Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches...
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Sketches in Higher Category Theories and the Homotopy Hypothesis R´ emy Tuy´ eras Macquarie University August 8th, 2016 Sketches in Higher Category Theories and the H.H. R´ emy Tuy´ eras 1
Transcript of Sketches in Higher Category Theories and the Homotopy ...tuyeras/th/CT2016_slides.pdf · Sketches...
Sketches in Higher Category Theories and the HomotopyHypothesis
Remy Tuyeras
Macquarie University
August 8th, 2016
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 1
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 2
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 3
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 4
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 5
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 6
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 7
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 8
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 9
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 10
Models and Theories
All mathematical theories are born from inductive reasonings:
Observation ⇒ Pattern ⇒ Hypothesis ⇒ Theory
Once a theory is obtained, one then uses deductive reasonings to explainother observations:
Theory ⇒ Hypothesis ⇒ Observation ⇒ Confirmation
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 11
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 12
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 13
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 14
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 15
Refinement of Theories
When a theory does not imply an observation ...
Theory ��ZZ⇒ Confirmation
... One usually decides to refine the theory by adding more axioms ...
Possible Theories$,
dl Covered Examples
... One thus gets a back-and-forth between
. the poset of theories (ordered by the level of abstraction)
. the poset of classes of Examples (ordered by the inclusion)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 16
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 17
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 18
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 19
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 20
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 21
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 22
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right
^(•$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 23
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right
^(•$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 24
Refinement of Theories
Informally, we have an adjunction of order-preserving relations
Possible Theories
Right
^(•$,
dlLeft
Covered Examples
For the Examples: We want to find a greastest fixed point
For the Theories: We want to find the right level of abstraction
... the refinement of a theory can be done in two ways:
. Either one piles up axioms, and we get a long list of properties• •_
. Or one can starts over the inductive process with the hope of finding
a better set of axioms• •^
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 25
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 26
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 27
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 28
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Properness properties;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 29
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 30
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 31
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 32
Quillen’s Model Categories
For instance, in 1967, D. Quillen came up with a set of axioms for theconcept of model category on the base of a particular set of examples.
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom
But, using the concept of model category is generally not enough toretrieve the combinatorial properties of these examples; e.g.
. Left or right Properness;
. The idea of smallness;
. The concept of CW-complex;
. The Whitehead theorem ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 33
Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
Induction (Left)'/
em
Deduction (Right)
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 34
Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 35
Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 36
Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 37
Cofibrantly Generated Model Categories
So, we add more axioms on the theoretical side (without losing too manyexamples though)
sSetTopChR
. . .
((((((hhhhhhInduction (Left)
'/em
Deduction (Right)
^(•
(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
This need to further restrict our theoretical setting suggests that we havenot reached the right level of abstraction ...
... and thus have failed to see the existence of a more fundamentalsetting...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 38
A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 39
A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 40
A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 41
A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 42
A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 43
A More Fundamental Setting
For instance, in cofibrantly generated model categories, we distinguishbetween
- the genuine structure (i.e. the given classes of arrows);
- the generating structure (i.e. some sets of cofibrations);
But the genuine structure is entirely determined by the generatingstructure via lifting properties:
S
��
∀ // X
f��
D∀//
∃??
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 44
A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
∀// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 45
A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
∀// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 46
A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
∀// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 47
A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 48
A More Fundamental Setting
Similarly, in tractable model categories, we can characterise the weakequivalences of the model category from the generating structure ...
These can be characterised by using a lifting property with respect tocommutative squares of generating structure as follows:
S
!!
��
∀ // X
��
D1
∃
>>
��
D2
!!
// Y
D′∃
>>
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 49
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 50
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 51
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 52
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 53
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 54
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 55
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 56
Generating Structure = Internal Logic
What really happens is that these generating arrows form an internallogic of the model category.
... and as in every logic, the commutativity relations satisfied by thesegenerating arrows form the axioms of the internal logic...
Long list of Axioms(a) 2/3-property(b) Lifting properties(c) Weak factorisations(d) Retract axiom(e) Set of cofibrations(f) . . .
Deduction +3
Internal LogicGenerating arrowsCommutative squaresCommutativity relations. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 57
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 58
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 59
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 60
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 61
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 62
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 63
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 64
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 65
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 66
Opposite Point of View
So, to avoid the Long List of Axioms ...
... we could take inspiration on these internal logics and take theirassociated internal language as a starting point for a brand newHomotopy Theory ...
Internal LogicSet of arrows. . .
Deduction +3
Homotopy Theories. Model categories. Fibrant categories. Homotopical categories. . . .
But what kind of axioms should we set for this new Homotopy logic?
...well, this will be determined by Induction from the world of Examples.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 67
Loading .
... After looking into how the internal logics work ...
... we notice that a very elementary structure appears constantly ...
... in any formulation of the internal axioms ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 68
Loading ..
... After looking into how the internal logics work ...
... we notice that a very elementary structure appears constantly ...
... in any formulation of the internal axioms ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 69
Loading ...
... After looking into how the internal logics work ...
... we notice that a very elementary structure appears constantly ...
... in any formulation of the internal axioms ...
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 70
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 71
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 72
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 73
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 74
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 75
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 76
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 77
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 78
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 79
Elementary Structure for Homotopy Theory
This elementary structure is made of :
. a pushout square;
. and an additional arrow as follows.
• //
��
x
•
��• // • // •
This diagram encodes nothing but the internal logic of a cell or disc:
• &&88⇓ •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 80
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 81
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 82
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
������• // • //// •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 83
Elementary Structure for Homotopy Theory
We can also retrieve the representatives of the so-called generatingstructure seen previously as follows:
. generating cofibrations;
. generating trivial cofibrations;
. generating squares;
• //
��x•
����• //
88• // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 84
Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 85
Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 86
Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 87
Examples
In the category of topological spaces Top:
Sn−1 //
��
x
Dn
��
Dn // Sn // Dn+1
In the category of simplicial sets sSet:
∂∆n−1//
��
xΛkn
��
∆n−1// ∂∆n
// ∆n
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 88
Examples
In the topos Sh(C∞Ringop):
{0} //
��
D
��
D // D(2) // D
... many other examples exist.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 89
Examples
In the topos Sh(C∞Ringop):
{0} //
��
D
��
D // D(2) // D
... many other examples exist.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 90
Examples
In the topos Sh(C∞Ringop):
{0} //
��
D
��
D // D(2) // D
... many other examples exist.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 91
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 92
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 93
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 94
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 95
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
• //
��x
•
��• // • // •
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 96
A Mini-Homotopy Theory
By taking inspiration on the (previously seen) characterisations of
. fibrations;
. trivial fibrations;
. weak equivalences;
we can define a mini-homotopy theory for each structure of the form
S //
��x
D1
��
D2// S′ // D′
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 97
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a trivial fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 98
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a trivial fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 99
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 100
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 101
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 102
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 103
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 104
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 105
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a trivial fibration if •
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 106
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ �Z− // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 107
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 108
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 109
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
∀// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 110
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 111
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 112
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 113
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 114
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 115
A Mini-Homotopy Theory
So, for every structure S //
��
x D1
��
D2// S′ // D′
we say that a morphismf : X → Y is ...
a fibration if D1
��
∀ // X
f��
D′∀//
∃>>
Y
a����XXXXtrivial fibration ifa pseudo fibration
•
��
∀ // X
f��
•∀//
∃??
Y
a surtraction if S!!
��
∀ // X
��
D1∃
==
��
D2
!!
// Y
D′ ∃
>>
an intraction if S′
��
∀ // X
f��
D′∀//
∃>>
Y
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 116
A Mini-Homotopy Theory
weak equivalence = surtraction + intraction
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 117
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 118
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 119
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 120
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 121
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 122
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 123
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 124
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 125
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 126
A Mini-Homotopy Theory
And now, let the magic happen... It follows from the definitions that:
isomorphisms are intractions;
f ◦ g intraction ⇒ g intraction;
f and g intractions ⇒ f ◦ g intraction;
f ◦ g surtraction and f intraction ⇒ g surtraction;
isomorphisms are * fibrations;
* fibrations are stable under pullbacks;
* fibrations/surtractions/intractions are stable under retracts;
* = pseudo / trivial / ∅. . .
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 127
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 128
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 129
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 130
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 131
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 132
A Mini-Homotopy Theory
trivial fibrations ⊆ pseudo fibrations
fibrations ∩ surtractions ⊆ pseudo fibrations
trivial fibrations ⊆ fibrations ∩ intractions
The last two items remind of:
fibrations ∩weak equivalence = trivial fibrations
But we cannot really prove more if we do not require more...
It is possible to obtain more properties if we add give some axioms to ourinternal logic.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 133
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 134
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 135
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 136
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��⊆
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 137
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations��
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 138
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 139
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 140
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 141
Axioms
Recall that S //
��
x D1
��
D2// S′ // D′
= •##
;;⇓ • so if we encode:
. an identity cell • ..00}� • we obtain:
If r ◦ i = idX where r is an intraction, then i is a weak equivalence.
fibrations ∩weak equivalences�
&&
oo ? _ trivial fibrations
pseudofibrations
. a composition of cells ⇓•
��
EE// •⇓
we obtain:
weak equivalences are stable under composition.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 142
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 143
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 144
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 145
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 146
Axioms
Also, to be able to finish proving the 2/3-property, we need to “draw”more complex axioms than identities or compositions
In fact we need to draw and fully understand the internal logic of highercoherence theory
Basically, the main operations that I define are:
whiskerings • // •""
<<⇓ • // • in all dimensions;
a very weak notion of invertible cell •∼= // • in all dimensions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 147
How do we encode dimensions?
... and we need to extend the elementary structures S //
��
D1��
D2// S′ // D′
to
longer ones made out of the elementary ones:
S0
��
//
x
D01
��
D02
// S1 . . .Sn //
��
x
Dn1
��
Dn2
// Sn+1// D′
... this encodes the dimension of a cell!Sketches in Higher Category Theories and the H.H. Remy Tuyeras 148
How do we encode dimensions?
... and we need to extend the elementary structures S //
��
D1��
D2// S′ // D′
to
longer ones made out of the elementary ones:
S0
��
//
x
D01
��
D02
// S1 . . .Sn //
��
x
Dn1
��
Dn2
// Sn+1// D′
... this encodes the dimension of a cell!Sketches in Higher Category Theories and the H.H. Remy Tuyeras 149
How do we encode dimensions?
... and we need to extend the elementary structures S //
��
D1��
D2// S′ // D′
to
longer ones made out of the elementary ones:
S0
��
//
x
D01
��
D02
// S1 . . .Sn //
��
x
Dn1
��
Dn2
// Sn+1// D′
... this encodes the dimension of a cell!Sketches in Higher Category Theories and the H.H. Remy Tuyeras 150
Vertebrae & Spines
I call every elementary structure · //
��
x
·��
· // · // ·
a vertebra and I call every
structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 151
Vertebrae & Spines
I call every elementary structure · //
��
x
·��
· // · // ·
a vertebra and I call every
structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 152
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure•��
//
x
•��
• // • //
��
x
•��
· // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 153
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure•��
//
x
•��
• // • //
��
x
·��
• // ·��
//
x
·��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 154
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // • //
��
x
•��
• // •��
//
x
•��
· // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 155
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // • //
��
x
•��
• // •��
//
x
·��
• // · . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 156
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // · //
��
x
·��
· // •��
//
x
•��
• // • . . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 157
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // ·. . .· //
��
x
·��
· // · // ·
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 158
Vertebrae & Spines
I call every elementary structure • //
��
x
•��
• // • // •
a vertebra and I call
every structure·��
//
x
·��
· // · //
��
x
·��
· // ·��
//
x
·��
· // · . . .• //
��
x
•��
• // • // •
a spine (which is made of vertebrae)
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 159
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 160
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 161
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 162
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 163
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 164
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 165
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 166
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 167
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 168
Spinal Categories
If you now take
a category C;
a bunch of vertebrae in C equipped with spine structures;
+
the axioms I previously described on the vertebrae and spines, we get thenotion of
SPINAL CATEGORY
equipped with three classes of morphisms (defined vertebra-wise)
weak equivalences;
fibrations trivial cofibrations;
trivial fibrations cofibrations;
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 169
Spinal Categories Model Categories
“acyclic fibrations” = cofibrations + weak equivalences
Theorem
If pushouts of acyclic cofibrations are weak equivalences (and somesmallness condition holds), then a spinal category is a model category.
The proof does not even need the Jeff Smith’s criteria as all the desiredproperties come from my definitions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 170
Spinal Categories Model Categories
“acyclic fibrations” = cofibrations + weak equivalences
Theorem
If pushouts of acyclic cofibrations are weak equivalences (and somesmallness condition holds), then a spinal category is a model category.
The proof does not even need the Jeff Smith’s criteria as all the desiredproperties come from my definitions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 170
Spinal Categories Model Categories
“acyclic fibrations” = cofibrations + weak equivalences
Theorem
If pushouts of acyclic cofibrations are weak equivalences (and somesmallness condition holds), then a spinal category is a model category.
The proof does not even need the Jeff Smith’s criteria as all the desiredproperties come from my definitions.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 170
More General Structures
... more general structures are sometimes needed ...
Local (e.g. stacks) non-algebraic algebraic
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 171
More General Structures
... more general structures are sometimes needed ...
Local (e.g. stacks) non-algebraic algebraic
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 172
More General Structures
... more general structures are sometimes needed ...
Local (e.g. stacks) non-algebraic algebraic
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Sketches in Higher Category Theories and the H.H. Remy Tuyeras 173
Homotopy Hypothesis
Can we apply our theory to the category of Grothendieck’s ∞-groupoids∞-Grp?
It follows from the definition of a Grothendieck’s ∞-groupoid that any pairof parallel arrows in ∞-Grp ...
Dn+1∃
!!
Dn
f //
g//
sn
OO
tn
OO
B
...can be filled by a homotopy.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 174
Homotopy Hypothesis
Can we apply our theory to the category of Grothendieck’s ∞-groupoids∞-Grp?
It follows from the definition of a Grothendieck’s ∞-groupoid that any pairof parallel arrows in ∞-Grp ...
Dn+1∃
!!
Dn
f //
g//
sn
OO
tn
OO
B
...can be filled by a homotopy.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 175
Homotopy Hypothesis
Can we apply our theory to the category of Grothendieck’s ∞-groupoids∞-Grp?
It follows from the definition of a Grothendieck’s ∞-groupoid that any pairof parallel arrows in ∞-Grp ...
Dn+1∃
!!
Dn
f //
g//
sn
OO
tn
OO
B
...can be filled by a homotopy.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 176
Homotopy Hypothesis
Equivalently, this amount to saying that ...
Sn−1
x
γn//
γn
��
Dn
δn1��
g
$$Dn δn2
//
f
44Sn //γn+1
// Dn+1// B
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 177
Homotopy Hypothesis
Equivalently, this amount to saying that ...
Sn−1
x
γn//
γn
��
Dn
δn1��
g
$$Dn δn2
//
f
44Sn //Dn+1 B
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 178
Homotopy Hypothesis
Equivalently, this amount to saying that ...
Sn−1
x
γn//
γn
��
Dn
δn1��
g
$$Dn δn2
//
f
44Snγn+1
// Dn+1// B
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 179
Homotopy Hypothesis
On the one hand, the operations on ∞-groupoids come from theexistence of a certain homotopy filling of a certain pair of parallelarrows going to a certain globular sum.
On the other hand, all the operations defined on vertebrae and spinescome from existence of a certain vertebra filling a certain givencommutative square of parallel arrows going to a certain globular sum.
Theorem
The category of Grothendieck’s ∞-groupoids admits a spinal structure.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 180
Homotopy Hypothesis
On the one hand, the operations on ∞-groupoids come from theexistence of a certain homotopy filling of a certain pair of parallelarrows going to a certain globular sum.
On the other hand, all the operations defined on vertebrae and spinescome from existence of a certain vertebra filling a certain givencommutative square of parallel arrows going to a certain globular sum.
Theorem
The category of Grothendieck’s ∞-groupoids admits a spinal structure.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 180
Homotopy Hypothesis
On the one hand, the operations on ∞-groupoids come from theexistence of a certain homotopy filling of a certain pair of parallelarrows going to a certain globular sum.
On the other hand, all the operations defined on vertebrae and spinescome from existence of a certain vertebra filling a certain givencommutative square of parallel arrows going to a certain globular sum.
Theorem
The category of Grothendieck’s ∞-groupoids admits a spinal structure.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 180
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
Conclusion
Key points of this talk:
internal logic of homotopy theory Spinal categories;
Category of Grothendieck’s ∞-groupoids admits a spinal structure;
What remains to be proven:
pushouts of acyclic cofibrations in ∞-Grp are weak equivalences;
Consider the canonical adjunction
TopU
//
oo F
⊥ ∞-Grp
For every X such that there exists some Y for which∞-Grp(X ,U(Y )) 6= ∅, the unit X ⇒ FU(X ) is a component-wiseweak equivalence.
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 181
The End
Thank you!
References:
[1] R. Tuyeras, Sketches in Higher Category Theory, version 1.01,http://www.normalesup.org/~tuyeras/th/thesis.html.
[2] R. Tuyeras, Elimination of quotients in various localisation of modelsinto premodels, arXiv:1511.09332
Sketches in Higher Category Theories and the H.H. Remy Tuyeras 182
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