a full subcategory $\mathcal{O}^{\mathrm{el}} \subset \mathcal{O}^{\mathrm{int}}$, whose objects are called elementary objects,

subject to the condition that for all $f:X \rightarrow X'$ in $\mathcal{O}$, the space of factorizations of $f$ of the form $X \xrightarrow{i} Y \xrightarrow{a} X'$ with $i \in \mathcal{O}^{\mathrm{int}}$ and $r \in \mathcal{O}^{\mathrm{act}}$ is contractible.

The pair $(\mathcal{O}^{\mathrm{int}},\mathcal{O}^{\mathrm{act}})$ is a factorization system, with no assumptions made about orthogonality.

It is customary to abusively refer to the quadruple $(\mathcal{O},\mathcal{O}^{\mathrm{int}},\mathcal{O}^{\mathrm{act}}, \mathcal{O}^{\mathrm{el}})$ simply as $\mathcal{O}$, and to decorate inert morphisms as $\mapsto$ and active morphisms as $\rightsquigarrow$. These are so defined in order to define Segal objects.

Definition

Let $\mathcal{O}$ be an algebraic pattern and $\mathcal{C}$ a complete (∞,1)-category. A Segal $\mathcal{O}$-object in $\mathcal{C}$ is a functor

$$F:\mathcal{O} \rightarrow \mathcal{C}$$

subject to the condition that, for all $Z \in \mathcal{O}$, the canonical morphism

$$F(Z) \rightarrow \lim_{\stackrel{E \mapsto Z}{E \in \mathcal{O}^{\mathrm{el}}}}F(E)$$

is an equivalence.

By Lemma 2.9 of Chu-Hangseng, these are equivalently given as functors $F:\mathcal{O} \rightarrow \mathcal{C}$ whose restriction to $\mathcal{O}^{\mathrm{int}}$ are right Kan extended from $\mathcal{O}^{\mathrm{el}}$.

One example of this is the pattern $\mathbb{F}_*^{\flat}$, whose underlying category is the category $\mathbb{F}_*$ of finite pointed sets with

• inert morphisms given by those $\phi:\langle n \rangle \rightarrow \langle m \rangle$ with $|\varphi^{-1}(j)| = 1$ for all $j \neq *$,

• active morphisms given by those $\phi$ with $\phi^{-1}(*) = \{*\}$, and

• elementary objects spanned by $\langle 1 \rangle$.

The resulting (∞,1)-category of Segal $\mathbb{F}_*^{\flat}$-objects in $\mathcal{C}$ consists of the Γ-objects in $\mathcal{C}$; hence they are equivalent to E∞ algebras in $\mathcal{C}$ with the cartesian symmetric monoidal structure.

In particular, applying this to the category $\mathrm{Cat}_\infty$ of small (∞,1)-categories and applying unstraightening yields a distinguished subcategory of cocartesian fibrations over $\mathcal{O}$ satisfying Segal conditions. These are sometimes called Segal fibrations.

Relaxing the condition that Segal fibrations have underlying cocartesian fibrations then yields the following notion, originally called weak Segal fibrations.

Definition

Let $\mathcal{O}$ be an algebraic pattern. A functor $\pi:\mathcal{P} \rightarrow \mathcal{O}$ is called a fibrous pattern if it satisfies the following conditions:

• $\mathcal{P}$ has $\pi$-cocartesian lifts for inert morphisms in $\mathcal{O}$.

• For all $Z \in \mathcal{O}$, the canonical map

  $$\mathcal{P} \times_{\mathcal{O}} \mathcal{O}^{\mathrm{act}}_{/Z} \rightarrow \mathcal{O}^{\mathrm{act}}_{/Z} \times_{ \lim_{E \mapsto Z} \mathcal{O}^{\mathrm{act}}_{/E}} \left(\lim_{E \mapsto Z} \mathcal{P} \times_{\mathcal{O}} \mathcal{O}^{\mathrm{act}}_{/E},\right)$$

  is an equivalence, where the limits run across all inert maps $E \mapsto Z$ with elementary domain.

One of the main reasons to use algebraic patterns over the categorical patterns of Higher Algbera is the relative ease of constructing functoriality in families of examples. The central concept driving this is that of a Segal morphism.

Definition

A functor of algebraic patterns $F:\mathcal{O} \rightarrow \mathcal{P}$ is called a

• Segal morphism if the pullback functor $F^*:\mathrm{Fun}(\mathcal{P}, \mathcal{C}) \rightarrow \mathrm{Fun}(\mathcal{O}, \mathcal{C})$ preserves Segal objects for all $\mathcal{C}$, and a

• strong Segal morphism if for all $Z \in \mathcal{O}$, the associated functor $\mathcal{O}^{\mathrm{el}}_{/Z} \rightarrow \mathcal{O}^{\mathrm{el}}_{/F(Z)}$ is initial.

By Lemma 4.5 of Chu-Haugseng, for a functor of algebraic patterns to be a Segal morphism, it suffices to check that $F^*$ preserves Segal spaces. As developed in Barkan-Haugseng-Steinebrunner, (strong) Segal morphisms induce functors on fibrous patterns in the presence of mild categorical “soundness” or “extendability” conditions on the patterns involved.

Examples

• By Proposition 4.1.7 of Barkan-Haugseng-Steinebrunner, $\mathbb{F}_*^{\flat}$-fibrous patterns are precisely the (∞,1)-operads; defining the pattern structure $\mathbb{F}_*^{\natural}$ on $\mathbb{F}_*$ with the usual inert and active morphisms together with $\emptyset, \langle 1 \rangle$ as elementary objects, the $\mathbb{F}_*^{\natural}$-fibrous patterns are the generalized (∞,1)-operads.

• More generally, when G is a finite group, the G-category of finite pointed G-sets admits a similar pattern structure $\underline{\mathbb{F}}_{G,*}^{\flat}$, whose Segal objects are the G-commutative monoids and whose fibrous patterns are precisely the G-∞-operads.

Both of these examples are more easily presented via span patterns. The theorem relating these is Proposition 5.2.14 of Barkan-Haugseng-Steinebrunner. They define a pattern $\mathrm{Span}(\mathbb{F}_G)$ whose inert maps are backwards, active maps are forwards, and elementary objects are transitive G-sets.

Theorem

The forgetful functor $\mathbb{F}_{G,*} \rightarrow \mathrm{Span}(\mathbb{F}_G)$ lifts to a Segal morphism of patterns, which induces an equivalence on the categories of Segal obejcts and fibrous patterns.

• The category $\Delta^{\op}$ has two pattern structures $\Delta^{\op,\natural}, \Delta^{\op, \flat}$, which parameterize Segal spaces (i.e. ∞-categories) and E1 algebras, respectively.

• The $n$-fold product $\Delta^{\op, n, \natural}$ parameterizes $n$-fold Segal spaces, i.e. n-fold ∞-categories; the $n$-fold product $\Delta^{\op, n, \flat}$ models $\mathbb{E}_n$-spaces.

• The $n$-fold wreath product $\Theta_n := \Delta \wr \cdots \wr \Delta$ has on its opposite two interesting pattern structures; the pattern $\Theta_n^{\op,\natural}$ models (∞,n)-categories, and the pattern $\Theta_n^{\op,\flat}$ models $\mathbb{E}_n$-spaces.

• Let $\mathcal{S}_m$ denote the $\infty$-category of m-finite spaces. Then, the pattern $\mathrm{Span}(\mathcal{S}_{m})$ models m-commutative monoids.

• The opposite $\Omega^{\op}$ of the dendroidal category of Moerdijk-Weiss has a pattern structure $\Omega^{\op,\natural}$ whose segal objects are the Dendroidal Segal spaces of Cisinski-Moerdijk, hence they are operads.

Related concepts

• Segal condition

• (∞,n)-categories,

• operad

• lawvere theory

• Parametrized Higher Category Theory and Higher Algebra

References

Original references:

• Hongyi Chu, Rune Haugseng, Homotopy-coherent algebra via Segal conditions, (2019) (arXiv:1907.03977)

• Shaul Barkan, Rune Haugseng, Jan Steinebrunner, Envelopes for Algebraic Patterns, (2022) (arXiv:2208.07183)

• Hongyi Chu, Rune Haugseng, Enriched homotopy-coherent structures, (2023) (arxiv:2308.11502)

Slice patterns and morita equivalences appear in:

• Shaul Barkan, Arity Approximation of ∞-Operads, (2022) (arXiv:2207.07200)