Accessible model structures

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Damian Hamilton
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1 École Polytechnique Fédérale de Lausanne Joint work with K. Hess, E. Riehl and B. Shipley September 13, 2016

2 Plan

3 Model structure Additional structure on a category: allows one to do homotopy theory

4 Model structure Additional structure on a category: allows one to do homotopy theory A model category (M,F,C,W): bicomplete M a class W + 2-of-3 a pair of weak factorization systems (C W,F) (C,F W)

5 There are different ways to put a model structure on a category D.

6 There are different ways to put a model structure on a category D. choose your clasess of maps + prove the axioms (good luck with that)

7 There are different ways to put a model structure on a category D. choose your clasess of maps + prove the axioms (good luck with that) find a different category M with a model structure and try to use it (transfer it): M L D

8 Structured objects C L Objects with additional structure in C

9 Structured objects C For example: L C Objects with additional structure in C L R mod in C

10 Structured objects C For example: L C Objects with additional structure in C L R mod in C There are classical theorems [Kan, Quillen]: Model structure on C = model structure on R mod in C Important assumption: C cofibrantly generated m.c.

11 Structured objects, dual situation Dual situation is much more difficult: C Objects with additional structure in C

12 Structured objects, dual situation Dual situation is much more difficult: C For example: C Objects with additional structure in C R comod in C

13 Structured objects, dual situation Dual situation is much more difficult: C For example: C Objects with additional structure in C R comod in C Recent theorems based on [Makkai, Rosický]. Important assumption: C cofibrantly generated model category and locally presentable

14 Structured objects, dual situation Dual situation is much more difficult: C For example: C Objects with additional structure in C R comod in C Recent theorems based on [Makkai, Rosický]. Important assumption: C cofibrantly generated model category and locally presentable Even more difficult: several adjoint pairs from C to Objects with additional structure in C (Bialgebras in C)

15 Structured objects, dual situation Dual situation is much more difficult: C For example: C Objects with additional structure in C R comod in C Recent theorems based on [Makkai, Rosický]. Important assumption: C cofibrantly generated model category and locally presentable Even more difficult: several adjoint pairs from C to Objects with additional structure in C (Bialgebras in C) We will relax assumption of cofibrantly generated but keep locally presentable

16 Accessible model structure Locally presentable category - built from a set of objects via colimits (ssets, Ch R, Sp Σ (ssets) but not Top)

17 Accessible model structure Locally presentable category - built from a set of objects via colimits (ssets, Ch R, Sp Σ (ssets) but not Top) Definition [HKRS] A model structure on a locally presentable category is accessible if it is algebraic and both (functorial) factorisations preserve sufficiently large filtered colimits

18 Accessible model structure Locally presentable category - built from a set of objects via colimits (ssets, Ch R, Sp Σ (ssets) but not Top) Definition [HKRS] A model structure on a locally presentable category is accessible if it is algebraic and both (functorial) factorisations preserve sufficiently large filtered colimits IMPORTANT: All cofibrantly generated model structures on locally presentable categories are accessible

19 Accessible model structure Locally presentable category - built from a set of objects via colimits (ssets, Ch R, Sp Σ (ssets) but not Top) Definition [HKRS] A model structure on a locally presentable category is accessible if it is algebraic and both (functorial) factorisations preserve sufficiently large filtered colimits IMPORTANT: All cofibrantly generated model structures on locally presentable categories are accessible Methods to obtain an accessible model structure: via enriched algebraic small object argument [Riehl]

20 An example Hurewicz model structure on Ch R : W = chain homotopy equivalences C = levelwise split mono F = levelwise split epi Not cofibrantly generated, but accessible.

21 Transfer (M,F,C,W) model category K M L C C and K are bicomplete.

22 Transfer (M,F,C,W) model category K M L C C and K are bicomplete. If they exist: right-induced model structure on C: ( 1 F,LLP, 1 W ), left-induced model structure on K: ( RLP, 1 C, 1 W ).

23 Implications of Burke s and Garner s theorem Suppose (M,F,C,W) is an accessible model category, C and K are locally presentable, and K M L C

24 Implications of Burke s and Garner s theorem Suppose (M,F,C,W) is an accessible model category, C and K are locally presentable, and K M L C Then there exist weak factorization systems: On K: ( 1 (C),RLP) and ( 1 (C W),RLP) On C: (LLP, 1 (F)) and (LLP, 1 (F W))

25 Acyclicity theorem [HKRS] Suppose (M,F,C,W) is an accessible model category, C and K are locally presentable, and K M L C. The right-induced model structure exists on C LLP( 1 F) 1 W The left-induced model structure exists on K RLP( 1 C) 1 W

26 Acyclicity theorem [HKRS] Suppose (M,F,C,W) is an accessible model category, C and K are locally presentable, and K M L C. The right-induced model structure exists on C LLP( 1 F) 1 W The left-induced model structure exists on K RLP( 1 C) 1 W These are again accessible model structures, we can repeat this process! Even cofibrantly generated if M was!

27 : 2-out of-6 Dual to Quillen s path object argument Square Theorem

28 2-out of-6 W in any model category satisfy 2-out of-6: A f B g C h D if gf and hg W = f,g,h, and hgf W

29 Example: right induced to Alg R To check: LLP( 1 F) 1 W L Ch R,Hur Alg R

30 Example: right induced to Alg R To check: LLP( 1 F) 1 W i : A B LLP( 1 F) L Ch R,Hur Alg R

31 Example: right induced to Alg R To check: LLP( 1 F) 1 W i : A B LLP( 1 F) L Ch R,Hur Alg R i A B 1W A F (B s 1 B) B i+q 1 F

32 Example: right induced to Alg R To check: LLP( 1 F) 1 W i : A B LLP( 1 F) L Ch R,Hur Alg R i A B 1W A F (B s 1 B) B i+q 1 F

33 Example: right induced to Alg R To check: LLP( 1 F) 1 W i : A B LLP( 1 F) L Ch R,Hur Alg R A 1W A F (B s 1 B) i B 2-of-6 = i 1 W B i+q 1 F = the right-induced model structure on Alg R exists

34 Dual to Quillen s path object argument [HKRS] (M,C,F,W) an accessible model category K locally presentable and K M Define W K := 1 (W) and C K := 1 (C).

35 Dual to Quillen s path object argument [HKRS] (M,C,F,W) an accessible model category K locally presentable and K M Define W K := 1 (W) and C K := 1 (C). If 1 every object in K is cofibrant and 2 good cylinder objects X X Cyl(X ) X exist in K for all objects X K, then the AC holds on K = the left-induced model structure exists

36 Example: left induced to Coalg R Coalg R Ch R,Hur

37 Example: left induced to Coalg R Coalg R Ch R,Hur 1 In Ch R,Hur every object is cofibrant = in Coalg R too 2 We can use cylinder object in Ch R to get cylinder object in Coalg R = the left-induced model structure on Coalg R exists

38 Square Theorem [HKRS] M,C,K,P locally presentable, (M,C,F,W) a model category M L C K L P If = and if L = L, then there exist id P right P left. id

39 Square Theorem [HKRS] M,C,K,P locally presentable, (M,C,F,W) a model category M L C right K left L P

40 Square Theorem [HKRS] M,C,K,P locally presentable, (M,C,F,W) a model category M L C right K left L P If = and if L = L, then there exist id P right id P left

41 Square Theorem [HKRS] M,C,K,P locally presentable, (M,C,F,W) a model category M L C right K left L P If = and if L = L, then there exist id P right id P left We have a more general version of Square Theorem.

42 Example: Bialg R We apply Square Theorem to the square Ch R,Hur Coalg R F F Alg R Bialg R = Two model structures on Bialg R Two different m.s. Fibrant/ cofibrant replacements given by Bar-Cobar and Cobar-Bar constructions.

43 Example: Bialg R We apply Square Theorem to the square Ch R,Hur Coalg R,left F F Alg R,right Bialg R

44 Example: Bialg R We apply Square Theorem to the square Ch R,Hur Coalg R,left F F Alg R,right Bialg R = Two model structures on Bialg R

45 Example: Bialg R We apply Square Theorem to the square Ch R,Hur Coalg R,left F F Alg R,right Bialg R = Two model structures on Bialg R Two different model structures. Fibrant / cofibrant replacements given by Bar-Cobar and Cobar-Bar constructions.

46 Motivic Galois extensions We get model structures on categories of Equivariant motivic spaces : spre(g Sch) and Equivariant motivic spectra: Sp Σ spre(gsch)

47 Motivic Galois extensions We get model structures on categories of and Equivariant motivic spaces : spre(g Sch) Equivariant motivic spectra: Sp Σ spre(gsch) where fibrant replacement of X is Hom(EG +,X ) or Hom(E G +,X ) needed for Galois extension examples (Joint work with A.Beaudry, K.Hess, M.Merling and.stojanoska).

48 (Generalized) Reedy R a Reedy category R + a direct wide subcategory R an inverse wide subcategory M obr R M R L L M R+ R M R

49 (Generalized) Reedy R a Reedy category R + a direct wide subcategory R an inverse wide subcategory M obr R M R L L M R+ R M R Reedy model structure on M R is left-right induced and right-left induced from levelwise model structure on M obr

50 To sum up accessible model structures are great!:

51 To sum up accessible model structures are great!: it is relatively easy to transfer them via left or right adjoints repeatedly.

52 To sum up accessible model structures are great!: it is relatively easy to transfer them via left or right adjoints repeatedly. All you need to check is the acyclicity condition

53 To sum up accessible model structures are great!: it is relatively easy to transfer them via left or right adjoints repeatedly. All you need to check is the acyclicity condition and there are methods to do it!

54 To sum up accessible model structures are great!: it is relatively easy to transfer them via left or right adjoints repeatedly. All you need to check is the acyclicity condition and there are methods to do it! We get many new model structures both in dg and spectral settings... and

55 To sum up accessible model structures are great!: it is relatively easy to transfer them via left or right adjoints repeatedly. All you need to check is the acyclicity condition and there are methods to do it! We get many new model structures both in dg and spectral settings... and We can understand better existing model structures: a Reedy model structure.

Algebraic model structures

Algebraic model structures Emily Riehl Harvard University http://www.math.harvard.edu/~eriehl 18 September, 2011 Homotopy Theory and Its Applications AWM Anniversary Conference ICERM Emily Riehl (Harvard