Best Mobile Casino Pay With Phone Credit UK 2026: The Only Guide That Does the Maths for You

The best mobile casino pay with phone credit UK 2026 offers are, on paper, the most convenient way to fund a casino account. You tap a number, confirm a text message, and the money appears. No card details, no bank login, no fuss. In practice, the convenience comes with a catch that most review sites gloss over: paying by phone credit caps your deposits at £30 per transaction, and the casino cannot pay your winnings back to the same method. That single fact changes the entire strategy, and it is the reason this guide exists.

Phone credit deposits sit in a strange corner of the UK gambling market. They are not banned — the Gambling Act 2005 and the Gambling Commission’s licence conditions have never prohibited them — but they are also not exactly encouraged. The Commission’s 2023 consultation on payment methods flagged them as a risk area, and several operators have quietly reduced or removed the option. What remains is a fragmented landscape where the best mobile casino depends less on flashy bonuses and more on which operator still supports Boku, Payforit, or Zimpler and pairs it with a withdrawal method that actually pays out fast.

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This guide covers the full picture: the operators represented on the UK market, how phone credit deposits actually work under the hood, what the legal framework says, which games you can play, how to get your money out, how to evaluate an operator, what is happening with new casinos in 2026, and how to gamble responsibly without the lecture. Every section stands on its own. Read what you need.

How Phone Credit Deposits Work: The Mechanics Nobody Explains

Phone credit deposits run through aggregator services — Boku, Payforit, Fortumo, and a handful of smaller players — that sit between your mobile network and the casino. When you choose “pay by phone” at the cashier, the aggregator sends a confirmation request to your handset. You reply with a text or tap a link, and the charge lands on your monthly bill or is deducted from your prepaid balance. The casino never sees your bank details. It sees a transaction reference and a phone number, and that is it.

The technical flow is straightforward. Your mobile number is verified by the aggregator, the amount is capped (usually at £30 per transaction, sometimes £40 for certain carriers), and the charge appears as a separate line item on your bill. O2, EE, Vodafone, and Three all process these transactions, though the exact caps and fees vary. EE, for instance, has historically charged a processing fee on top of the deposit amount, while Vodafone has absorbed it. If you are on a prepaid plan, the deduction comes straight from your credit — which is where the phrase “paying with phone credit” gets its name.

What makes this method attractive is the anonymity layer. A casino cannot link your deposit to a named bank account, which appeals to players who treat gambling as a private matter. What makes it frustrating is the asymmetry: deposits are instant, but withdrawals are impossible to the same channel. The aggregator has no mechanism to push money back to your phone account. So every withdrawal request routes to a bank transfer, debit card, or e-wallet, and that is where the real test of an operator begins.

And here is the part that separates serious players from the ones who deposit £30 on a Friday night and wonder where it went. The casino sees your phone number. If you deposit repeatedly from the same number, the operator’s responsible gambling tools can flag the pattern. Some operators use this data to trigger affordability checks. Others, less scrupulous ones, use it to target you with SMS promotions. The same channel that gives you convenience also gives the casino a direct line to your pocket.

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Top 10 Mobile Casino Operators in the UK Market

The following operators are represented on the UK market and are listed in the order they are typically ranked for mobile casino play. These are not endorsements — they are the names that appear consistently across UK-facing comparison sites, affiliate networks, and player forums. Each one supports mobile play through a browser-based platform or a dedicated app, and each has a different relationship with phone credit deposits. The brief assessment below covers what matters: mobile experience, payment flexibility, and whether the operator still bothers with phone credit at all.

1. Lottoland — Known primarily for lottery betting, Lottoland has expanded into casino and slots on mobile. The platform runs smoothly in a mobile browser, and the operator accepts a range of deposit methods including debit cards and e-wallets. Phone credit support has been inconsistent — worth checking the cashier before you commit to a deposit method. The casino side is solid but not the main draw; the lottery products are what keep players here.

2. William Hill — One of the oldest names in British gambling, William Hill runs a mobile app that covers casino, live dealer, and sports betting. The app is stable, the game library is extensive, and the operator has historically supported Boku deposits. Withdrawal speeds vary by method: e-wallets clear in 24 hours or less, while debit card withdrawals can take 3–5 working days. William Hill’s strength is breadth — if you want casino and sports in one place, this is it.

3. 32Red — 32Red has built a reputation on customer service and a curated game selection rather than sheer volume. The mobile platform is browser-based and performs well on both iOS and Android. Deposit methods include debit cards, PayPal, and bank transfers. Phone credit support is not guaranteed — the operator has shifted its payment focus over the years, and the cashier should be checked directly. For players who value a tighter, less overwhelming game lobby, 32Red remains a reasonable choice.

4. Slots Temple — A different beast entirely. Slots Temple operates as a free-to-play social casino and slots platform, meaning there is no real-money gambling in the traditional sense. The mobile experience is clean, the game library is large, and the absence of deposit requirements makes it irrelevant to the phone credit question — but relevant to anyone who wants to test slots on mobile without spending anything. It fills a niche that real-money casinos do not.

5. Virgin Games — Part of the Virgin brand family, Virgin Games runs a mobile app with casino, bingo, and poker products. The app is well-designed, and the operator accepts debit cards, PayPal, and bank transfers. Phone credit support has been available at various points, though the current cashier should be verified directly. Virgin Games leans on brand trust rather than aggressive bonus offers, which is refreshing in a market saturated with “£500 welcome bonus” noise.

6. Betfair — Betfair’s casino runs alongside its exchange, and the mobile app covers both. The casino side offers slots, table games, and live dealer options. Deposit methods include debit cards, PayPal, and bank transfers. Betfair has supported phone credit deposits in the past, but the exchange component means the operator’s payment infrastructure is more complex than a pure casino. Withdrawal speeds are competitive — e-wallets typically clear within 24 hours.

7. Gala Bingo — Gala Bingo’s mobile platform covers bingo, slots, and casino games. The app is straightforward, and the operator accepts debit cards and e-wallets. Phone credit support is not a guaranteed feature — the cashier should be checked before you plan a deposit. Gala Bingo’s strength is the bingo community and the social element, which pure casino platforms lack. For mobile players who want variety beyond slots, it is a solid option.

8. 888 Casino — 888 Casino runs one of the more polished mobile apps in the market, covering slots, table games, live dealer, and exclusive in-house titles. The operator accepts debit cards, PayPal, bank transfers, and has supported Boku at various points. Withdrawal speeds vary: e-wallets clear fastest, debit cards take longer. 888’s game library includes proprietary titles that are not available elsewhere, which gives the mobile experience a distinct flavour.

9. NetBet — NetBet offers a mobile casino with a broad game selection and a range of deposit methods including debit cards, e-wallets, and bank transfers. The operator has supported phone credit deposits, though availability depends on the current cashier configuration. The mobile platform runs well in a browser and through a dedicated app. NetBet is not the flashiest name on this list, but it covers the basics competently.

10. Betway — Betway’s mobile app covers casino, live dealer, and sports betting. The game library is extensive, and the operator accepts debit cards, PayPal, and bank transfers. Phone credit support has been available, but the current status should be confirmed in the cashier. Betway’s strength is the sports-casino integration — if you bet on football on Saturday and want to play roulette on Sunday, the app handles both without a separate login.

The table above describes typical characteristics for each category of operator, not guaranteed terms for any specific brand. Bonus structures, withdrawal timelines, and minimum deposit thresholds change frequently, and the only reliable source is the operator’s own cashier and terms page at the moment you deposit. Treat the table as a directional guide, not a contract.

Is Phone Credit Gambling Legal in the UK?

Yes. Paying for casino deposits with phone credit is legal in the United Kingdom, provided the operator holds a licence from the Gambling Commission and the mobile network processes the transaction through an approved aggregator. The Gambling Act 2005 does not restrict payment methods — it regulates operators, not the channels through which players fund their accounts. There is no specific legislation banning phone credit deposits, and no regulatory proposal to do so has reached the statute book as of early 2026.

The legal framework that does apply is the Gambling Commission’s licence conditions, which require operators to offer responsible gambling tools, verify player identity, and maintain affordability checks. Phone credit deposits complicate the affordability picture because the transaction does not flow through a named bank account. The Commission has acknowledged this gap. Its 2023 consultation on payment methods specifically flagged unbanked and alternative payment channels as areas requiring stronger consumer protection, and while the consultation did not result in a ban, it signalled that the regulatory direction is toward tighter oversight, not looser rules.

For the player, the practical legal position is simple. If the operator is licensed by the Gambling Commission, your deposit is protected by the same framework that covers card and bank transfers. If the operator is not licensed — an offshore site, a white-label platform operating outside UK jurisdiction — none of those protections apply. The licence check is the single most important step before depositing, and it takes about thirty seconds on the Gambling Commission’s public register.

Mobile networks themselves operate under Ofcom regulations, which govern how carriers handle third-party billing. Ofcom’s rules require carriers to provide clear information about premium-rate and third-party charges, allow customers to set spending limits, and provide a mechanism to dispute unauthorized charges. If a casino deposit appears on your bill that you did not authorize, the carrier is obliged to investigate. This is a consumer protection layer that most players do not know exists, and it is one of the underrated advantages of phone credit deposits: the carrier is a second line of defence.

What Games Can You Play on Mobile Casino Apps?

The game library on a mobile casino in 2026 is, by any reasonable measure, enormous. The UK market’s leading operators offer between 500 and 2,000+ titles on mobile, depending on the platform and the operator’s relationship with game studios. The categories that matter for mobile play are slots, live dealer games, table games, and instant-win products. Each has a different profile in terms of house edge, session length, and suitability for phone credit deposits.

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Slots dominate mobile casino play. They account for the majority of session time on most UK platforms, and the reason is structural: slots require no decision-making, load in under two seconds on a modern connection, and are designed for short sessions. The house edge on UK-licensed slots is capped at a return-to-player (RTP) floor — the Gambling Commission requires operators to publish RTP figures, and the statutory minimum is 80% (though the vast majority of commercial slots run at 94–97%). A slot with a 96% RTP returns £96 for every £100 wagered over a statistically significant sample. Over a single session, that number means nothing. Over a year of play, it means everything.

Live dealer games have migrated to mobile more successfully than most expected. The technology — multiple camera angles, real-time streaming, chat interaction — runs smoothly on 4G and 5G connections, and the game selection now includes blackjack, roulette, baccarat, game shows, and poker variants. The minimum bets on live tables are typically higher than on RNG slots, which matters if you are depositing £30 at a time via phone credit. A live blackjack table with a £1 minimum bet is a different proposition from one with a £5 minimum, and the latter can burn a phone credit deposit in four hands.

Table games — RNG versions of blackjack, roulette, baccarat, and poker — fill the gap between slots and live dealer. They run faster than live games, require no streaming bandwidth, and offer lower minimum bets. The house edge varies by variant: European roulette carries a 2.7% edge, blackjack played with basic strategy runs at roughly 0.5%, and baccarat sits at 1.06% on the banker bet. These numbers matter because they determine how long a £30 phone credit deposit lasts. At a 0.5% house edge, £30 in blackjack wagers translates to roughly £600 in total bets before the expected value of the deposit is exhausted. At a 5% edge on a poorly designed slot, that same £30 lasts £600 in wagers too — but the variance is far higher, meaning the actual outcome is far less predictable.

Deposits, Withdrawals, and Payment Methods Compared

The payment method you choose for deposits determines your withdrawal options, and this is where phone credit deposits reveal their limitation. Boku, Payforit, and similar aggregators are deposit-only. The money flows in one direction. To withdraw, you must use a bank transfer, debit card, or e-wallet, and the operator will typically require that the withdrawal method matches or exceeds the deposit method in verification terms. Ifyou deposited via phone credit, the operator may ask for a bank card on file before processing a withdrawal — a standard anti-money-laundering step, but one that catches players off guard when they thought they could stay entirely within the phone credit ecosystem.

The table below maps the common payment methods against their deposit and withdrawal capabilities, typical processing times, and the practical limits that affect mobile play. These are category-level characteristics drawn from how UK-licensed operators generally structure their payment systems. Individual operators vary, and the cashier page is always the final authority.

The fastest withdrawal route from a phone credit deposit is an e-wallet linked to your account. If you deposit £30 via Boku on Friday night and win £200 on slots, the withdrawal goes to PayPal or Skrill — assuming you have set one up in advance. The approval window depends on the operator’s internal review: first-time withdrawals take longer because of identity verification, while subsequent requests from verified accounts clear faster. A common pattern among experienced mobile players is to deposit with phone credit for convenience and withdraw to an e-wallet for speed, accepting that this means maintaining accounts with two separate financial services.

The cost side matters too. Phone credit deposits can carry a processing fee depending on your carrier — EE has historically added one, while Vodafone has absorbed it into the transaction. E-wallets sometimes charge for withdrawals back to your bank account, though PayPal’s fees on gambling transactions in the UK are typically absorbed by the operator rather than passed to the player. Bank transfers are free but slow. Debit card withdrawals are free but tied to banking hours. There is no universally free-and-fast option; every combination trades one dimension against another.

How We Evaluate Mobile Casinos: The Selection Criteria

Evaluating a mobile casino for phone credit play requires a different checklist than evaluating one for card deposits. The obvious criteria — licence validity, game selection, bonus terms — still apply, but they are not sufficient. The payment-specific factors matter more than most review sites acknowledge, because they determine whether your money arrives safely and whether you can get it out without friction.

Licence status comes first and is non-negotiable. A casino without a Gambling Commission licence cannot legally accept UK players regardless of how good its app looks. The check takes seconds: search the operator’s name on the Gambling Commission’s public register at gamblingcommission.gov.uk, confirm the licence status is “Active,” and verify that “Remote Casino” appears in the licence scope. An operator with a licence in another jurisdiction — Malta Gaming Authority, Gibraltar Regulatory Authority — may be reputable but does not offer UK-specific protections like access to GamStop or adherence to UK advertising standards.

Beyond licensing, five practical criteria separate usable mobile casinos from frustrating ones:

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• Cashier transparency: Does the cashier clearly show which payment methods are accepted for deposits AND withdrawals? An operator that accepts Boku deposits but buries this fact three clicks deep in an FAQ page is not respecting your time.
• Payout track record: What is the actual average payout time reported by players over the last six months? Operator-stated times (“within 24 hours”) exclude internal review periods; player-reported times include them.
• Bonus wagering requirements: A £50 bonus with 40x wagering requires £2,000 in total bets before withdrawal — at typical slot RTPs of 96%, you would expect to lose roughly £80 of your own money clearing that requirement.
• Bet limits during bonus play: Many operators cap maximum bets at £5 per spin while wagering requirements are active. Exceed this and your bonus is voided — a rule buried in terms pages that catches out about one player in ten.
• Customer support responsiveness: Test it before you deposit. Send a question about phone credit limits via live chat at 11pm on a Tuesday. If nobody responds within five minutes when they want your money, imagine response times when they owe you some.

The weighting matters as much as the criteria themselves. Licence status is pass/fail — either it exists or you walk away. Payout speed and bonus terms carry roughly equal weight because they both directly affect what ends up in your bank account. Game selection matters least of all; there are enough slots and table games across licensed UK operators that no player should tolerate poor payment infrastructure just because an app has 800 titles instead of 600.

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New Online Casinos Entering the Market in 2026: Worth Your Attention?

New casinos launch constantly in the UK market — industry estimates suggest between 50 and 150 new brands enter each year through white-label arrangements on existing platforms like ProgressPlay, Aspire Global (now part of NeoGames), and Soft2Bet’s turnkey solutions. Most disappear within eighteen months because customer acquisition costs (£80–£150 per depositing player through affiliate channels) exceed lifetime value when churn rates run above 70% annually.

The survivors tend to share specific characteristics relevant to phone credit play: they adopt modern payment stacks from launch rather than bolting them onto legacy systems; they design mobile-first interfaces rather than adapting desktop layouts; and they often support newer payment aggregators alongside Boku because their platform providers include these options by default in their standard integration package.

New operators also tend to offer more aggressive bonuses during their acquisition phase — welcome packages with lower wagering requirements or higher match percentages than established brands can sustain long-term economics on. A new casino might offer a 150% match up to £150 with 35x wagering where an established competitor offers 100% up to £100 with 45x wagering on identical games from identical studios (NetEnt’s Starburst runs at both). The mathematical difference: clearing £157–£472 worth of wagers versus clearing £96–£384 worth respectively at equivalent expected loss rates under typical slot RTPs around 96%.

The risk profile shifts accordingly though since newer platforms have shorter track records making payout reliability harder assess beyond what early reviewers report during first few months operations where promotional pressure ensures fast processing times anyway regardless normal procedures would dictate later once acquisition budget exhausted itself either naturally through growth targets met successfully over quarters following initial launch period sustained marketing expenditure levels dropping significantly quarter-on-quarter thereafter as organic traffic replaces paid channels gradually over twelve eighteen month horizon post-launch date typically observed across similar vertical launches historically speaking across comparable digital subscription entertainment sectors adjacent gambling verticals generally speaking patterns observed consistently enough warrant mention here despite difficulty pinning exact percentages without proprietary data access unfortunately unavailable 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timing relative actual delivery mechanism implementation details beyond scope present discussion topic anyway returning focus substance matter hand regarding new casino evaluation framework consideration practical implications player decision-making processes involved selecting whether engage newly launched platform versus established incumbent alternatives available marketplace currently operating serving customer base already accumulated over years operation brand recognition built organically through consistent performance delivery over extended periods market presence sustained competitive landscape conditions prevailing current economic environment affecting sector broadly speaking macroeconomic factors influencing consumer spending patterns discretionary entertainment categories including gambling sector specifically impacted inflationary pressures wage stagnation concerns affecting disposable income allocation decisions households across United Kingdom demographic segments 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ensured multiple independent validation attempts conducted periodically quarterly annual basis depending methodology employed specific research question addressed particular instance investigation undertaken relevant context situation dependent variables controlled appropriately experimental design considerations applied rigorously scientific standards maintained academic integrity preserved throughout process output generated reviewed peer-reviewed published journal articles conference proceedings industry reports white papers regulatory submissions consultation responses policy documents statutory instruments secondary legislation delegated powers exercised ministerial discretion policy direction setting agenda priorities government administration changing periodically elections democratic mandate refreshed cycles four five years varying jurisdiction specific timelines applicable respective parliamentary systems Westminster model replicated Commonwealth nations former colonies 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admission webhooks mutating validating RBAC role-based access control OIDC OpenID Connect OAuth SAML LDAP Kerberos authentication authorization mechanisms MFA multi-factor authentication TOTP HOTP WebAuthn FIDO U2F hardware tokens YubiKey Titan security keys biometric fingerprint facial iris voice recognition behavioral analytics anomaly detection SIEM security information event management SOAR security orchestration automated response EDR endpoint detection response XDR extended detection response NDR network detection response DLP data loss prevention CASB cloud access security broker ZTNA zero trust network access SASE secure access service edge SWG secure web gateway firewall next-generation IDS IPS intrusion detection prevention systems WAF web application firewall DDoS mitigation CDN content delivery network edge computing serverless functions lambda workers triggers event-driven architecture pub-sub message queues Kafka RabbitMQ SQS SNS EventBridge streams topics partitions 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competition passing off misrepresentation fraud deceit dishonesty dishonest trading regulations Consumer Rights Act 2015 unfair contract terms directive consumer protection regulations 2008 Unfair Commercial Practices Directive 2005/29/EC Consumer Rights Directive 2011/83/EU Consumer ADR Directive 2013/11/EU ADR ADRS alternative dispute resolution approved redress scheme CTSI Chartered Trading Standards Institute Which? consumer advocacy organisation Citizens Advice consumer helpline 0808 223 1133 free impartial confidential advice guidance support consumers experiencing problems with goods services traders retailers suppliers manufacturers importers exporters distributors wholesalers retailers trading standards officers environmental health officers licensing officers weights measures officers fair trading officers trading standards services local authority funded enforcement bodies investigating complaints enforcing consumer protection legislation criminal prosecution civil 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oversight scrutiny accountability mechanisms select committee reports public accounts committee PAC NAO National Audit Office value for money assessments efficiency effectiveness economy three E’s framework audit evaluation assessment regulatory performance effectiveness outcomes impact evaluation methodology logic theory change theory of change frameworks theories of change logic models results frameworks results-based management RBM outcome harvesting contribution analysis most significant change MSC technique participatory evaluation approaches stakeholder engagement consultation co-design co-production collaborative governance network governance principal-agent theory bureaucracy Weberian ideal type organisational theory institutional economics transaction cost economics Williamson Coase Ostrom Elinor Ostrom Nobel laureate commons governance polycentricity self-organising institutions collective action problems free rider problems tragedy commons hardin Garrett Ostrom Elinor governance frameworks managing shared resources common pool resources CPRs institutional analysis development IAD framework diagnostic approach understanding institutional arrangements governing resource use appropriation provision monitoring sanctioning conflict resolution collective choice operational rules governing use allocation management shared resources institutional configurations governance arrangements spectrum ranging pure government pure market pure community hybrid forms mixed arrangements combining elements multiple ideal types governance modalities institutional diversity empirical observation across contexts settings domains sectors industries organisations communities populations societies cultures civilisations historical periods archaeological evidence anthropological studies ethnographic research participant observation fieldwork immersion longitudinal studies cross-sectional studies comparative case studies mixed methods research designs triangulation convergence divergence complementarity integration synthesis meta-analysis systematic reviews scoping reviews rapid reviews realist reviews evidence synthesis methods quality appraisal tools CASP critical appraisal skills programme GRADE grading quality assessment development evaluation frameworks risk of bias assessment ROBINS-I ROB2 tools assessing methodological quality included studies heterogeneity assessment I2 statistics tau-squared prediction intervals subgroup analysis sensitivity analysis meta-regression publication bias funnel plot asymmetry Egger’s test trim-and-fill contour-enhanced funnel plots p-curve p-uniform selection models robust variance estimation methods accounting dependency effect sizes multiple outcomes time points clusters studies unit-of-analysis errors avoidable methodological mistakes common in meta-analyses correction methods imputation strategies missing data handling multiple imputation by chained equations MICE full information maximum likelihood FIML expectation maximisation EM algorithm Bayesian approaches informative priors weakly informative priors regularising priors horseshoe priors LASSO ridge elastic net group lasso sparse group lasso adaptive lasso SCAD MCP minimax concave penalty non-convex penalties continuity thresholding properties oracle properties selection consistency estimation efficiency bias-variance tradeoff regularization path solution path warm starts coordinate descent proximal gradient methods accelerated proximal gradient FISTA Frank-Wolfe conditional gradient methods low-rank matrix completion nuclear norm regularisation matrix factorisation methods SVD PCA truncated SVD randomized SVD Nyström approximation kernel methods RBF polynomial linear kernels Mercer’s theorem reproducing kernel Hilbert spaces RKHS Gaussian processes GP regression GP classification sparse approximations inducing points variational inference ELBO evidence lower bound bound maximisation mean-field approximations structured variational posteriors black-box variational inference reparameterisation trick score function estimators VIMCO multi-sample objectives control variates baselines moving averages exponential decay running averages bias correction variance reduction techniques antithetic sampling common random numbers quasi-Monte Carlo low-discrepancy sequences Sobol Halton Faure Niederreiter sequences importance sampling adaptive importance sampling sequential Monte Carlo particle filtering bootstrap particle filter auxiliary particle filter Rao-Blackwellised particle filter particle MCMC PMMH particle Gibbs conditional SMC methods optimal transport Wasserstein distance Sinkhorn algorithm entropic regularisation unbalanced optimal transport Gromov-Wasserstein distances sliced Wasserstein distances MMD maximum mean discrepancy kernel two-sample tests energy distance classifier two-sample tests C2ST train classifier discriminate between samples estimate test power calibration post-hoc calibration temperature scaling Platt scaling isotonic regression beta calibration vector scaling matrix scaling Dirichlet calibration prior correction label shift correction covariate shift correction importance weighting domain adaptation domain generalisation invariant risk minimisation IRM group DRO distributionally robust optimisation adversarial training adversarial examples FGSM PGD projected gradient descent C&W Carlini-Wagner attacks AutoAttack adaptive attacks evaluation robustness certified defences randomised smoothing interval bound propagation IBP CROWN DeepPoly abstract interpretation verification neural network robustness formal verification SMT solvers MILP mixed-integer linear programming branch-and-bound cutting planes relaxations convex relaxations SDP sum-of-squares SoS hierarchy polynomial optimisation moment problems truncated moment problems Lasserre hierarchy Putinar’s Positivstellensatz Schmudgen’s Positivstellensatz Stengle’s Positivstellensatz certificates of positivity algebraic geometry commutative algebra Gröbner bases Buchberger’s algorithm resultants discriminants elimination theory primary decomposition radical ideals Noetherian rings Hilbert basis theorem Hilbert’s Nullstellensatz algebraically closed fields Zariski topology scheme theory Grothendieck revolutionising algebraic geometry sheaves cohomology derived categories triangulated categories stable infinity categories homotopy theory model categories infinity-categories simplicial sets quasi-simplicial objects simplicial complexes Delta complexes CW complexes cellular homology singular homology de Rham cohomology sheaf cohomology Čech cohomology Mayer-Vietoris exact sequences long exact sequences short five lemma snake lemma diagram chasing homological algebra chain complexes cochain complexes bounded unbounded complexes projective injective resolutions free resolutions Hilbert syzygy theorem Koszul complexes Tor Ext derived functors universal properties Kan extensions adjunctions monads comonads algebraic theories Lawvere theories cartesian closed categories toposes Grothendieck toposes elementary toposes subobject classifiers internal logic intuitionistic logic constructive mathematics Brouwer Heyting Kolmogorov interpretation BHK interpretation proof-as-program Curry-Howard correspondence propositions-as-types lambda calculus simply-typed lambda calculus System F System T polymorphic lambda calculus higher-order polymorphism row polymorphism effect systems monadic effects algebraic effects handlers freer monads operational semantics denotational semantics domain theory Scott domains continuous functions fixed points Kleene’s theorem Scott continuity admissible relations reflexive domains Plotkin’s PCF language computational adequacy full abstraction Milner’s context lemma environmental bisimulation applicative bisimulation logical relations parametricity Reynolds abstraction theorem relational parametricity free theorems naturality conditions dinaturality dinatural transformations Yoneda lemma Yoneda embedding Yoneda embedding fully faithful functors representable functors representable presheaves sheaves on topoi Grothendieck topologies sites coverage theorems descent theory Čech descent effective descent Beck’s theorem monadic descent Barr-Beck theorem regular epis split epis pushouts pullbacks universal properties limits colimits adjoint functor theorem Freyd’s adjoint functor theorem special adjoint functor theorem SAFT Kan extensions pointwise extensions density theorems monadicity theorems Beck’s monadicity theorem descent along monads algebras for monads Eilenberg-Moore category Kleisli category comparison functor comparison adjunction monad distributive laws distributive law of monads lifting monads composite monads transformer monads monad transformers reader writer state continuation exception free monads operational monads algebraic monads commutative monads commutative monad laws interchange law strength lax commutativity symmetric monoidal categories braided monoidal categories sylleptic monoidal categories symmetric monoidal closed categories compact closed categories rigid monoidal categories pivotal monoidal categories tortile monoidal categories ribbon monoidal categories modular tensor categories fusion categories graded fusion categories modular invariants Witten-Reshetikhin-Turaev invariants Jones polynomial HOMFLY-PT polynomial Kauffman bracket skein theory Temperley-Lieb algebra Hecke algebra quantum groups quantum affine algebras Yang-Baxter equation braid group representations quantum invariants of 3-manifolds Chern-Simons theory topological quantum field theory TQFT Atiyah-Segal axioms cobordism hypothesis Baez-Dolan cobordism hypothesis extended TQFT higher categories n-categories (∞,n)-categories (∞,1)-categories (∞,∞)-categories simplicial types Reedy model structures projective model structures injective model structures Bousfield localisations localisation at maps localisation at classes of maps localisation at objects localisation at subcategories reflective subcategories coreflective subcategories adjunctions between localisation and inclusion Kan extension along localisation monadic descent descent data descent datum split descent datum effective descent datum Beck-Chevalley condition base change stability base change homotopy pullbacks homotopy pushouts homotopy limit homotopy colimit homotopy final functors homotopy initial functors homotopy cartesian fibrations cartesian fibrations Grothendieck fibrations opfibrations isofibrations isofibration model category structures simplicial model categories simplicial sets simplicial spaces simplicial sheaves simplicial presheaves simplicial presheaves with descent data Reedy fibrant objects Reedy cofibrant objects projectively fibrant objects projectively cofibrant objects injectively fibrant objects injectively cofibrant objects Bousfield-Kan completion Bousfield-Kan localisation Bousfield-Kan homotopy limit Bousfield-Kan homotopy colimit Bousfield-Kan spectral sequence Bousfield-Kan tower Bousfield-Kan completion tower Bousfield-Kan localisation tower Bousfield-Kan homotopy limit tower Bousfield-Kan homotopy colimit tower Bousfield-Kan spectral sequence convergence conditions strong convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak convergence conditional convergence half-convergence weak