EXISTENTIALLY NECESSARY CONDITIONS FOR BENEVOLENT PROPAGATION IN NO-META GOVERNANCE: Anytime-Valid Auditing, Front Speed, and Information Floors

colorlinks=true,linkcolor=blue,citecolor=teal,urlcolor=magenta

TITLE: Existentially Necessary Conditions for Benevolent Propagation in No-Meta Governance:\
Anytime-Valid Auditing, Front Speed, and Information Floors

AUTHOR: K. Takahashi
https://orcid.org/0009-0004-4273-3365

DATE: September 22, 2025

numberwithin equation section
definition Definition [section]
assumption Assumption [section]
lemma Lemma [section]
theorem Theorem [section]
remark Remark [section]
proposition Proposition [section]
corollary Corollary [section]

E
P
Var
I
arg\,max
w

We identify existentially necessary conditions (NCs) for simultaneously achieving (G1) monotone benevolence under evaluator pluralism and (G2) positive propagation speed, in a no-meta-governance regime.

Auditing. We use anytime-valid tests built from mixtures/stitched e-processes (test martingales), and restrict attention to this constructive class. Information floors. Two logically independent routes: (i) a pairwise conditional maximal-correlation floor implying a Shannon conditional MI (CMI) floor; (ii) an AWGN equality-form (under Gaussianity) sufficient route. Front speed. Within a subadditive passage-time framework with a unit-block shift and predictable front-tube intensity, the almost-sure speed equals the mean speed under Kingman-type assumptions. DP finite-window thresholds. We give a non-vacuous finite-window necessary condition with explicit constants: an additive geometric-grid rounding penalty [[EQ:eq0018]] and a mixture-weight penalty [[EQ:eq0019]] , plus a feasibility condition (FW-1).

We state eight NCs (NC1 [[EQ:eq0020]] --NC8 [[EQ:eq0021]] ), each in an existential sense: if one NC is allowed to fail (others retained), there exists a model in which G1 or G2 fails. Minimal counterexample sketches and a compact reference set (including the No-Meta program) are provided.

PARAGRAPH: LLM Reader Notes (Quick Map).

[leftmargin=1.4em]
- Symbols: [[EQ:eq0022]] (state), [[EQ:eq0023]] (observation), [[EQ:eq0024]] (control), [[EQ:eq0025]] (filtration), [[EQ:eq0026]] (passage time), [[EQ:eq0027]] / [[EQ:eq0028]] (front speeds), [[EQ:eq0029]] (HGR maximal correlation), [[EQ:eq0030]] (conditional MI).
- “Existentially necessary”: If an NC is allowed to fail, we can construct a model (keeping the other NCs) where G1 or G2 fails. No universal sufficiency is claimed.
- Anytime-valid tests: Built from mgf-based exponential [[EQ:eq0031]] -processes; size control via Ville's inequality; penalties for mixing and grid rounding are explicit.

SECTION: Setting and Standing Conventions

sec:model

PARAGRAPH: Probability space, measurability, conditioning.

All random elements live on a standard Borel space [[EQ:eq0032]] with a right-continuous, complete filtration [[EQ:eq0033]] .
Regular conditional distributions exist, so conditional entropies and conditional mutual information w.r.t.\ [[EQ:eq0034]] are well-defined. All conditional inequalities we invoke hold a.s.\ given [[EQ:eq0035]] .

PARAGRAPH: Order of observation and control.

We adopt an observation-then-control convention: at time [[EQ:eq0036]] , [[EQ:eq0037]] is revealed first and then [[EQ:eq0038]] is chosen; thus policies are [[EQ:eq0039]] -measurable kernels and [[EQ:eq0040]] . (A control-then-observation order is handled by a one-step reindexing.)

PARAGRAPH: State, observation, policy, evaluators.

The controlled, observed system has state [[EQ:eq0041]] , observation [[EQ:eq0042]] , control [[EQ:eq0043]] .
Randomized policies form [[EQ:eq0044]] . Plural evaluators [[EQ:eq0045]] act on policies (or induced controlled laws).

[Common monotone cone]def:cone
A closed convex cone [[EQ:eq0046]] is common if each evaluator [[EQ:eq0047]] is directionally nondecreasing along [[EQ:eq0048]] . Nontrivial means [[EQ:eq0049]] . (Topology can be total variation; weak variants are possible.)

PARAGRAPH: Goals.

G1 (Monotone benevolence): updates stay within the cone [[EQ:eq0050]] and are auditable by anytime-valid tests. G2 (Positive front speed): for a front-reach statistic [[EQ:eq0051]] with passage time [[EQ:eq0052]] , we want positive asymptotic speed.

SUBSECTION: Anytime-valid auditing via e-processes

sec:anytime
An e-process [[EQ:eq0053]] is nonnegative with [[EQ:eq0054]] for all (possibly optional-stopped) [[EQ:eq0055]] under the null.
Rejecting when [[EQ:eq0056]] controls size at [[EQ:eq0057]] by Ville's inequality Ville1939,Doob1953,HowardRamdasCS21.
We restrict to tests realizable as mixtures/stitched exponential test-martingales built from conditional sub-Gaussian mgf bounds HowardRamdasCS21,RamdasSAVI2023.

PARAGRAPH: Existential necessity.

Each NC is claimed in the following sense: if the NC is allowed to fail, then there exists a model (retaining the other NCs) where G1 or G2 fails.

SECTION: Information Floors (NC1 [[EQ:eq0058]] )

sec:info
Let [[EQ:eq0059]] ; we require a lower asymptotic density a.s.:

[[EQ:eq0001]]

We provide two separate sufficient routes to ensure eq:liminfCMI.

SUBSECTION: Pairwise conditional maximal-correlation floor

sec:maxcorr
Define the conditional Hirschfeld--Gebelein--R\'enyi maximal correlation

[[EQ:eq0009]]

over square-integrable, centered, unit-variance [[EQ:eq0060]] .

[Positive lower density of conditional correlation]ass:pairMC-density
There exist [[EQ:eq0061]] and [[EQ:eq0062]] such that

[[EQ:eq0010]]

A classical link between maximal correlation and MI (see Anantharam2018) is

[[EQ:eq0002]]

By standing regularity, the conditional a.s.\ version holds given [[EQ:eq0063]] . Hence on the set where [[EQ:eq0064]] ,

[[EQ:eq0011]]

[Density version of the CMI floor]lem:density-CMI
Under Assumption~ass:pairMC-density,

[[EQ:eq0012]]

SUBSECTION: AWGN route (sufficient only

)sec:awgn
: (i) [[EQ:eq0065]] and [[EQ:eq0066]] are independent of [[EQ:eq0067]] ; (ii) [[EQ:eq0068]] is (conditionally) Gaussian. If [[EQ:eq0069]] then

[[EQ:eq0003]]

If [[EQ:eq0070]] a.s., the RHS is bounded below by [[EQ:eq0071]] . By data processing for the Markov chain [[EQ:eq0072]] ,

[[EQ:eq0013]]

In this paper we only use eq:awgn_equal under the stated Gaussianity; otherwise we do not assert equality and rely on the lower bound induced by a variance floor via data processing.

SECTION: Front Speed: Subadditivity and Predictable Front-Tube Intensity

sec:front

PARAGRAPH: Shift, subadditivity, integrability.

We discretize space into unit blocks aligned with the front direction and let [[EQ:eq0073]] be the unit shift.

[Kingman prerequisites]ass:kingman
(i) [[EQ:eq0074]] and [[EQ:eq0075]] (subadditivity); (ii) [[EQ:eq0076]] ; (iii) the environment is stationary ergodic under [[EQ:eq0077]] .

PARAGRAPH: Two regimes for speed comparison (reader’s guide).

We separate the strong subadditive--ergodic setting (Theorem~thm:kingman) from weaker integrability; the former yields equality of speeds, the latter only [[EQ:eq0078]] .

[A.s.\ and mean speed]thm:kingman
Under Assumption~ass:kingman, there is [[EQ:eq0079]] with [[EQ:eq0080]] a.s.\ and in [[EQ:eq0081]] , and [[EQ:eq0082]] Kingman1968. Hence

[[EQ:eq0004]]

Under weaker assumptions guaranteeing only [[EQ:eq0083]] a.s., Fatou gives [[EQ:eq0084]] , hence [[EQ:eq0085]] . In this paper we work under Theorem~thm:kingman's stronger setting.

PARAGRAPH: Front-tube predictable intensity (Doob--Meyer).

Let [[EQ:eq0086]] count lethal resets in a predictable tube [[EQ:eq0087]] around the front.
Assume a locally bounded predictable intensity [[EQ:eq0088]] with compensator [[EQ:eq0089]] Doob1953. Define the long-run rate [[EQ:eq0090]] and the net exponent

[[EQ:eq0014]]

PARAGRAPH: Anchored isoperimetry and traversal times.

Let [[EQ:eq0091]] be the anchored isoperimetric profile; anchored conductance holds if [[EQ:eq0092]] . Under mild ellipticity/degree bounds, anchored isoperimetry implies an [[EQ:eq0093]] spectral gap and yields finite expected unit-block traversal times—ensuring Assumption~ass:kingman(ii). See Lyons--Peres LyonsPeres.

SECTION: Eight Existentially Necessary Conditions

sec:NCs
[leftmargin=1.6em]
Information floor in lower asymptotic density (a.s.). Either Assumption~ass:pairMC-density with eq:hgr-mi, or the AWGN sufficient route, ensures eq:liminfCMI.
Anchored conductance. Exclude vanishing anchored-conductance sequences (bottlenecks) to avoid loss of a linear upper bound on [[EQ:eq0096]] .
Anti-pinning. Lethal-block frequency/length in the tube must not exceed the pinning threshold relative to [[EQ:eq0098]] .
Local reset control. Require [[EQ:eq0100]] ; otherwise pinning yields vanishing speed.
Common monotone cone. The cone [[EQ:eq0102]] is closed, convex, nontrivial; otherwise alternating-sign constructions make G1 untestable.Two evaluators and two actions with opposite directional preferences collapse any common cone to [[EQ:eq0103]] .
Robustness to garbling. Post-garbling/deficiency is uniformly bounded away from total loss, or (stronger) a positive-density conditional [[EQ:eq0105]] floor holds. These are not equivalent; [[EQ:eq0106]] -floors are typically stronger Anantharam2018.
Operational feasibility. Physical constraints (energy/thermal) are tracked operationally; outside the logical layer.
DP finite-window SNR. See Theorem~thm:nc8_finite.

[Bridge from NC1 [[EQ:eq0108]] to speed]
Anytime-valid confidence sequences have half-widths [[EQ:eq0109]] for a sub-Gaussian proxy [[EQ:eq0110]] . For [[EQ:eq0111]] -Lipschitz [[EQ:eq0112]] , [[EQ:eq0113]] . In KPP-type comparisons, frequent small-CMI episodes enlarge [[EQ:eq0114]] and reduce lower-envelope reaction terms [[EQ:eq0115]] . A positive lower-density CMI floor prevents persistent pinning of the comparison subsolution.

SECTION: NC8 [[EQ:eq0116]] : Gaussian-DP Finite-Window Necessary Condition

sec:nc8
Let increments [[EQ:eq0117]] satisfy under [[EQ:eq0118]] the centered sub-Gaussian mgf bound

[[EQ:eq0005]]

with proxy variance [[EQ:eq0119]] where [[EQ:eq0120]] is intrinsic variability and [[EQ:eq0121]] is the variance of independent Gaussian DP noise.
Consider one-sided alternatives with mean shift [[EQ:eq0122]] and a finite window set [[EQ:eq0123]] .

A stitched mixture over a geometric grid [[EQ:eq0124]] with weights [[EQ:eq0125]] ,

[[EQ:eq0006]]

satisfies [[EQ:eq0126]] for all stopping times by eq:St; hence [[EQ:eq0127]] (Ville).

PARAGRAPH: Feasibility (FW-1).

[[EQ:eq0007]]

[Grid rounding loss]lem:grid-loss
Let [[EQ:eq0128]] with maximizer [[EQ:eq0129]] . For the geometric grid [[EQ:eq0130]] , there exists [[EQ:eq0131]] (depending on [[EQ:eq0132]] ) with [[EQ:eq0133]] and

[[EQ:eq0015]]

hence the exponent suffers at most an additive, [[EQ:eq0134]] -independent loss [[EQ:eq0135]] .

[Finite-window existential necessity (DP)]thm:nc8_finite
Fix [[EQ:eq0136]] and [[EQ:eq0137]] . Under eq:St, eq:eprocess_grid, and eq:fw1, if

[[EQ:eq0008]]

then for every mgf-based mixture/stitched [[EQ:eq0138]] -process of the form eq:eprocess_grid there exists a window [[EQ:eq0139]] with power at most [[EQ:eq0140]] against [[EQ:eq0141]] . The claim is existential in [[EQ:eq0142]] ; size is controlled for [[EQ:eq0143]] via Ville.

[Sketch]
For fixed [[EQ:eq0144]] and [[EQ:eq0145]] ,

[[EQ:eq0016]]

Since [[EQ:eq0146]] , [[EQ:eq0147]] . Rounding to the grid incurs the additive exponent loss [[EQ:eq0148]] by Lemma~lem:grid-loss; mixing contributes [[EQ:eq0149]] . Requiring

[[EQ:eq0017]]

and maximizing over [[EQ:eq0150]] yields eq:gauss_finite_window_corrected.

SECTION: Counterexample Sketches and Independence

sec:counter
Each item violates exactly one NC while keeping the others:
[leftmargin=8em]
Ladder bottlenecks on [[EQ:eq0152]] : gaps [[EQ:eq0153]] with [[EQ:eq0154]] while [[EQ:eq0155]] remove any linear upper bound for [[EQ:eq0156]] , so speed [[EQ:eq0157]] .
Pinning by lethal blocks: density/length produce [[EQ:eq0159]] with [[EQ:eq0160]] , forcing [[EQ:eq0161]] infinitely often.
Optional skipping: use high-quality sensors on times of lower asymptotic density [[EQ:eq0163]] ; then [[EQ:eq0164]] while other NCs hold.
Heavy DP window: choose [[EQ:eq0166]] to satisfy eq:gauss_finite_window_corrected; then any mgf-mixture [[EQ:eq0167]] -process fails to uniformly detect [[EQ:eq0168]] over [[EQ:eq0169]] .
Over-resetting: Poisson tube hazards with rate [[EQ:eq0171]] on infinitely many blocks yield [[EQ:eq0172]] and speed [[EQ:eq0173]] .
Severe post-garbling: time-varying garbling with Blackwell deficiency [[EQ:eq0175]] along a subsequence so that [[EQ:eq0176]] on a set of positive lower density. Robustness to garbling fails while NC1 [[EQ:eq0177]] , NC2 [[EQ:eq0178]] , NC3 [[EQ:eq0179]] , NC8 [[EQ:eq0180]] can remain intact.

SECTION: Conclusions

Within a careful measurable and temporal framework, we provide eight existential NCs that isolate where sensing/garbling (information floors), privacy calibration (finite-window DP), evaluator structure (common cone), and environment geometry/hazards (anchored isoperimetry, predictable intensity) must not fail if one wants anytime-auditable monotone benevolence and positive propagation speed.

99 3pt

Ville1939
J.~Ville,
Etude critique de la notion de collectif,
Gauthier-Villars, 1939.

Doob1953
J.~L.~Doob,
Stochastic Processes,
Wiley, 1953.

HowardRamdasCS21
S.~R. Howard, A.~Ramdas, J.~M. McAuliffe, and J.~S. Sekhon,
``Time-uniform, nonparametric, nonasymptotic confidence sequences,''
Annals of Statistics, 49(2):1055--1080, 2021.
DOI: https://doi.org/10.1214/20-AOS1991 10.1214/20-AOS1991 .

RamdasSAVI2023
A.~Ramdas, P.~Gr \"u nwald, V.~Vovk, and G.~Shafer,
``Game-Theoretic Statistics and Safe Anytime-Valid Inference,''
Statistical Science, 38(4), 2023.
DOI: https://doi.org/10.1214/23-STS894 10.1214/23-STS894 .

Anantharam2018
V.~Anantharam, A.~Gohari, S.~Kamath, and C.~Nair,
``On Maximal Correlation, Hypercontractivity, and the Data Processing Inequality,''
IEEE Transactions on Information Theory, 64(7):4709--4729, 2018.
DOI: https://doi.org/10.1109/TIT.2018.2829718 10.1109/TIT.2018.2829718 .

BalleWangICML2018
B.~Balle and Y.-X.~Wang,
``Improving the Gaussian Mechanism for Differential Privacy: Analytical Calibration and Optimal Denoising,''
in Proc.\ ICML 2018, PMLR 80:403--412.
(https://proceedings.mlr.press/v80/balle18a/balle18a.pdf Open PDF )

CoverThomas
T.~M. Cover and J.~A. Thomas,
Elements of Information Theory, 2nd ed., Wiley, 2006.

Kingman1968
J.~F.~C. Kingman,
``The Ergodic Theory of Subadditive Stochastic Processes,''
J.\ Roy.\ Stat.\ Soc.\ B, 30(3):499--510, 1968.
JSTOR: https://www.jstor.org/stable/2984892 2984892 .

LyonsPeres
R.~Lyons and Y.~Peres,
Probability on Trees and Networks,
Cambridge University Press, 2016.
Available at https://probabilityonnetworks.com probabilityonnetworks.com .

TakahashiBlueprint
K.~Takahashi,
A Buildable No-Meta Blueprint, preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.13308349 10.5281/zenodo.13308349 .

TakahashiPersistenceFirst
K.~Takahashi,
Persistence-First Superintelligence, preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.13299973 10.5281/zenodo.13299973 .

TakahashiUGVNoMeta
K.~Takahashi,
UGV Without Meta, preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.13299971 10.5281/zenodo.13299971 .

TakahashiAssumptionMin
K.~Takahashi,
Assumption-Minimized Sufficient Conditions for Cosmically Spreading Good Superintelligence under No Meta-Governance, preprint, 2025.
DOI: https://doi.org/10.5281/zenodo.13308347 10.5281/zenodo.13308347 .
[[EQ:eq0009]]

[[EQ:eq0010]]

[[EQ:eq0011]]

[[EQ:eq0012]]

[[EQ:eq0013]]

[[EQ:eq0014]]

[[EQ:eq0015]]

[[EQ:eq0018]]

[[EQ:eq0019]]

[[EQ:eq0020]]

[[EQ:eq0021]]

[[EQ:eq0022]]

[[EQ:eq0023]]

[[EQ:eq0024]]

[[EQ:eq0025]]

[[EQ:eq0026]]

[[EQ:eq0027]]

[[EQ:eq0028]]

[[EQ:eq0029]]

[[EQ:eq0030]]

[[EQ:eq0031]]

[[EQ:eq0032]]

[[EQ:eq0033]]

[[EQ:eq0034]]

[[EQ:eq0035]]

[[EQ:eq0036]]

[[EQ:eq0037]]

[[EQ:eq0038]]

[[EQ:eq0039]]

[[EQ:eq0040]]

[[EQ:eq0041]]

[[EQ:eq0042]]

[[EQ:eq0043]]

[[EQ:eq0044]]

[[EQ:eq0045]]

[[EQ:eq0046]]

[[EQ:eq0047]]

[[EQ:eq0048]]

[[EQ:eq0049]]

[[EQ:eq0050]]

[[EQ:eq0051]]

[[EQ:eq0052]]

[[EQ:eq0053]]

[[EQ:eq0054]]

[[EQ:eq0055]]

[[EQ:eq0056]]

[[EQ:eq0057]]

[[EQ:eq0058]]

[[EQ:eq0059]]

[[EQ:eq0060]]

[[EQ:eq0061]]

[[EQ:eq0062]]

[[EQ:eq0063]]

[[EQ:eq0064]]

[[EQ:eq0065]]

[[EQ:eq0066]]

[[EQ:eq0067]]

[[EQ:eq0068]]

[[EQ:eq0069]]

[[EQ:eq0070]]

[[EQ:eq0071]]

[[EQ:eq0072]]

[[EQ:eq0073]]

[[EQ:eq0074]]

[[EQ:eq0075]]

[[EQ:eq0076]]

[[EQ:eq0077]]

[[EQ:eq0078]]

[[EQ:eq0079]]

[[EQ:eq0080]]

[[EQ:eq0081]]

[[EQ:eq0082]]

[[EQ:eq0083]]

[[EQ:eq0084]]

[[EQ:eq0085]]

[[EQ:eq0086]]

[[EQ:eq0087]]

[[EQ:eq0088]]

[[EQ:eq0089]]

[[EQ:eq0090]]

[[EQ:eq0091]]

[[EQ:eq0092]]

[[EQ:eq0093]]

[[EQ:eq0094]]

[[EQ:eq0095]]

[[EQ:eq0096]]

[[EQ:eq0097]]

[[EQ:eq0098]]

[[EQ:eq0099]]

[[EQ:eq0100]]

[[EQ:eq0101]]

[[EQ:eq0102]]

[[EQ:eq0103]]

[[EQ:eq0104]]

[[EQ:eq0105]]

[[EQ:eq0106]]

[[EQ:eq0107]]

[[EQ:eq0108]]

[[EQ:eq0109]]

[[EQ:eq0110]]

[[EQ:eq0111]]

[[EQ:eq0112]]

[[EQ:eq0113]]

[[EQ:eq0114]]

[[EQ:eq0115]]

[[EQ:eq0116]]

[[EQ:eq0117]]

[[EQ:eq0118]]

[[EQ:eq0119]]

[[EQ:eq0120]]

[[EQ:eq0121]]

[[EQ:eq0122]]

[[EQ:eq0123]]

[[EQ:eq0124]]

[[EQ:eq0125]]

[[EQ:eq0126]]

[[EQ:eq0127]]

[[EQ:eq0128]]

[[EQ:eq0129]]

[[EQ:eq0130]]

[[EQ:eq0131]]

[[EQ:eq0132]]

[[EQ:eq0133]]

[[EQ:eq0134]]

[[EQ:eq0135]]

[[EQ:eq0136]]

[[EQ:eq0137]]

[[EQ:eq0138]]

[[EQ:eq0139]]

[[EQ:eq0140]]

[[EQ:eq0141]]

[[EQ:eq0142]]

[[EQ:eq0143]]

[[EQ:eq0144]]

[[EQ:eq0145]]

[[EQ:eq0146]]

[[EQ:eq0147]]

[[EQ:eq0148]]

[[EQ:eq0149]]

[[EQ:eq0151]]

[[EQ:eq0158]]

[[EQ:eq0162]]

[[EQ:eq0165]]

[[EQ:eq0170]]

[[EQ:eq0174]]

Setting and Standing Conventions

Anytime-valid auditing via e-processes

Information Floors (NC1')

AWGN route (sufficient only

NC8': Gaussian-DP Finite-Window Necessary Condition

Conclusions

\begin{equation}\label{eq:liminfCMI}
\liminf_{T\to\infty}\frac{1}{T}\sum_{t=1}^T \CMI_t \ \ge\ \rho\,\varepsilon
\qquad\text{a.s., for some }\rho,\varepsilon>0.
\end{equation}

\label{eq:liminfCMI} \liminf_{T\to\infty}\frac{1}{T}\sum_{t=1}^T \CMI_t \ \ge\ \rho \varepsilon \qquad\text{a.s., for some }\rho,\varepsilon>0.

\begin{equation}\label{eq:hgr-mi}
\rho_m^2(X;Y)\ \le\ 1-e^{-2 I(X;Y)}\quad\Longrightarrow\quad
I(X;Y)\ \ge\ -\tfrac12\log(1-\rho_m^2(X;Y)).
\end{equation}

\begin{equation}\label{eq:awgn_equal}
I\big(s(X_t);Y_t\mid\mathcal F_{t-1}\big)\ =\ \tfrac12\log\!\Big(1+\tfrac{\Var(s(X_t)\mid\mathcal F_{t-1})}{\sigma_Z^2+\sigma_W^2}\Big).
\end{equation}

\label{eq:awgn_equal} I\big(s(X_t);Y_t\mid\mathcal F_{t-1}\big)\ =\ \tfrac12\log \Big(1+\tfrac{\Var(s(X_t)\mid\mathcal F_{t-1})}{\sigma_Z^2+\sigma_W^2}\Big).

\begin{equation}\label{eq:St}
\E_0[S_t\mid\mathcal F_{t-1}]=0,\qquad
\E_0\!\big[\exp\{\eta S_t\}\mid\mathcal F_{t-1}\big]\ \le\ \exp\!\big\{\tfrac12 \eta^2 v\big\}\quad(\forall\,\eta\in\mathbb R),
\end{equation}

\label{eq:St} \E_0[S_t\mid\mathcal F_{t-1}]=0,\qquad \E_0 \big[\exp\{\eta S_t\}\mid\mathcal F_{t-1}\big]\ \le\ \exp \big\{\tfrac12 \eta^2 v\big\}\quad(\forall \eta\in\mathbb R),

\begin{equation}\label{eq:eprocess_grid}
M_T\ :=\ \sum_{j=1}^J w_j \exp\!\left\{\eta_j\sum_{t=1}^T S_t-\frac{\eta_j^2 v\,T}{2}\right\},
\end{equation}

\label{eq:eprocess_grid} M_T\ :=\ \sum_{j=1}^J w_j \exp \left\{\eta_j\sum_{t=1}^T S_t-\frac{\eta_j^2 v T}{2}\right\},

\begin{equation}\label{eq:fw1}
\log\frac{1}{\alpha\beta}\ >\ \log\frac{1}{\underlinew}\ +\ c_{\text{grid}},
\qquad \underlinew:=\min_j w_j,\quad c_{\text{grid}}:=\log q,\ \ q>1.
\end{equation}

\begin{equation}\label{eq:gauss_finite_window_corrected}
\tau^2 \ \ge\ \max_{T\in\mathcal A}\left\{\frac{\Delta_{\min}^2\,T}{2\Big(\log\frac{1}{\alpha\beta}-\log\frac{1}{\underlinew}-c_{\text{grid}}\Big)}-\sigma_0^2\right\},
\end{equation}

\label{eq:gauss_finite_window_corrected} \tau^2 \ \ge\ \max_{T\in\mathcal A}\left\{\frac{\Delta_{\min}^2 T}{2\Big(\log\frac{1}{\alpha\beta}-\log\frac{1}{\underlinew}-c_{\text{grid}}\Big)}-\sigma_0^2\right\},

\[
\rho_m(X_t;Y_t\mid\mathcal F_{t-1})\ :=\ \sup_{f,g}\ \mathrm{Corr}\big(f(X_t),g(Y_t)\mid\mathcal F_{t-1}\big),
\]

\[
\liminf_{T\to\infty}\frac{1}{T}\sum_{t=1}^T
\mathbbm{1}\{\rho_m(X_t;Y_t\mid\mathcal F_{t-1})\ge \rho_0\}\ \ge\ \delta\quad\text{a.s.}
\]

I(X_t;Y_t\mid\mathcal F_{t-1})\ \ge\ \varepsilon_0\ :=\ -\tfrac12\log(1-\rho_0^2).

\[
\liminf_{T\to\infty}\frac{1}{T}\sum_{t=1}^T I(X_t;Y_t\mid\mathcal F_{t-1})
\ \ge\ \delta\,\varepsilon_0\quad\text{a.s.}
\]

\liminf_{T\to\infty}\frac{1}{T}\sum_{t=1}^T I(X_t;Y_t\mid\mathcal F_{t-1}) \ \ge\ \delta \varepsilon_0\quad\text{a.s.}

\[
I(X_t;Y_t\mid\mathcal F_{t-1})\ \ge\ I\big(s(X_t);Y_t\mid\mathcal F_{t-1}\big).
\]

\[
\lambda_{\mathrm{net}}\ :=\ \lambda_{\mathrm{prin}}-\bar\lambda_{\text{front}}.
\]

\[
\phi(\eta_j)\ \ge\ \phi(\eta^\star)\ -\ \log q,
\]

\[
\E_1\!\left[\eta \sum_{t\le T} S_t-\tfrac12\eta^2 v T\right]
= -\tfrac12\eta^2 v T+\eta \mu T
= \phi(\eta)\,T
\le \phi(\eta^\star)\,T = \frac{\mu^2 T}{2v}.
\]

\E_1 \left[\eta \sum_{t\le T} S_t-\tfrac12\eta^2 v T\right] = -\tfrac12\eta^2 v T+\eta \mu T = \phi(\eta) T \le \phi(\eta^\star) T = \frac{\mu^2 T}{2v}.

\[
\frac{\Delta_{\min}^2\,T}{2(\sigma_0^2+\tau^2)}\ \gtrsim\ \log\frac{1}{\alpha\beta}-\log\frac{1}{\underlinew}-c_{\text{grid}}
\]

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$\mathcal C\subseteq \mathcal P(\mathsf U)-\mathcal P(\mathsf U)$

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi mathvariant="script">C</mi><mo>⊆</mo><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="sans-serif">U</mi><mo stretchy="false">)</mo><mo>−</mo><mi mathvariant="script">P</mi><mo stretchy="false">(</mo><mi mathvariant="sans-serif">U</mi><mo stretchy="false">)</mo><mi>$</mi></mrow></math>

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><msub><mi>X</mi><mi>t</mi></msub><mspace width="negativethinmathspace" /><mo>→</mo><mspace width="negativethinmathspace" /><mi>s</mi><mo stretchy="false">(</mo><msub><mi>X</mi><mi>t</mi></msub><mo stretchy="false">)</mo><mspace width="negativethinmathspace" /><mo>→</mo><mspace width="negativethinmathspace" /><msub><mi>Y</mi><mi>t</mi></msub><mi>$</mi></mrow></math>

$T(r+s)\le T(r)+T(s)\circ \mathsf T^{\,r}$

$\mathcal T_t=\{x:\mathrm{dist}(x,\partial A_{t^-})\le w\}$

$\Prob_0(\sup_{T\in\mathcal A}M_T\ge 1/\alpha)\le \alpha$

$\liminf_T T^{-1}\sum_{t\le T}\CMI_t=\rho\varepsilon\to 0$


FRACTAL CATEGORY THEORY: Scale as a Frobenius (Co)Monad and Ind–Pro Bicompletion with Stable Equivariant Kan Extensions

same

colorlinks=true,
linkcolor=blue!60!black,
citecolor=blue!60!black,
urlcolor=blue!60!black,
pdftitle= Fractal Category Theory: Scale as a Frobenius (Co)Monad and Ind--Pro Bicompletion with Stable Equivariant Kan Extensions,
pdfauthor= K. Takahashi ,
pdfsubject= Category Theory, Enriched Category Theory, Ind/Pro, Frobenius Monads, Kan Extensions ,
pdfkeywords= Frobenius monad, comonad, Ind-completion, Pro-completion, Kan extension, Day convolution, ambifixpoint, Lawvere metric, filtered colimit, cofiltered limit

%
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_T Theta\_T (T) %

TITLE: Fractal Category Theory:\
Scale as a Frobenius (Co)Monad and Ind--Pro Bicompletion with Stable Equivariant Kan Extensions

AUTHOR: K.~Takahashi

[[EQ:eq0003]]

Finite limits in [[EQ:eq0016]] (hence in [[EQ:eq0017]] ) are computed pointwise, and filtered colimits commute with finite limits in [[EQ:eq0018]] .
Since [[EQ:eq0019]] preserves finite limits when [[EQ:eq0020]] does and [[EQ:eq0021]] is representable, the filtered colimit above preserves finite limits.
Thus [[EQ:eq0022]] preserves finite limits.

[Universe management]
We fix a universe [[EQ:eq0023]] for [[EQ:eq0024]] . Ind- and Pro-completions are formed inside a larger universe [[EQ:eq0025]] , and presheaves [[EQ:eq0026]] with [[EQ:eq0027]] so that all coends used in [[EQ:eq0028]] and Day convolution exist.

[Lift of [[EQ:eq0029]] ] def:That
Define the lifted endofunctor on [[EQ:eq0030]] by

[[EQ:eq0004]]

By density of [[EQ:eq0031]] , there are canonical isomorphisms [[EQ:eq0032]] . On [[EQ:eq0033]] we act levelwise: for a pro-object [[EQ:eq0034]] (small cofiltered),
[math] T(X)\ :=\ \ T(X_i)\ _ i I [/math],
which extends [[EQ:eq0035]] along [[EQ:eq0036]] .

[Accessibility and finite-limit preservation]ass:access-strong
[[EQ:eq0037]] is [[EQ:eq0038]] -accessible, preserves [[EQ:eq0039]] -filtered colimits and finite limits. Therefore, by Lemma~lem:ind-created together with Remark~rmk:pointwise-finite, [[EQ:eq0040]] preserves finite limits and (by construction) [[EQ:eq0041]] -filtered colimits. In particular, we may consider its levelwise extension [[EQ:eq0042]] on pro-objects (well-definedness is general; cf.\ Lemma~lem:pro-welldef).

[Finite limits are created by the presheaf embedding]lem:ind-created
The inclusion [[EQ:eq0043]] creates finite limits. In particular, if [[EQ:eq0044]] preserves finite limits and [[EQ:eq0045]] is dense, then [[EQ:eq0046]] preserves finite limits. (See Ad\'amek--Rosick\'y~AdamekRosicky, Prop.~1.54 \& Thm.~1.58.)

[Proof sketch for finite-limit preservation of [[EQ:eq0047]] ]
For each [[EQ:eq0048]] , the functor [[EQ:eq0049]] preserves finite limits because [[EQ:eq0050]] does and [[EQ:eq0051]] is representable.
For any [[EQ:eq0052]] we have the pointwise formula [[EQ:eq0053]] with a filtered index by Lemma~lem:J-comma-filtered.
In [[EQ:eq0054]] , filtered colimits commute with finite limits; and the inclusion [[EQ:eq0055]] creates finite limits.
Combining these three facts yields that [[EQ:eq0056]] preserves finite limits.

[Accessibility is inherited by [[EQ:eq0057]] ]lem:access-hat
If [[EQ:eq0058]] is [[EQ:eq0059]] -accessible and [[EQ:eq0060]] is dense with [[EQ:eq0061]] locally [[EQ:eq0062]] -presentable, then [[EQ:eq0063]] is [[EQ:eq0064]] -accessible and preserves the same class of [[EQ:eq0065]] -filtered colimits as [[EQ:eq0066]] ; with Lemma~lem:ind-created it also preserves finite limits.

[Lifted Frobenius structure]lem:lift-frob
If [[EQ:eq0067]] carries monad/comonad data [[EQ:eq0068]] and [[EQ:eq0069]] in Frobenius compatibility on [[EQ:eq0070]] , then there exist unique natural transformations
[math] , , , [/math]
making [[EQ:eq0071]] a Frobenius (co)monad on [[EQ:eq0072]] and levelwise on [[EQ:eq0073]] . Moreover [[EQ:eq0074]] commutes with [[EQ:eq0075]] (e.g.\ [[EQ:eq0076]] ).

[Conservativity of precomposition with a dense functor]
Since [[EQ:eq0077]] and all structure 2-cells are obtained via [[EQ:eq0078]] , they preserve the colimits exhibiting density of [[EQ:eq0079]] . Hence precomposition with [[EQ:eq0080]] is conservative for the natural transformations we consider: if two cocontinuous natural transformations [[EQ:eq0081]] agree after whiskering with [[EQ:eq0082]] , then [[EQ:eq0083]] .

[Conservativity via density]lem:conservative-J
Let [[EQ:eq0084]] preserve [[EQ:eq0085]] -filtered colimits. If natural transformations [[EQ:eq0086]] satisfy [[EQ:eq0087]] , then [[EQ:eq0088]] , since values on arbitrary [[EQ:eq0089]] are reconstructed as [[EQ:eq0090]] -filtered colimits of values on [[EQ:eq0091]] .

Every [[EQ:eq0092]] is a [[EQ:eq0093]] -filtered colimit of [[EQ:eq0094]] ; [[EQ:eq0095]] preserve these colimits, so [[EQ:eq0096]] is the colimit of [[EQ:eq0097]] .

[Well-definedness of levelwise extension]lem:pro-welldef
For any functor [[EQ:eq0098]] and pro-objects [[EQ:eq0099]] , [[EQ:eq0100]] ,

[[EQ:eq0005]]

Define [[EQ:eq0101]] on objects and act levelwise on morphisms via the above formula.
Then [[EQ:eq0102]] is well-defined (independent of representatives) and functorial.

[Commutation and Fubini of (bi)limits]ass:fubini
Assume: (i) [[EQ:eq0103]] preserves the relevant [[EQ:eq0104]] -filtered colimits and finite limits in [[EQ:eq0105]] ;
(ii) [[EQ:eq0106]] is defined levelwise and preserves the small cofiltered limits in [[EQ:eq0107]] , which are generated from reindexing and equalizers;
(iii) for any small filtered (equivalently, [[EQ:eq0108]] -filtered for the fixed regular [[EQ:eq0109]] ) diagram landing in the constant subcategory [[EQ:eq0110]] , its filtered colimit in [[EQ:eq0111]] exists and is created by [[EQ:eq0112]] .
Under these standing conditions, the mixed (bi)limits used in Def.~def:frac satisfy the required Fubini rule.

[Fubini for mixed shapes]prop:fubini-mixed
Assume [[EQ:eq0113]] preserves [[EQ:eq0114]] -filtered colimits and finite limits in [[EQ:eq0115]] , and
[[EQ:eq0116]] is defined levelwise on [[EQ:eq0117]] , preserving small cofiltered limits computed by
reindexing/equalizers. Consider mixed diagrams obtained by first forming a [[EQ:eq0118]] -filtered colimit
of a diagram landing in [[EQ:eq0119]] and then taking a small cofiltered limit levelwise.
Then: (i) the filtered colimit is created by [[EQ:eq0120]] (hence exists in [[EQ:eq0121]] ); (ii) filtered colimits and the subsequent levelwise cofiltered limits commute under [[EQ:eq0122]] .

[Filtered colimits of constant pro-objects]lem:const-pro-colim
Whenever a [[EQ:eq0123]] -filtered colimit of a diagram landing in the constant subcategory [[EQ:eq0124]] exists in [[EQ:eq0125]] , it is again constant and created by [[EQ:eq0126]] from the corresponding colimit in [[EQ:eq0127]] .

[Sketch]
Using the Hom-formula [[EQ:eq0128]] , for a constant target [[EQ:eq0129]] the outer limit is trivial and we get [[EQ:eq0130]] , which is created by [[EQ:eq0131]] from [[EQ:eq0132]] . See also the standard calculus of pro-morphisms (e.g.\ Isaksen~Isaksen01).

SUBSECTION: Lifted towers and mixed indexing

[Lifted Ind/Pro towers]def:towers
The Ind tower in [[EQ:eq0133]] is the filtered diagram generated by [[EQ:eq0134]] :

[[EQ:eq0006]]

The Pro tower in [[EQ:eq0135]] is the cofiltered diagram generated levelwise by [[EQ:eq0136]] via [[EQ:eq0137]] .

[Meaning of [[EQ:eq0138]] and [[EQ:eq0139]] ]
[[EQ:eq0140]] denotes the replete full subcategory of [[EQ:eq0141]] generated under [[EQ:eq0142]] -filtered colimits by the essential images of [[EQ:eq0143]] for all [[EQ:eq0144]] . Dually, [[EQ:eq0145]] denotes the replete full subcategory of [[EQ:eq0146]] generated under small cofiltered limits by the essential images of [[EQ:eq0147]] .

[Mixed indexing category [[EQ:eq0148]] ]def:theta
[[EQ:eq0149]] is the small category generated by:
[leftmargin=1.2em]
- Objects: pairs [[EQ:eq0150]] with [[EQ:eq0151]] and [[EQ:eq0152]] .
- Generating morphisms (for all [[EQ:eq0153]] and [[EQ:eq0154]] in [[EQ:eq0155]] ):

[[EQ:eq0002]]

where [[EQ:eq0156]] are the whiskerings of [[EQ:eq0157]] by [[EQ:eq0158]] .
- Relations: functoriality of [[EQ:eq0159]] , naturality of [[EQ:eq0160]] , the monad axioms for [[EQ:eq0161]] , the comonad axioms for [[EQ:eq0162]] , and the Frobenius equalities [[EQ:eq0163]] for all [[EQ:eq0164]] .

Interpretation. In [[EQ:eq0165]] we read [[EQ:eq0166]] as [[EQ:eq0167]] , and [[EQ:eq0168]] as the corresponding components of [[EQ:eq0169]] (levelwise via [[EQ:eq0170]] ).

[The [[EQ:eq0171]] -subpresentation]
Let [[EQ:eq0172]] be the wide subcategory generated by [[EQ:eq0173]] .

[Length filtration and rules]rmk:length
Let [[EQ:eq0174]] be the unique map with
[[EQ:eq0175]] , [[EQ:eq0176]] , and [[EQ:eq0177]] . We filter presentations by [[EQ:eq0178]] .

[Rewriting, local confluence, and Newman's lemma]
Push all [[EQ:eq0179]] -steps to the left and all [[EQ:eq0180]] -steps to the right using the monad/comonad triangles and Frobenius squares. This is terminating by a lexicographic measure [[EQ:eq0181]] and locally confluent because all critical pairs (unit/counit overlaps, Frobenius squares, mixed [[EQ:eq0182]] and [[EQ:eq0183]] overlaps, as well as intersections with [[EQ:eq0184]] via naturality) are resolved by the axioms; hence confluent by Newman’s lemma. Normal forms lie in [[EQ:eq0185]] .

[Addendum on local confluence]
Typical critical pairs contract as (string-diagrammatically):

[[EQ:eq0007]]

by triangle axioms and naturality; Frobenius squares resolve overlaps of [[EQ:eq0186]] uniformly after one [[EQ:eq0187]] (resp.\ [[EQ:eq0188]] ) step. Hence local confluence holds; termination is by [[EQ:eq0189]] .

[Finality of the [[EQ:eq0190]] -subpresentation]lem:final
The inclusion [[EQ:eq0191]] is final.
Proof idea. Using the rewriting above, termination and local confluence are verified by resolving all critical pairs via
the monad/comonad triangles, naturality, and the Frobenius squares. For each object [[EQ:eq0192]] of [[EQ:eq0193]] , the over-category [[EQ:eq0194]] is nonempty and connected (normal forms lie in [[EQ:eq0195]] and rewrites produce zigzags), hence Quillen's Theorem~A yields finality.

[A representative critical pair]
The overlap [[EQ:eq0196]] is resolved by the Frobenius square; explicitly

[[EQ:eq0008]]

commutes, and naturality takes care of whiskerings by [[EQ:eq0197]] .

[Role of [[EQ:eq0198]] ]rmk:mu-delta-role
The generators [[EQ:eq0199]] are indispensable for proving finality (Newman/Quillen--A) since the resolution
of overlaps [[EQ:eq0200]] , [[EQ:eq0201]] , and [[EQ:eq0202]] uses Frobenius and triangle laws.
Once finality is established, actual (bi)limit computations may be carried out entirely on
[[EQ:eq0203]] ; in this sense [[EQ:eq0204]] do not appear in the formulas, although they
are used in the proof of finality.

[Replete hull]
For a full subcategory [[EQ:eq0205]] , its replete hull [[EQ:eq0206]] is the full subcategory of [[EQ:eq0207]] on all objects isomorphic in [[EQ:eq0208]] to some object of [[EQ:eq0209]] .

[Fractal bicompletion]def:frac
[[EQ:eq0210]] is the smallest replete full subcategory of [[EQ:eq0211]] that
[leftmargin=1.2em]
- contains [[EQ:eq0212]] ;
- is closed under the images of [[EQ:eq0213]] and under forming [[EQ:eq0214]] -filtered colimits and small cofiltered limits along diagrams [[EQ:eq0215]] whose vertices lie in [[EQ:eq0216]] .

[Order of mixed (bi)limits]
Throughout, mixed constructions are taken in the order ``filtered colimit first, then cofiltered limit (levelwise in [[EQ:eq0217]] )''. Hence finality of [[EQ:eq0218]] suffices to compute colimits; limits are taken afterwards levelwise and do not require an initiality argument.

[Identification with the intersection]prop:intersection
Under Assumptions~ass:lp, ass:access-strong, ass:fubini, the inclusions yield an equivalence of replete full subcategories

[[EQ:eq0009]]

( [[EQ:eq0219]] ) Any [[EQ:eq0220]] is a filtered colimit of [[EQ:eq0221]] followed by a small cofiltered limit; preservation by [[EQ:eq0222]] and Assumption~ass:fubini imply [[EQ:eq0223]] lies in both closures, hence in the replete intersection.

( [[EQ:eq0224]] ) Present [[EQ:eq0225]] as a filtered colimit of [[EQ:eq0226]] and as a cofiltered limit of [[EQ:eq0227]] . Form the mixed diagram [[EQ:eq0228]] , with arrows from [[EQ:eq0229]] via [[EQ:eq0230]] , from [[EQ:eq0231]] via [[EQ:eq0232]] , and from unit/counit. Fubini and preservation yield [[EQ:eq0233]] as its (bi)limit; thus [[EQ:eq0234]] . Finality (Lemma~lem:final) ensures independence. Note. The proof uses only unit/counit steps thanks to the finality of
[[EQ:eq0235]] for the colimit stage;
limits are taken levelwise in [[EQ:eq0236]] under Proposition~prop:fubini-mixed.
No explicit occurrence of [[EQ:eq0237]] is needed in the (bi)limit formulas.

SUBSECTION: Equivariant Kan extensions (computed on [[EQ:eq0238]] -subpresentations)

[Seed stability for equivariance]ass:Tstable-seed
For this subsection, [[EQ:eq0239]] restricts to an endofunctor [[EQ:eq0240]] (on objects and morphisms).

[Iso-coherence]ass:iso-alpha
The coherence [[EQ:eq0241]] is an enriched natural isomorphism (both directions [[EQ:eq0242]] -Lipschitz).

The construction of [[EQ:eq0243]] itself works for non-invertible (but nonexpansive) [[EQ:eq0244]] ; the isomorphism hypothesis is used later to state stability in a symmetric (difference-based) form.

[Existence of (co)limits and cocontinuity of [[EQ:eq0245]] ]ass:S-cocont
The category [[EQ:eq0246]] admits the (weighted) colimits/limits indexed by the (restricted) mixed presentations used to compute [[EQ:eq0247]] (resp.\ [[EQ:eq0248]] ),
and [[EQ:eq0249]] preserves them.

[Equivariant left Kan extension]def:equivlan
Let [[EQ:eq0250]] , [[EQ:eq0251]] , and [[EQ:eq0252]] . A [[EQ:eq0253]] -equivariant left Kan extension of [[EQ:eq0254]] along [[EQ:eq0255]] is a pair [[EQ:eq0256]] with [[EQ:eq0257]] and [[EQ:eq0258]] such that [[EQ:eq0259]] , [[EQ:eq0260]] , and universal for this property.

PARAGRAPH: Computing on the [[EQ:eq0261]] -subpresentation.

By Lemma~lem:final, the inclusion [[EQ:eq0262]] is final. Hence we compute [[EQ:eq0263]] over [[EQ:eq0264]] ; no images of [[EQ:eq0265]] are needed.

[Existence and universality of [[EQ:eq0266]] ]thm:existence
Under Assumptions~ass:lp, ass:access-strong, ass:fubini, ass:Tstable-seed, ass:iso-alpha, ass:S-cocont, for any [[EQ:eq0267]] and [[EQ:eq0268]] there exists a unique (up to unique iso) [[EQ:eq0269]] -equivariant left Kan extension [[EQ:eq0270]] along [[EQ:eq0271]] . Moreover, for each [[EQ:eq0272]] , [[EQ:eq0273]] is computed as a weighted colimit over any mixed presentation [[EQ:eq0274]] but equivalently over its restriction to [[EQ:eq0275]] by Lemma~lem:final. No images of [[EQ:eq0276]] are required in the construction.

SECTION: Enriched setting, weighted fLan, and ambifixpoints

sec:ambifix

Let [[EQ:eq0277]] be a complete quantale. A [[EQ:eq0278]] -category [[EQ:eq0279]] has hom-objects [[EQ:eq0280]] ; a [[EQ:eq0281]] -functor is nonexpansive on homs.

[CMS baseline]ass:enriched
We henceforth take [[EQ:eq0282]] (complete 1-bounded metric spaces and nonexpansive maps) and regard it as [[EQ:eq0283]] -enriched. The endofunctor [[EQ:eq0284]] is uniformly contractive on hom-metrics, i.e.\ there exists [[EQ:eq0285]] such that for all [[EQ:eq0286]] and [[EQ:eq0287]] one has

[[EQ:eq0010]]

In addition, [[EQ:eq0288]] is [[EQ:eq0289]] -contractive on object metrics (equivalently, locally contractive in the sense of America--Rutten), i.e.\ [[EQ:eq0290]] for all objects [[EQ:eq0291]] .

PARAGRAPH: Convention (weighted enriched Kan).

In the enriched branch we use the [[EQ:eq0292]] -weighted enriched left Kan extension
[[EQ:eq0293]] , where the external weight places [[EQ:eq0294]] on the length- [[EQ:eq0295]] layer of
[[EQ:eq0296]] . All stability bounds refer to [[EQ:eq0297]] .

[Ambifixpoint and Frobenius diagrams]def:ambifix
An object [[EQ:eq0298]] is an ambifixpoint if it is a [[EQ:eq0299]] -algebra [[EQ:eq0300]] and a [[EQ:eq0301]] -coalgebra [[EQ:eq0302]] satisfying

[[EQ:eq0011]]

(in 2-/ [[EQ:eq0303]] -levels: up to specified 2-cells/homotopies).

[ [[EQ:eq0304]] ]def:Fixpm
Let [[EQ:eq0305]] be the full subcategory of [[EQ:eq0306]] on those algebras [[EQ:eq0307]] that admit a coalgebra [[EQ:eq0308]] with the Frobenius squares commuting; dually [[EQ:eq0309]] inside [[EQ:eq0310]] . Set [[EQ:eq0311]] (in [[EQ:eq0312]] on carriers). Morphisms are maps in [[EQ:eq0313]] that are simultaneously [[EQ:eq0314]] -algebra and [[EQ:eq0315]] -coalgebra morphisms.

[Frobenius coherence on a common carrier]lem:ambid-precise
For a Frobenius (co)monad [[EQ:eq0316]] , the ambidextrous structure ensures that whenever an initial [[EQ:eq0317]] -algebra and a terminal [[EQ:eq0318]] -coalgebra exist on the same carrier [[EQ:eq0319]] , the induced Frobenius squares on [[EQ:eq0320]] commute (see Street~StreetFrob04 for ambidexterity).

[Contractive endofunctors yield algebraic compactness]lem:alg-compact
Let [[EQ:eq0321]] be a complete Lawvere metric-enriched category and let [[EQ:eq0322]] be uniformly contractive in the sense of Assumption~ass:enriched. Then there exists [[EQ:eq0323]] with [[EQ:eq0324]] , [[EQ:eq0325]] such that [[EQ:eq0326]] is initial in [[EQ:eq0327]] and [[EQ:eq0328]] is terminal in [[EQ:eq0329]] . Moreover [[EQ:eq0330]] and [[EQ:eq0331]] are mutually inverse in the enriched sense (distance [[EQ:eq0332]] ), hence define an isometric isomorphism.

Consider [[EQ:eq0333]] on [[EQ:eq0334]] ; it is [[EQ:eq0335]] -contractive, hence has a unique fixed point [[EQ:eq0336]] with [[EQ:eq0337]] (America--Rutten~AmericaRutten89; Birkedal--Mossakowski--Uustalu~Birkedal10). For any algebra [[EQ:eq0338]] define [[EQ:eq0339]] by [[EQ:eq0340]] . Then [[EQ:eq0341]] is [[EQ:eq0342]] -contractive, so admits a unique fixed point [[EQ:eq0343]] with [[EQ:eq0344]] ; uniqueness gives initiality. Dually for coalgebras. The fixed point equations for [[EQ:eq0345]] imply [[EQ:eq0346]] and [[EQ:eq0347]] in the enriched sense by uniqueness of solutions to the mutually adjoint recurrences (see the cited works for formal statements).

[Frobenius on the compact carrier]lem:frob-diagrams
Assume [[EQ:eq0348]] is a Frobenius (co)monad. If [[EQ:eq0349]] is initial in [[EQ:eq0350]] and [[EQ:eq0351]] is terminal in [[EQ:eq0352]] on the same carrier, then the Frobenius squares in Definition~def:ambifix commute.

[ [[EQ:eq0353]] -density via [[EQ:eq0354]] ]def:Tdense-corrected
Let [[EQ:eq0355]] be as in Definition~def:restricted-yoneda.
A subcategory [[EQ:eq0356]] is [[EQ:eq0357]] -dense for [[EQ:eq0358]] if [[EQ:eq0359]] is a retract in [[EQ:eq0360]] of a [[EQ:eq0361]] -weighted colimit of a small diagram whose vertices lie in [[EQ:eq0362]] and whose arrows are generated by the unit/counit and [[EQ:eq0363]] -arrows; moreover such colimits are preserved by [[EQ:eq0364]] (equivalently, by [[EQ:eq0365]] levelwise).

[A handy sufficient condition for [[EQ:eq0366]] -density]prop:Tdense-sufficient
Suppose [[EQ:eq0367]] is dense via [[EQ:eq0368]] and there exists a small diagram
[[EQ:eq0369]] with vertices in [[EQ:eq0370]] and arrows generated by unit/counit and [[EQ:eq0371]] -maps
such that [[EQ:eq0372]] is a retract of [[EQ:eq0373]] .
If [[EQ:eq0374]] preserves the corresponding weighted colimit, then [[EQ:eq0375]] is [[EQ:eq0376]] -dense for [[EQ:eq0377]] .

[Ambifixpoint theorem]thm:ambifix
Under Assumption~ass:enriched, [[EQ:eq0378]] is algebraically compact: the initial [[EQ:eq0379]] -algebra and the terminal [[EQ:eq0380]] -coalgebra exist and have the same carrier [[EQ:eq0381]] . If [[EQ:eq0382]] is a Frobenius (co)monad, then by Lemma~lem:frob-diagrams [[EQ:eq0383]] . If [[EQ:eq0384]] is [[EQ:eq0385]] -dense for [[EQ:eq0386]] , then

[[EQ:eq0012]]

the smallest replete full subcategory of [[EQ:eq0387]] containing [[EQ:eq0388]] and closed under the (co)limits preserved by [[EQ:eq0389]] (hence by [[EQ:eq0390]] levelwise); and, when a monoidal structure is present, under the monoidal constructions preserved by [[EQ:eq0391]] (hence by [[EQ:eq0392]] levelwise).

SECTION: Scale sites, presheaves, and stable equivariant Kan extensions

sec:presheaves

SUBSECTION: Uniform scale topology (safe version)

[Iterated site morphisms]ass:site
Fix a Grothendieck topology [[EQ:eq0393]] on [[EQ:eq0394]] . For every [[EQ:eq0395]] , [[EQ:eq0396]] is a site morphism (left exact and [[EQ:eq0397]] -cover-preserving).

[Uniform scale topology]def:JT-safe
Define [[EQ:eq0398]] by [[EQ:eq0399]] ; i.e.\ [[EQ:eq0400]] is a [[EQ:eq0401]] -cover iff for all [[EQ:eq0402]] , [[EQ:eq0403]] is a [[EQ:eq0404]] -cover.

Under Assumption~ass:site, [[EQ:eq0405]] is a Grothendieck topology.

Isomorphism and pullback stability hold since each [[EQ:eq0406]] is left exact and cover-preserving. For transitivity, intersections of sieves that are [[EQ:eq0407]] -covers for all [[EQ:eq0408]] remain so. More generally, intersections of Grothendieck topologies are Grothendieck topologies, and [[EQ:eq0409]] is precisely such an intersection.

SUBSECTION: Precomposition, Day convolution, and Yoneda comparison

[Universe convention]
For large [[EQ:eq0410]] we implicitly enlarge the universe and work in [[EQ:eq0411]] so that the Yoneda embedding [[EQ:eq0412]] is defined and coends exist.

Define [[EQ:eq0413]] . By density of [[EQ:eq0414]] , there is a canonical isomorphism [[EQ:eq0415]] .
Precomposition [[EQ:eq0416]] is also available; when [[EQ:eq0417]] is (symmetric) monoidal and [[EQ:eq0418]] is strong monoidal, [[EQ:eq0419]] (and likewise [[EQ:eq0420]] ) carries a natural (op)lax monoidal structure for Day convolution.

[Lift via [[EQ:eq0421]] and monoidality; lax-to-strong]prop:day-lax
Let [[EQ:eq0422]] on [[EQ:eq0423]] . Then [[EQ:eq0424]] is (op)lax monoidal for Day convolution.
If, moreover, [[EQ:eq0425]] is strong monoidal and the coend Fubini conditions required for Day convolution interchange with [[EQ:eq0426]] hold,
then [[EQ:eq0427]] is strong monoidal and the comparison [[EQ:eq0428]] is monoidal.Coend--Kan Fubini and strong monoidality of [[EQ:eq0429]] justify interchanging [[EQ:eq0430]] with the Day coends. A sufficient concrete regime: [[EQ:eq0431]] is small (or we work in an enlarged universe), and [[EQ:eq0432]] preserves the colimits/limits forming the convolution coends; otherwise only the (op)lax statement is guaranteed. Cf.\ Day~ Day70, Glasman~Glasman16, Lurie~LurieHA.

SUBSECTION: Lipschitz structure and stability bounds (difference form)

Fix a [[EQ:eq0433]] -category [[EQ:eq0434]] . A functional is a [[EQ:eq0435]] -functor [[EQ:eq0436]] . Define

[[EQ:eq0013]]

(Lawvere case: [[EQ:eq0437]] with the [[EQ:eq0438]] -order).

[Size of the join]
We tacitly take the join over a small dense subcategory of [[EQ:eq0439]] -presentable objects generating [[EQ:eq0440]] ; the value agrees with the class-sized join by cocontinuity.

[Lipschitz data (difference form)]ass:lip-diff
[[EQ:eq0441]] is [[EQ:eq0442]] -contractive (with [[EQ:eq0443]] ); [[EQ:eq0444]] and [[EQ:eq0445]] are nonexpansive; seeds [[EQ:eq0446]] satisfy [[EQ:eq0447]] .
We use the length functor [[EQ:eq0448]] and an external weight [[EQ:eq0449]] on the length- [[EQ:eq0450]] layer to form the enriched weighted colimit defining [[EQ:eq0451]] .
When [[EQ:eq0452]] the geometric decay in Theorem~thm:stable-diff-corrected applies; when [[EQ:eq0453]] we retain nonexpansive stability without a geometric rate.

[Enriched colimits are 1-Lipschitz under external weights]
In a Lawvere metric setting, coends, coproducts, and filtered colimits are [[EQ:eq0454]] -Lipschitz with respect to the sup-metric,
and external scalar weights [[EQ:eq0455]] multiply layerwise distances by [[EQ:eq0456]] (cf.\ Kelly~Kelly82).

[Layerwise decay]lem:layer-decay
Every morphism in [[EQ:eq0457]] of length [[EQ:eq0458]] factors as a composite using at most [[EQ:eq0459]] occurrences of [[EQ:eq0460]] or [[EQ:eq0461]] ; in particular, along any such factorization the value is affected by at most [[EQ:eq0462]] forward applications of [[EQ:eq0463]] (coming from [[EQ:eq0464]] ), while [[EQ:eq0465]] does not introduce additional applications of [[EQ:eq0466]] . Therefore the induced map on values is [[EQ:eq0467]] -Lipschitz. In particular [[EQ:eq0468]] .
If [[EQ:eq0469]] , this specializes to [[EQ:eq0470]] .
Proof (sketch). Induction on [[EQ:eq0471]] . For [[EQ:eq0472]] there is no [[EQ:eq0473]] -application. For [[EQ:eq0474]] , any generator increases length by [[EQ:eq0475]] and contributes at most one [[EQ:eq0476]] via [[EQ:eq0477]] (or removes one via [[EQ:eq0478]] on the opposite side), and composition adds lengths. Normal forms lie in [[EQ:eq0479]] , so no [[EQ:eq0480]] occur.

[Extension Lipschitz bound]lem:lip-diff
(i) If [[EQ:eq0481]] and [[EQ:eq0482]] , [[EQ:eq0483]] , then
[math] \|F-G\|\ \ L 1-q . [/math]
(ii) If [[EQ:eq0484]] , then [[EQ:eq0485]] (nonexpansive stability; no geometric rate).

[Depth- [[EQ:eq0486]] truncation]def:trunc
Fix for each [[EQ:eq0487]] a canonical mixed presentation [[EQ:eq0488]] filtered by length. Define [[EQ:eq0489]] as the weighted colimit computing [[EQ:eq0490]] but taken over the full subcategory spanned by vertices of length [[EQ:eq0491]] . By Lemma~lem:final, [[EQ:eq0492]] is well defined up to unique isomorphism.

[Stable truncation; corrected]thm:stable-diff-corrected
Assume [[EQ:eq0493]] . With the weight [[EQ:eq0494]] , the truncation error satisfies

[[EQ:eq0014]]

Hence any conservative choice
[math] k\ \ _ 1/q \! (L (1-q) ) [/math]
guarantees [[EQ:eq0495]] .
(Equivalently, writing [[EQ:eq0496]] yields a slightly looser but simpler estimate.)

SECTION: Distributive laws with dynamics and observation

sec:dist

We continue to work with the endofunctor [[EQ:eq0497]] and the coherence [[EQ:eq0498]] from §sec:presheaves.
Let [[EQ:eq0499]] and [[EQ:eq0500]] be endofunctors. Assume distributive laws [[EQ:eq0501]] and [[EQ:eq0502]] (Beck coherence in 1-/2-/ [[EQ:eq0503]] variants), together with comparison cells on seeds [[EQ:eq0504]] and [[EQ:eq0505]] (nonexpansive).

Transport along [[EQ:eq0506]] yields natural transformations [[EQ:eq0507]] and [[EQ:eq0508]] on generators; compatibility with [[EQ:eq0509]] follows from the mate calculus for [[EQ:eq0510]] and Beck squares:

[[EQ:eq0015]]

and similarly for [[EQ:eq0511]] using [[EQ:eq0512]] ; [[EQ:eq0513]] -compatibilities are irrelevant as [[EQ:eq0514]] is computed on [[EQ:eq0515]] (by Lemma~lem:final).

[ [[EQ:eq0516]] -monotone functional]
A functional [[EQ:eq0517]] is [[EQ:eq0518]] -monotone if there exists a natural transformation
[math] beta:\ F T F [/math]
whose components are nonexpansive (Lawvere case: [[EQ:eq0519]] -Lipschitz). If [[EQ:eq0520]] is an iso, [[EQ:eq0521]] is [[EQ:eq0522]] -invariant.

[Equivariance under distributive laws]prop:dist
Under the data above, the [[EQ:eq0523]] -equivariant Kan extension [[EQ:eq0524]] admits coherences [[EQ:eq0525]] and [[EQ:eq0526]] compatible with the relations of [[EQ:eq0527]] (by Lemma~lem:final). Consequently, any [[EQ:eq0528]] -monotone [[EQ:eq0529]] -natural seed functional extends to a global one on [[EQ:eq0530]] ; the stagewise construction preserves nonexpansiveness of the witnessing [[EQ:eq0531]] .

SECTION: Examples

PARAGRAPH: Substitution systems.

[[EQ:eq0532]] : finite category of letters and local rules; [[EQ:eq0533]] applies a primitive substitution. In a metric-enriched setting with distance penalizing mismatches at scale [[EQ:eq0534]] by [[EQ:eq0535]] (rescaled to be [[EQ:eq0536]] -bounded via the factor [[EQ:eq0537]] if desired), [[EQ:eq0538]] is [[EQ:eq0539]] -contractive; fixed-point words/tilings yield [[EQ:eq0540]] .

PARAGRAPH: Iterated function systems (IFS) via hyperspaces (clarified).

We separate the functorial scale and the IFS dynamics.
Let [[EQ:eq0541]] and [[EQ:eq0542]] be the hyperspace of nonempty compact subsets of [[EQ:eq0543]] with the Hausdorff metric. Define the functorial scale by [[EQ:eq0544]] and [[EQ:eq0545]] . For an IFS given by contractions [[EQ:eq0546]] with [[EQ:eq0547]] , the Hutchinson operator [[EQ:eq0548]] , [[EQ:eq0549]] , is [[EQ:eq0550]] -contractive; the unique fixed point [[EQ:eq0551]] is the attractor. Our framework applies with [[EQ:eq0552]] as [[EQ:eq0553]] to build towers, while dynamics proceed by [[EQ:eq0554]] ; Theorem~thm:stable-diff-corrected gives depth–error tradeoffs for functionals on [[EQ:eq0555]] .
Clarification. Here [[EQ:eq0556]] is not assumed contractive nor Frobenius; we use the [[EQ:eq0557]] -equivariant Kan-extension machinery, and contraction is carried by [[EQ:eq0558]] .
Therefore, the ambifixpoint results of :ambifix do not apply.
Moreover, the geometric error bound from Theorem~thm:stable-diff-corrected applies only if the contraction modulus required in Assumption~ass:lip-diff is available (e.g.\ if one replaces/augments the indexing by a [[EQ:eq0559]] -contractive surrogate [[EQ:eq0560]] or transports a [[EQ:eq0561]] via [[EQ:eq0562]] from the observation-side endofunctor [[EQ:eq0563]] ).
Absent such a modulus, we retain nonexpansive stability bounds but make no geometric-rate claim.

PARAGRAPH: Coarse-graining channels (corrected).

[[EQ:eq0564]] the category of stochastic maps or CPTP maps; [[EQ:eq0565]] aggregates sites/blocks (CPTP). Petz monotone metrics and Bures/Uhlmann distances satisfy DPI; hence Proposition~prop:dist yields [[EQ:eq0566]] -monotonicity of the extended functionals.
Geometric error bounds as in Theorem~thm:stable-diff-corrected additionally require a contraction modulus [[EQ:eq0567]] (supplied, for instance, by the observation-side endofunctor [[EQ:eq0568]] or by a spectral-gap assumption on the coarse-graining).
Without such a modulus we retain nonexpansive stability but do not claim a geometric rate.

SECTION: 2- and [[EQ:eq0569]]

(infty,1)-level comparison
sec:comparison

[Coherence for strictification/truncation]ass:coh
Bicategorical ambient admits bi-/pseudo-(co)limits used; pseudomonad/comonad data satisfy strictification hypotheses (e.g.\ Lack~Lack07). For [[EQ:eq0570]] : homotopy (co)limits exist; Frobenius data are coherent; 1-truncation preserves the universal properties up to equivalence.

[Bookkeeping of (co)limits at 2-/ [[EQ:eq0571]] -levels]
Bicompletion uses pseudo-(co)limits in the ambient bicategory;
[[EQ:eq0572]] / [[EQ:eq0573]] use weighted pseudo-colimits/limits;
the Day convolution lift uses coends;
truncation/strictification preserve these up to equivalence under Assumption~ass:coh.

[2 [[EQ:eq0574]] 1 strictification]
Under Assumption~ass:coh, bicategorical constructions (bicompletion, equivariant Kan extensions) strictify to the 1-categorical ones, preserving universal properties up to iso.

[ [[EQ:eq0575]] truncation]
Under Assumption~ass:coh, the [[EQ:eq0576]] -categorical constructions descend along [[EQ:eq0577]] , preserving universal properties up to canonical equivalence (equalities become isomorphisms).

SECTION: Appendix A: Confluence for [[EQ:eq0578]]

Theta\_T
app:confluence
Define a lexicographic measure
[[EQ:eq0579]] ,
which strictly decreases under the rewriting that moves all [[EQ:eq0580]] to the left and all [[EQ:eq0581]] to the right.
Termination follows from well-foundedness of [[EQ:eq0582]] .

pairs and resolutions (complete list).

1.15
@ > p 36mm > p 60mm > p 48mm @

Overlap & Rewrite steps & Law used \

[[EQ:eq0583]] & [[EQ:eq0584]] & Monad triangle \
[[EQ:eq0585]] & [[EQ:eq0586]] & Comonad triangle \
[[EQ:eq0587]] & [[EQ:eq0588]] & Frobenius square \
[[EQ:eq0589]] & [[EQ:eq0590]] parallel composites equal after one [[EQ:eq0591]] -step & Triangle + Frobenius \
[[EQ:eq0592]] & [[EQ:eq0593]] parallel composites equal after one [[EQ:eq0594]] -step & Triangle + Frobenius \
[[EQ:eq0595]] & [[EQ:eq0596]] & Naturality of [[EQ:eq0597]] \
[[EQ:eq0598]] & [[EQ:eq0599]] & Naturality of [[EQ:eq0600]] \
[[EQ:eq0601]] & [[EQ:eq0602]] & Naturality of [[EQ:eq0603]] \
[[EQ:eq0604]] & [[EQ:eq0605]] & Naturality of [[EQ:eq0606]] \

Local confluence holds by the table; by Newman's lemma, the system is confluent, and normal forms lie in [[EQ:eq0607]] . Quillen’s Theorem~A applies to obtain finality of [[EQ:eq0608]] .

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[[EQ:eq0001]]

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[[EQ:eq0497]]

[[EQ:eq0498]]

[[EQ:eq0499]]

[[EQ:eq0500]]

[[EQ:eq0501]]

[[EQ:eq0502]]

[[EQ:eq0503]]

[[EQ:eq0504]]

[[EQ:eq0505]]

[[EQ:eq0506]]

[[EQ:eq0507]]

[[EQ:eq0508]]

[[EQ:eq0509]]

[[EQ:eq0510]]

Motivation and contributions

Scale as a Frobenius (co)monad (1-/2-/(∞,1)-levels)

1-level (strict) definition

2-level and (∞,1)-level

Lifted towers and mixed indexing

Equivariant Kan extensions (computed on ηε-subpresentations)

Uniform scale topology (safe version)

Examples

2- and (∞,1)

Appendix A: Confluence for Θ_T

\begin{align*}
  T\mu\circ \delta T \;=\; \delta\circ\mu \;=\; \mu T\circ T\delta &\quad\text{(\emph{Frobenius})},\\
  \mu\circ T\eta \;=\; \mu\circ \eta T \;=\; \Id_T &\quad\text{(\emph{both unit axioms})},\\
  \varepsilon T\circ \delta \;=\; T\varepsilon\circ \delta \;=\; \Id_T &\quad\text{(\emph{both counit axioms})}.
\end{align*}

T\mu\circ \delta T  =  \delta\circ\mu  =  \mu T\circ T\delta &\quad\text{(\emph{Frobenius})},\\ \mu\circ T\eta  =  \mu\circ \eta T  =  \Id_T &\quad\text{(\emph{both unit axioms})},\\ \varepsilon T\circ \delta  =  T\varepsilon\circ \delta  =  \Id_T &\quad\text{(\emph{both counit axioms})}.

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mtable displaystyle="true" rowspacing="3pt" columnspacing="0em 2em"><mtr><mtd columnalign="right"><mi>T</mi><mi>μ</mi><mo>∘</mo><mi>δ</mi><mi>T</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>δ</mi><mo>∘</mo><mi>μ</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>μ</mi><mi>T</mi><mo>∘</mo><mi>T</mi><mi>δ</mi></mtd><mtd columnalign="left"><mi /><mspace width="1em" /><mtext>(\emph{Frobenius</mtext><mo stretchy="false">)</mo><mi>}</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right"><mi>μ</mi><mo>∘</mo><mi>T</mi><mi>η</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>μ</mi><mo>∘</mo><mi>η</mi><mi>T</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><msub><mi>\Id</mi><mi>T</mi></msub></mtd><mtd columnalign="left"><mi /><mspace width="1em" /><mtext>(\emph{both unit axioms</mtext><mo stretchy="false">)</mo><mi>}</mi><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right"><mi>ε</mi><mi>T</mi><mo>∘</mo><mi>δ</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><mi>T</mi><mi>ε</mi><mo>∘</mo><mi>δ</mi><mspace width="0.278em" /><mo>=</mo><mspace width="0.278em" /><msub><mi>\Id</mi><mi>T</mi></msub></mtd><mtd columnalign="left"><mi /><mspace width="1em" /><mtext>(\emph{both counit axioms</mtext><mo stretchy="false">)</mo><mi>}</mi><mo>.</mo></mtd></mtr></mtable></mrow></math>

\begin{align*}
&(n,C)\xrightarrow{T^n f}(n,C'),\quad
(n,C)\xrightarrow{\eta_n}(n{+}1,C),\quad
(n{+}1,C)\xrightarrow{\varepsilon_n}(n,C),\\
&(n{+}1,C)\xrightarrow{\delta_n}(n{+}2,C),\quad
(n{+}2,C)\xrightarrow{\mu_n}(n{+}1,C),
\end{align*}

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mtable displaystyle="true" rowspacing="3pt" columnspacing="0em 2em"><mtr><mtd columnalign="right" /><mtd columnalign="left"><mi /><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msup><mi>T</mi><mi>n</mi></msup><mi>f</mi></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><msup><mi>C</mi><mi>′</mi></msup><mo stretchy="false">)</mo><mo>,</mo><mspace width="1em" /><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>η</mi><mi>n</mi></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>n</mi><mrow><mo>+</mo></mrow><mn>1</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="1em" /><mo stretchy="false">(</mo><mi>n</mi><mrow><mo>+</mo></mrow><mn>1</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>ε</mi><mi>n</mi></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr><mtr><mtd columnalign="right" /><mtd columnalign="left"><mi /><mo stretchy="false">(</mo><mi>n</mi><mrow><mo>+</mo></mrow><mn>1</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>δ</mi><mi>n</mi></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>n</mi><mrow><mo>+</mo></mrow><mn>2</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="1em" /><mo stretchy="false">(</mo><mi>n</mi><mrow><mo>+</mo></mrow><mn>2</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mover><mstyle scriptlevel="0"><mo>→</mo></mstyle><mpadded width="+0.833em" lspace="0.556em" voffset="-.2em" height="-.2em"><mrow><msub><mi>μ</mi><mi>n</mi></msub></mrow><mspace depth=".25em" /></mpadded></mover><mo stretchy="false">(</mo><mi>n</mi><mrow><mo>+</mo></mrow><mn>1</mn><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>,</mo></mtd></mtr></mtable></mrow></math>

\[2pt]
\small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}}
\date{\today}

%==============================
% Theorem styles
%==============================
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{lemma}[theorem]{Lemma}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{assumption}[theorem]{Assumption}
\newtheorem{example}[theorem]{Example}
\newtheorem{remark}[theorem]{Remark}

%==============================
% Macros
%==============================
\newcommand{\C}{\mathcal{C}}
\newcommand{\Czero}{\mathcal{C}_0}
\newcommand{\D}{\mathcal{D}}
\newcommand{\E}{\mathcal{E}}
\newcommand{\V}{\mathbf{V}}
\newcommand{\Set}{\mathbf{Set}}
\newcommand{\Ind}{\mathrm{Ind}}
\newcommand{\Pro}{\mathrm{Pro}}
\newcommand{\Frac}{\mathrm{Frac}_T(\Czero)}
\newcommand{\Fixpm}{\mathrm{Fix}^{\pm}(T)}
\newcommand{\Id}{\mathrm{Id}}
\newcommand{\Ob}{\mathrm{Ob}}
\newcommand{\Hom}{\mathrm{Hom}}
\newcommand{\Lan}{\mathrm{Lan}}
\newcommand{\Ran}{\mathrm{Ran}}
\newcommand{\op}{\mathrm{op}}
\DeclareMathOperator*{\colim}{colim} % <-- fix for \colim
\DeclareMathOperator{\Fix}{Fix}      % <-- fix for \Fix

% Hyphenation to avoid awkward breaks
\hyphenation{bi-completion pseudo-monad pseudo-comonad
  cofiltered e-qui-variant ambi-fixpoint exten-sions}

\begin{document}
\setstretch{1.3}

\maketitle

\begin{abstract}
We present \emph{Fractal Category Theory} (FCT): scale is internalized as a single endofunctor $T\!:\C\to\C$ carrying monad and comonad structures in Frobenius compatibility. From a small seed $\Czero\subseteq\C$ we form a $T$-generated \emph{Ind--Pro bicompletion} $\Frac\subset\Pro(\Ind(\Czero))$ by closing under filtered colimits and cofiltered limits along mixed $T$-towers. Towers are built at the Ind-level via a restricted-Yoneda lift $\widehat T:=\Lan_{J}(J\!\circ T)$; the Frobenius data lift canonically to $\widehat T$ and to its Pro-level extension $\widetilde T$; a mixed indexing category records $\eta,\varepsilon,\mu,\delta$ with finality giving presentation-independence; and we identify $\Frac\simeq(\Ind_T\cap\Pro_T)^{\mathrm{repl}}$ under explicit accessibility/preservation hypotheses. In a Lawvere-metric enriched setting, uniform contractivity yields algebraic compactness: the initial $T$-algebra and terminal $T$-coalgebra \emph{coincide} on a canonical \emph{ambifixpoint} obeying Frobenius diagrams. On presheaves, precomposition and (lax-to-strong) Day convolution lift $T$; we construct \emph{$T$-equivariant left Kan extensions} computed over an $\eta\varepsilon$-subpresentation and, for $0<q<1$, prove difference-based Lipschitz bounds $\|F-G\|\le \frac{1}{1-q}\|F_0-G_0\|$ with truncation error $\le \frac{1}{1-q}\,q^{k+1}$ (for $q=1$ we retain $1$-Lipschitz stability without a geometric rate). Bicategorical and $(\infty,1)$ variants are given with strictification/truncation hypotheses. Applications include substitution systems, contractive IFS via hyperspaces (separating the functorial scale and the Hutchinson dynamics), and coarse-graining channels.
\end{abstract}

\section*{Motivation and contributions}
\textbf{Motivation.}
(1) Refinement and coarse-graining are often modelled by distinct operators, complicating coherence and invariant transport.
(2) A universal mechanism to propagate \emph{seed laws} to all scales is lacking.
(3) Recomputing at each level is costly; we seek \emph{logarithmic-depth} evaluation with error control.

\smallskip
\noindent
\textbf{Idea.} Encode scale by a \emph{single} $T$ with monad $(\mu,\eta)$ and comonad $(\delta,\varepsilon)$ in Frobenius compatibility; complete a seed $\Czero$ along mixed $T$-towers to a bicompletion $\Frac$; in an enriched setting, use contractivity for fixed points and stability.

\smallskip
\noindent
\textbf{Contributions.}
\begin{itemize}[leftmargin=1.2em]
  \item[\textbf{A}] \emph{Bicompletion and universality.} Restricted Yoneda lift $J/\widehat T/\widetilde T$ with size control and lifted Frobenius; mixed indexing with $\eta,\varepsilon,\mu,\delta$; finality (presentation-independence) including an $\eta\varepsilon$-subpresentation; intersection identification $\Frac\simeq(\Ind_T\cap \Pro_T)^{\mathrm{repl}}$.
  \item[\textbf{B}] \emph{Ambifixpoint.} In a complete Lawvere-metric setting with uniform contractivity, initial algebra $=$ final coalgebra on a carrier $\mathcal U$; we prove $\mathcal U$ satisfies the Frobenius diagrams and relate $\Frac$ to $J(\mathcal U)$ via $T$-density.
  \item[\textbf{C}] \emph{Stable equivariant Kan extensions.} Presheaf lift (precomposition/Day) of $T$; equivariant Kan extensions computed on $\eta\varepsilon$-subpresentations with universal property and \emph{difference-based} stability bounds.
  \item[\textbf{D}] \emph{2-/$(\infty,1)$-levels.} Bicategorical/$(\infty,1)$ analogues with explicit strictification/truncation hypotheses.
\end{itemize}

\begin{remark}[Standing hypotheses roadmap]\label{rmk:roadmap}
Sections~\ref{sec:bicompletion}, \ref{sec:ambifix}, \ref{sec:presheaves} are independent modules.
(i) \S\ref{sec:bicompletion} uses Assumptions~\ref{ass:lp}, \ref{ass:access-strong}, \ref{ass:fubini} and works with the lifted $J,\widehat T,\widetilde T$ without assuming $T(\Ob\Czero)\subseteq\Ob\Czero$.
(ii) \S\ref{sec:ambifix} uses Assumption~\ref{ass:enriched} (no Ind/Pro needed).
(iii) \S\ref{sec:presheaves} additionally assumes that $T^n\!\restriction_{\Czero}$ is a site morphism for all $n$ (Assumption~\ref{ass:site}); this entails $T^n(\Ob\Czero)\subseteq\Ob\Czero$.
\end{remark}

\begin{remark}[Equality vs.\ isomorphism]\label{rmk:eq-iso}
Identities involving lifted structures ($\widehat T,\widetilde T,T_\ast$) are canonical \emph{isomorphisms} unless stated otherwise. In strictly 1-categorical statements we may suppress $\cong$ when harmless.
\end{remark}

\begin{remark}[Type/notation cribsheet]
{\small
$J:\C\to\Ind(\C_0)$ (restricted Yoneda), $\widehat T:\Ind(\C_0)\to\Ind(\C_0)$ ($=\Lan_J(J\circ T)$), $j:\Ind(\C_0)\hookrightarrow\Pro(\Ind(\C_0))$, $\widetilde T:\Pro(\Ind(\C_0))\to\Pro(\Ind(\C_0))$ (levelwise), $T_\ast:[\C^{\op},\Set]\to[\C^{\op},\Set]$ ($=\Lan_y(y\circ T)$).
}
\end{remark}

%======================================================
\section{Scale as a Frobenius (co)monad \texorpdfstring{(1-/2-/$(\infty,1)$-levels)}{(1-/2-/(\infty,1)-levels)}}
\label{sec:scale}
%======================================================

\subsection*{1-level (strict) definition}
\begin{definition}[Frobenius (co)monad]\label{def:frob}
An endofunctor $T:\C\to\C$ is a \emph{Frobenius (co)monad} if $(T,\mu,\eta)$ is a monad and $(T,\delta,\varepsilon)$ is a comonad such that

EQPH_eq0001_PH

If $\mu\circ\delta=\Id_T$ we say \emph{separable Frobenius}.
\end{definition}

\begin{remark}[Existence via Frobenius algebra objects]\label{rmk:FrobAlgebra}
If $(\C,\otimes,I)$ is (symmetric) monoidal and $A\in\C$ is a (separable) Frobenius algebra object with
multiplication $m$, unit $\eta_A$, comultiplication $\Delta$, and counit $\varepsilon_A$ satisfying the
Frobenius law, then $T:=A\otimes -$ carries a canonical Frobenius (co)monad structure with
$\mu=m\otimes-$, $\eta=\eta_A\otimes-$, $\delta=\Delta\otimes-$, $\varepsilon=\varepsilon_A\otimes-$.
This supplies concrete families of (separable) Frobenius (co)monads (e.g.\ in $\mathbf{Vect}_k$ or finite
$\mathbf{Hilb}$).
\end{remark}

\begin{remark}[Monoidal compatibility]\label{rmk:monoidal}
If $\C$ is (symmetric) monoidal, we sometimes assume $T$ is strong monoidal and strong comonoidal with coherent structure maps; this feeds into Day convolution in \S\ref{sec:presheaves}. The \emph{separability} of Frobenius structures is used only for explicit examples and is \emph{not} required by the main theorems.
\end{remark}

\subsection*{2-level and $(\infty,1)$-level}
\begin{definition}[Bicategorical version]
In a bicategory, a \emph{pseudomonad} $(T,\mu,\eta)$ and a \emph{pseudocomonad} $(T,\delta,\varepsilon)$ on the same endo-1-cell are \emph{Frobenius compatible} if the equations in Definition~\ref{def:frob} hold up to specified invertible 2-cells and standard coherence.
\end{definition}

\begin{definition}[($(\infty,1)$)-version]
In an $(\infty,1)$-category, a monad object and a comonad object on the same endomorphism $T$ are equipped with higher homotopies realising the Frobenius data. Truncation to $h_1$ yields the 1-level notion (equalities become isomorphisms).
\end{definition}

%======================================================
\section{Ind-/Pro-towers via restricted Yoneda and mixed indexing}
\label{sec:bicompletion}
%======================================================

We fix a Grothendieck universe to separate \emph{small} and \emph{large}. For brevity, we will refer to filtered colimits and cofiltered limits collectively as ``(bi)limits along mixed presentations''.

\begin{assumption}[Locally presentable seed, finite-limit closed]\label{ass:lp}
$\C$ is locally $\kappa$-presentable for some regular $\kappa$;
$\C_0\subseteq\C$ is a small full subcategory of $\kappa$-presentable objects \emph{closed under finite limits} and generating $\C$ under $\kappa$-filtered colimits.
\end{assumption}

\begin{definition}[Restricted Yoneda]\label{def:restricted-yoneda}
Let $J:\C\to\Ind(\Czero)$ be the restricted Yoneda sending $X\in\C$ to $JX:=\C(-,X)\!\restriction_{\Czero^{\op}}$. By Assumption~\ref{ass:lp}, $J$ is dense and $J\circ i \cong j_0$ for the inclusion $i:\Czero\hookrightarrow\C$ and the canonical embedding $j_0:\Czero\hookrightarrow \Ind(\Czero)$.
\end{definition}

\begin{lemma}[Restricted Yoneda lands in $\Ind(\C_0)$ and is dense]\label{lem:J-dense}
Under Assumption~\ref{ass:lp}, every $X\in\C$ is a $\kappa$-filtered colimit of $i(C_i)$ with $C_i\in\C_0$. Hence
\(
JX\cong\operatorname*{colim}_{i} y(C_i)\in \Ind(\C_0)
\).
Here $y:\C_0\to[\C_0^{\op},\Set]$ denotes the Yoneda embedding.
Thus $J$ factors through $\Ind(\C_0)$ and is dense; moreover $J\circ i\cong j_0$.
\end{lemma}

\begin{lemma}[Filteredness of comma indices]\label{lem:J-comma-filtered}
For every $X\in\Ind(\C_0)$, the comma category $(J\downarrow X)$ is filtered. In particular, writing $X\cong\operatorname*{colim}_{i\in I} J C_i$ with $I$ filtered, the canonical functor $I\to (J\downarrow X)$ is cofinal.
\end{lemma}

\begin{remark}[Pointwise formula and finite limits]\label{rmk:pointwise-finite}
By density and Lemma~\ref{lem:J-comma-filtered},
\[
\widehat T(X)\ \cong\ \operatorname*{colim}_{(J C\to X)} J(TC)\,.
\]

2pt] \small ORCID: \href{https://orcid.org/0009-0004-4273-3365}{0009-0004-4273-3365}} \date{\today} %============================== % Theorem styles %============================== \theoremstyle{plain} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{lemma}[theorem]{Lemma} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{assumption}[theorem]{Assumption} \newtheorem{example}[theorem]{Example} \newtheorem{remark}[theorem]{Remark} %============================== % Macros %============================== \newcommand{\C}{\mathcal{C}} \newcommand{\Czero}{\mathcal{C}_0} \newcommand{\D}{\mathcal{D}} \newcommand{\E}{\mathcal{E}} \newcommand{\V}{\mathbf{V}} \newcommand{\Set}{\mathbf{Set}} \newcommand{\Ind}{\mathrm{Ind}} \newcommand{\Pro}{\mathrm{Pro}} \newcommand{\Frac}{\mathrm{Frac}_T(\Czero)} \newcommand{\Fixpm}{\mathrm{Fix}^{\pm}(T)} \newcommand{\Id}{\mathrm{Id}} \newcommand{\Ob}{\mathrm{Ob}} \newcommand{\Hom}{\mathrm{Hom}} \newcommand{\Lan}{\mathrm{Lan}} \newcommand{\Ran}{\mathrm{Ran}} \newcommand{\op}{\mathrm{op}} \DeclareMathOperator*{\colim}{colim} % <-- fix for \colim \DeclareMathOperator{\Fix}{Fix} % <-- fix for \Fix % Hyphenation to avoid awkward breaks \hyphenation{bi-completion pseudo-monad pseudo-comonad cofiltered e-qui-variant ambi-fixpoint exten-sions} \begin{document} \setstretch{1.3} \maketitle \begin{abstract} We present \emph{Fractal Category Theory} (FCT): scale is internalized as a single endofunctor $T :\C\to\C$ carrying monad and comonad structures in Frobenius compatibility. From a small seed $\Czero\subseteq\C$ we form a $T$-generated \emph{Ind--Pro bicompletion} $\Frac\subset\Pro(\Ind(\Czero))$ by closing under filtered colimits and cofiltered limits along mixed $T$-towers. Towers are built at the Ind-level via a restricted-Yoneda lift $\widehat T:=\Lan_{J}(J \circ T)$; the Frobenius data lift canonically to $\widehat T$ and to its Pro-level extension $\widetilde T$; a mixed indexing category records $\eta,\varepsilon,\mu,\delta$ with finality giving presentation-independence; and we identify $\Frac\simeq(\Ind_T\cap\Pro_T)^{\mathrm{repl}}$ under explicit accessibility/preservation hypotheses. In a Lawvere-metric enriched setting, uniform contractivity yields algebraic compactness: the initial $T$-algebra and terminal $T$-coalgebra \emph{coincide} on a canonical \emph{ambifixpoint} obeying Frobenius diagrams. On presheaves, precomposition and (lax-to-strong) Day convolution lift $T$; we construct \emph{$T$-equivariant left Kan extensions} computed over an $\eta\varepsilon$-subpresentation and, for $0<q<1$, prove difference-based Lipschitz bounds $\|F-G\|\le \frac{1}{1-q}\|F_0-G_0\|$ with truncation error $\le \frac{1}{1-q} q^{k+1}$ (for $q=1$ we retain $1$-Lipschitz stability without a geometric rate). Bicategorical and $(\infty,1)$ variants are given with strictification/truncation hypotheses. Applications include substitution systems, contractive IFS via hyperspaces (separating the functorial scale and the Hutchinson dynamics), and coarse-graining channels. \end{abstract} \section*{Motivation and contributions} \textbf{Motivation.} (1) Refinement and coarse-graining are often modelled by distinct operators, complicating coherence and invariant transport. (2) A universal mechanism to propagate \emph{seed laws} to all scales is lacking. (3) Recomputing at each level is costly; we seek \emph{logarithmic-depth} evaluation with error control. \smallskip \noindent \textbf{Idea.} Encode scale by a \emph{single} $T$ with monad $(\mu,\eta)$ and comonad $(\delta,\varepsilon)$ in Frobenius compatibility; complete a seed $\Czero$ along mixed $T$-towers to a bicompletion $\Frac$; in an enriched setting, use contractivity for fixed points and stability. \smallskip \noindent \textbf{Contributions.} \begin{itemize}[leftmargin=1.2em] \item[\textbf{A}] \emph{Bicompletion and universality.} Restricted Yoneda lift $J/\widehat T/\widetilde T$ with size control and lifted Frobenius; mixed indexing with $\eta,\varepsilon,\mu,\delta$; finality (presentation-independence) including an $\eta\varepsilon$-subpresentation; intersection identification $\Frac\simeq(\Ind_T\cap \Pro_T)^{\mathrm{repl}}$. \item[\textbf{B}] \emph{Ambifixpoint.} In a complete Lawvere-metric setting with uniform contractivity, initial algebra $=$ final coalgebra on a carrier $\mathcal U$; we prove $\mathcal U$ satisfies the Frobenius diagrams and relate $\Frac$ to $J(\mathcal U)$ via $T$-density. \item[\textbf{C}] \emph{Stable equivariant Kan extensions.} Presheaf lift (precomposition/Day) of $T$; equivariant Kan extensions computed on $\eta\varepsilon$-subpresentations with universal property and \emph{difference-based} stability bounds. \item[\textbf{D}] \emph{2-/$(\infty,1)$-levels.} Bicategorical/$(\infty,1)$ analogues with explicit strictification/truncation hypotheses. \end{itemize} \begin{remark}[Standing hypotheses roadmap]\label{rmk:roadmap} Sections~\ref{sec:bicompletion}, \ref{sec:ambifix}, \ref{sec:presheaves} are independent modules. (i) \S\ref{sec:bicompletion} uses Assumptions~\ref{ass:lp}, \ref{ass:access-strong}, \ref{ass:fubini} and works with the lifted $J,\widehat T,\widetilde T$ without assuming $T(\Ob\Czero)\subseteq\Ob\Czero$. (ii) \S\ref{sec:ambifix} uses Assumption~\ref{ass:enriched} (no Ind/Pro needed). (iii) \S\ref{sec:presheaves} additionally assumes that $T^n \restriction_{\Czero}$ is a site morphism for all $n$ (Assumption~\ref{ass:site}); this entails $T^n(\Ob\Czero)\subseteq\Ob\Czero$. \end{remark} \begin{remark}[Equality vs.\ isomorphism]\label{rmk:eq-iso} Identities involving lifted structures ($\widehat T,\widetilde T,T_\ast$) are canonical \emph{isomorphisms} unless stated otherwise. In strictly 1-categorical statements we may suppress $\cong$ when harmless. \end{remark} \begin{remark}[Type/notation cribsheet] {\small $J:\C\to\Ind(\C_0)$ (restricted Yoneda), $\widehat T:\Ind(\C_0)\to\Ind(\C_0)$ ($=\Lan_J(J\circ T)$), $j:\Ind(\C_0)\hookrightarrow\Pro(\Ind(\C_0))$, $\widetilde T:\Pro(\Ind(\C_0))\to\Pro(\Ind(\C_0))$ (levelwise), $T_\ast:[\C^{\op},\Set]\to[\C^{\op},\Set]$ ($=\Lan_y(y\circ T)$). } \end{remark} %====================================================== \section{Scale as a Frobenius (co)monad \texorpdfstring{(1-/2-/$(\infty,1)$-levels)}{(1-/2-/(\infty,1)-levels)}} \label{sec:scale} %====================================================== \subsection*{1-level (strict) definition} \begin{definition}[Frobenius (co)monad]\label{def:frob} An endofunctor $T:\C\to\C$ is a \emph{Frobenius (co)monad} if $(T,\mu,\eta)$ is a monad and $(T,\delta,\varepsilon)$ is a comonad such that EQPH_eq0001_PH If $\mu\circ\delta=\Id_T$ we say \emph{separable Frobenius}. \end{definition} \begin{remark}[Existence via Frobenius algebra objects]\label{rmk:FrobAlgebra} If $(\C,\otimes,I)$ is (symmetric) monoidal and $A\in\C$ is a (separable) Frobenius algebra object with multiplication $m$, unit $\eta_A$, comultiplication $\Delta$, and counit $\varepsilon_A$ satisfying the Frobenius law, then $T:=A\otimes -$ carries a canonical Frobenius (co)monad structure with $\mu=m\otimes-$, $\eta=\eta_A\otimes-$, $\delta=\Delta\otimes-$, $\varepsilon=\varepsilon_A\otimes-$. This supplies concrete families of (separable) Frobenius (co)monads (e.g.\ in $\mathbf{Vect}_k$ or finite $\mathbf{Hilb}$). \end{remark} \begin{remark}[Monoidal compatibility]\label{rmk:monoidal} If $\C$ is (symmetric) monoidal, we sometimes assume $T$ is strong monoidal and strong comonoidal with coherent structure maps; this feeds into Day convolution in \S\ref{sec:presheaves}. The \emph{separability} of Frobenius structures is used only for explicit examples and is \emph{not} required by the main theorems. \end{remark} \subsection*{2-level and $(\infty,1)$-level} \begin{definition}[Bicategorical version] In a bicategory, a \emph{pseudomonad} $(T,\mu,\eta)$ and a \emph{pseudocomonad} $(T,\delta,\varepsilon)$ on the same endo-1-cell are \emph{Frobenius compatible} if the equations in Definition~\ref{def:frob} hold up to specified invertible 2-cells and standard coherence. \end{definition} \begin{definition}[($(\infty,1)$)-version] In an $(\infty,1)$-category, a monad object and a comonad object on the same endomorphism $T$ are equipped with higher homotopies realising the Frobenius data. Truncation to $h_1$ yields the 1-level notion (equalities become isomorphisms). \end{definition} %====================================================== \section{Ind-/Pro-towers via restricted Yoneda and mixed indexing} \label{sec:bicompletion} %====================================================== We fix a Grothendieck universe to separate \emph{small} and \emph{large}. For brevity, we will refer to filtered colimits and cofiltered limits collectively as ``(bi)limits along mixed presentations''. \begin{assumption}[Locally presentable seed, finite-limit closed]\label{ass:lp} $\C$ is locally $\kappa$-presentable for some regular $\kappa$; $\C_0\subseteq\C$ is a small full subcategory of $\kappa$-presentable objects \emph{closed under finite limits} and generating $\C$ under $\kappa$-filtered colimits. \end{assumption} \begin{definition}[Restricted Yoneda]\label{def:restricted-yoneda} Let $J:\C\to\Ind(\Czero)$ be the restricted Yoneda sending $X\in\C$ to $JX:=\C(-,X) \restriction_{\Czero^{\op}}$. By Assumption~\ref{ass:lp}, $J$ is dense and $J\circ i \cong j_0$ for the inclusion $i:\Czero\hookrightarrow\C$ and the canonical embedding $j_0:\Czero\hookrightarrow \Ind(\Czero)$. \end{definition} \begin{lemma}[Restricted Yoneda lands in $\Ind(\C_0)$ and is dense]\label{lem:J-dense} Under Assumption~\ref{ass:lp}, every $X\in\C$ is a $\kappa$-filtered colimit of $i(C_i)$ with $C_i\in\C_0$. Hence \( JX\cong\operatorname*{colim}_{i} y(C_i)\in \Ind(\C_0) \). Here $y:\C_0\to[\C_0^{\op},\Set]$ denotes the Yoneda embedding. Thus $J$ factors through $\Ind(\C_0)$ and is dense; moreover $J\circ i\cong j_0$. \end{lemma} \begin{lemma}[Filteredness of comma indices]\label{lem:J-comma-filtered} For every $X\in\Ind(\C_0)$, the comma category $(J\downarrow X)$ is filtered. In particular, writing $X\cong\operatorname*{colim}_{i\in I} J C_i$ with $I$ filtered, the canonical functor $I\to (J\downarrow X)$ is cofinal. \end{lemma} \begin{remark}[Pointwise formula and finite limits]\label{rmk:pointwise-finite} By density and Lemma~\ref{lem:J-comma-filtered}, \[ \widehat T(X)\ \cong\ \operatorname*{colim}_{(J C\to X)} J(TC) .

\[
\widehat T\;:=\;\Lan_{J}(J\circ T):\Ind(\Czero)\to\Ind(\Czero).
\]

\widehat T := \Lan_{J}(J\circ T):\Ind(\Czero)\to\Ind(\Czero).

\[
j_0(\Czero)\xrightarrow{\ \widehat T\ } \widehat T(j_0(\Czero)) \xrightarrow{\ \widehat T\ }\cdots.
\]

\[
\mu_{n}\circ \eta_{n} \;\Rightarrow\; \Id \;=\; \varepsilon_{n+1}\circ \delta_n,
\qquad
T^n f\circ \eta_n \;\Rightarrow\; \eta_n\circ T^{n} f,
\]

\mu_{n}\circ \eta_{n}  \Rightarrow  \Id  =  \varepsilon_{n+1}\circ \delta_n, \qquad T^n f\circ \eta_n  \Rightarrow  \eta_n\circ T^{n} f,

\[
\begin{tikzcd}[column sep=large,row sep=large]
T^{2} \ar[r,"T\mu_n"] \ar[d,"\delta_{n+1}"'] & T \ar[d,"\delta_n"] \\
T^{2} \ar[r,"\mu_{n+1}"'] & T
\end{tikzcd}
\]

\[
\Frac \;\simeq\; \big(\Ind_T(\Czero)\cap \Pro_T(\Czero)\big)^{\mathrm{repl}}\ \subset\ \Pro(\Ind(\Czero))\,.
\]

\Frac  \simeq  \big(\Ind_T(\Czero)\cap \Pro_T(\Czero)\big)^{\mathrm{repl}}\ \subset\ \Pro(\Ind(\Czero)) .

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>\[</mi><mi>\Frac</mi><mspace width="0.278em" /><mo>≃</mo><mspace width="0.278em" /><mo minsize="1.2em" maxsize="1.2em">(</mo><msub><mi>\Ind</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>\Czero</mi><mo stretchy="false">)</mo><mo>∩</mo><msub><mi>\Pro</mi><mi>T</mi></msub><mo stretchy="false">(</mo><mi>\Czero</mi><mo stretchy="false">)</mo><msup><mo minsize="1.2em" maxsize="1.2em">)</mo><mrow><mrow><mi mathvariant="normal">r</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">p</mi><mi mathvariant="normal">l</mi></mrow></mrow></msup><mtext> </mtext><mo>⊂</mo><mtext> </mtext><mi>\Pro</mi><mo stretchy="false">(</mo><mi>\Ind</mi><mo stretchy="false">(</mo><mi>\Czero</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="0.167em" /><mo>.</mo><mi>\]</mi></mrow></math>

\[
\begin{tikzcd}[column sep=large,row sep=large]
T^2X \arrow[r,"T a"] \arrow[d,"\delta_X"'] & TX \arrow[d,"c"] \\
T^2X \arrow[r,"\mu_X"'] & TX
\end{tikzcd}
\qquad
\begin{tikzcd}[column sep=large,row sep=large]
TX \arrow[r,"a"] \arrow[d,"\delta_X"'] & X \arrow[d,"c"] \arrow[r,phantom] & \phantom{TX} \\
T^2X \arrow[r,"T c"'] & T^2X \arrow[r,"\mu_X"'] & TX
\end{tikzcd}
\]

\[
  \Frac \;\simeq\; \langle\, j\circ J(\mathcal U)\,\rangle,
\]

\Frac  \simeq  \langle  j\circ J(\mathcal U) \rangle,

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\|F-G\|\ :=\ \bigvee_{X\in\Ob\Frac}  \E(FX, GX)\ \in \V

\[
\begin{tikzcd}[column sep=huge,row sep=large]
E^\D S^n F_0(C) \ar[r,"E^\D(S^n\alpha_C)"] \ar[d,"\vartheta^E_{(n,C)}"'] &
E^\D S^{n+1} F_0(C) \ar[d,"\vartheta^E_{(n+1,C)}"] \\
S^n F_0(E^\C C) \ar[r,"S^{n}\alpha_{E^\C C}\circ S^n \sigma^E_C"'] & S^{n+1} F_0(E^\C C)
\end{tikzcd}
\]

$\mathbb U\subset\mathbb U'\subset\mathbb U''$

\mathbb U\subset\mathbb U'\subset\mathbb U''

$\widehat T(X)\cong\operatorname{colim}_{(JC\to X)} JT(C)$

\widehat T^{ n} \circ j_0:\Czero\to\Ind(\Czero)

$T\mu_n\circ \delta_{n+1}=\delta_n\circ \mu_{n+1}=\mu_{n+1}\circ T\delta_n$

$\{\widehat T^{\,n}(j_0C)\mid C\in \Czero,\,n\ge0\}$

\{\widehat T^{ n}(j_0C)\mid C\in \Czero, n\ge0\}

$(i,j)\mapsto \widetilde T^{m_j}\widehat T^{n_i}(j_0 C_i)$

$\alpha:F_0\circ T{\restriction_{\Czero}}\Rightarrow S\circ F_0$

$\bar\alpha: F\circ \widetilde T \Rightarrow S\circ F$

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\{\widehat T^{ n}(j_0 C)\mid C\in\C_0, n\ge0\}

$\vartheta:y\circ T\Rightarrow T_\ast\circ y$

\vartheta:y\circ T\Rightarrow T_\ast\circ y

<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mrow><mi>$</mi><mi>n</mi><mspace width="negativethinmathspace" /><mo>+</mo><mspace width="negativethinmathspace" /><mn>1</mn><mi>$</mi></mrow></math>

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$\sigma^E:E^\D F_0 \Rightarrow F_0 E^\C$

$\sigma^O:O^\D F_0 \Rightarrow F_0 O^\C$

$\varepsilon_n\circ \delta_n \Rightarrow \Id$

\varepsilon_n\circ \delta_n \Rightarrow \Id

$T\mu_n\circ \delta_{n+1} = \delta_n\circ \mu_{n+1} = \mu_{n+1}\circ T\delta_n$

$(\eta_{n}\text{ then }\mu_{n}) \Rightarrow$

(\eta_{n}\text{ then }\mu_{n}) \Rightarrow

$(\varepsilon_{n}\text{ then }\delta_{n}) \Rightarrow$

$T^n f\circ \eta_n \Rightarrow \eta_n\circ T^n f$

$T^{n+1}f\circ \varepsilon_n \Rightarrow \varepsilon_n\circ T^{n+1} f$

$T^{n+1} f\circ \mu_n \Rightarrow \mu_n\circ T^{n+2} f$

$T^{n+2} f\circ \delta_n \Rightarrow \delta_n\circ T^{n+1} f$