in the last semester I learned homological algebra and higher category theory\homotopy theory

but I am kind of confused when I try to realy understate the link between the two subject ( this is really not my comfort zone ...)

Therefore I try to write(a kind of self exersice) a text on homological algebra and homotopy theory but realy introduce from $0$ the two subjets

I would like to introduce the following concepts in homological algebra  :

1. chain complex

1$\frac{1}{2}$.grothendieck group

2. homotopy of complex

3. derived category

4. t-structures

And also I would like to introduce the following concepts in homotopy theory  :

1. Model categories

2. Homotopy category of a model category

3. Derivation in the setting of model categories

4. Quasi categories

4.5. simplicial object in a category and homotopy in this context

5. Dold kan equivalence

Now the "hard" part start  :

how to organize this concept in a good way? for 1-3 (either in homology\homotopy) I think that I know how to do that but for 3-5 especially in homotopy I dont have any idea ...

This gives rise to my questions  :

1. how to give motivation infinite category or with more generallity homotopy theory\higher category theory but from a homological point of view I read somewhere a maybe good idea  :

For an abelian category $\mathcal{A}$

  the derived category $\mathcal{D(A)}$ is not defined by a universal property

I read somewhere that in some sense higher category theory resolve the problem,Okay but why  ? and do we need quasi categories or model categories will be sufficient for doing that?

2. and if someone have some idea to organize this text I open to any suggestion

I will be grateful if someone could give me some clues for doing this self exercise  .

In the last semester I learned homological algebra and higher category theory/homotopy theory.

But I am kind of confused when I try to really understand the link between the two subjects (this is really not my comfort zone ...)

Therefore I try to write  (a kind of self-exercise) a text on homological algebra and homotopy theory but really introduce from $0$ the two subjects.

I would like to introduce the following concepts in homological algebra:

1. chain complex

1$\frac{1}{2}$. Grothendieck group

2. homotopy of a complex

3. derived category

4. t-structures

And also I would like to introduce the following concepts in homotopy theory:

1. Model categories

2. Homotopy category of a model category

3. Derivation in the setting of model categories

4. Quasi-categories

4.5. simplicial object in a category and homotopy in this context

5. Dold-Kan equivalence

Now the "hard" part start:

How to organize these concepts in a good way? For 1-3 (either in homology/homotopy) I think that I know how to do that but for 3-5 especially in homotopy I don't have any idea ...

This gives rise to my questions:

1. How to motivate infinity categories, or more generally homotopy theory/higher category theory but from a homological point of view. I read somewhere a maybe good idea:

For an abelian category $\mathcal{A}$, the derived category $\mathcal{D(A)}$ is not defined by a universal property.

I read somewhere that in some sense higher category theory resolves the problem. Okay but why? And, do we need quasi categories, or would model categories be sufficient for doing that?

2. If someone have some idea to organize this text I open to any suggestion.

I will be grateful if someone could give me some clues for doing this self exercise.

in the last semester I learn homological algebra and higher category theory\homotopy theory

but I am kind of confused when I try to realy understate the link between the two subject ( this is really not my comfort zone ...)

Therefore I try to write(a kind of self exersice) a text on homological algebra and homotopy theory but realy introduce from $0$ the two subjets

I would like to introduce the following concepts in homological algebra :

1. chain complex

1$\frac{1}{2}$.grothendieck group

2. homotopy of complex

3. derived category

4. t-structures

And also I would like to introduce the following concepts in homotopy theory :

1. Model categories

2. Homotopy category of a model category

3. Derivation in the setting of model categories

4. Quasi categories

4.5. simplicial object in a category and homotopy in this context

5. Dold kan equivalence

Now the "hard" part start :

how to organize this concept in a good way? for 1-3 (either in homology\homotopy) I think that I know how to do that but for 3-5 especially in homotopy I dont have any idea ...

This gives rise to my questions :

1. how to give motivation infinite category or with more generallity homotopy theory\higher category theory but from a homological point of view I read somewhere a maybe good idea :

For an abelian category $\mathcal{A}$

the derived category $\mathcal{D(A)}$ is not defined by a universal property

I read somewhere that in some sense higher category theory resolve the problem,Okay but why ? and do we need quasi categories or model categories will be sufficient for doing that?

2. and if someone have some idea to organize this text I open to any suggestion

I will be grateful if someone could give me some clues for doing this self exercise .

in the last semester I learned homological algebra and higher category theory\homotopy theory

but I am kind of confused when I try to realy understate the link between the two subject ( this is really not my comfort zone ...)

Therefore I try to write(a kind of self exersice) a text on homological algebra and homotopy theory but realy introduce from $0$ the two subjets

I would like to introduce the following concepts in homological algebra :

1. chain complex

1$\frac{1}{2}$.grothendieck group

2. homotopy of complex

3. derived category

4. t-structures

And also I would like to introduce the following concepts in homotopy theory :

1. Model categories

2. Homotopy category of a model category

3. Derivation in the setting of model categories

4. Quasi categories

4.5. simplicial object in a category and homotopy in this context

5. Dold kan equivalence

Now the "hard" part start :

how to organize this concept in a good way? for 1-3 (either in homology\homotopy) I think that I know how to do that but for 3-5 especially in homotopy I dont have any idea ...

This gives rise to my questions :

1. how to give motivation infinite category or with more generallity homotopy theory\higher category theory but from a homological point of view I read somewhere a maybe good idea :

For an abelian category $\mathcal{A}$

the derived category $\mathcal{D(A)}$ is not defined by a universal property

I read somewhere that in some sense higher category theory resolve the problem,Okay but why ? and do we need quasi categories or model categories will be sufficient for doing that?

2. and if someone have some idea to organize this text I open to any suggestion

I will be grateful if someone could give me some clues for doing this self exercise .

in the last semester I learn homological algebra and higher category theory\homotopy theory

but I am kind of confused when I try to realy understate the link between the two subject ( this is really not my comfort zone ...)

Therefore I try to write(a kind of self exersice) a text on homological algebra and homotopy theory but starting at a low level (accesible for students which do basis courses in algebaic topology , commutative algebra and category theory )

I would like to introduce the following concepts in homological algebra :

1. chain complex

1$\frac{1}{2}$.grothendieck group

2. homotopy of complex

3. derived category

4. t-structures

And also I would like to introduce the following concepts in homotopy theory :

1. Model categories

2. Homotopy category of a model category

3. Derivation in the setting of model categories

4. Quasi categories

4.5. simplicial object in a category and homotopy in this context

5. Dold kan equivalence

Now the "hard" part start :

how to organize this concept in a good way? for 1-3 (either in homology\homotopy) I think that I know how to do that but for 3-5 especially in homotopy I dont have any idea ...

This gives rise to my questions :

1. how to give motivation infinite category or with more generallity homotopy theory\higher category theory but from a homological point of view I read somewhere a maybe good idea :

For an abelian category $\mathcal{A}$

the derived category $\mathcal{D(A)}$ is not defined by a universal property

I read somewhere that in some sense higher category theory resolve the problem,Okay but why ? and do we need quasi categories or model categories will be sufficient for doing that?

2. and if someone have some idea to organize this text I open to any suggestion

I will be grateful if someone could give me some clues for doing this self exercise .

in the last semester I learn homological algebra and higher category theory\homotopy theory

but I am kind of confused when I try to realy understate the link between the two subject ( this is really not my comfort zone ...)

Therefore I try to write(a kind of self exersice) a text on homological algebra and homotopy theory but realy introduce from $0$ the two subjets

I would like to introduce the following concepts in homological algebra :

1. chain complex

1$\frac{1}{2}$.grothendieck group

2. homotopy of complex

3. derived category

4. t-structures

And also I would like to introduce the following concepts in homotopy theory :

1. Model categories

2. Homotopy category of a model category

3. Derivation in the setting of model categories

4. Quasi categories

4.5. simplicial object in a category and homotopy in this context

5. Dold kan equivalence

Now the "hard" part start :

how to organize this concept in a good way? for 1-3 (either in homology\homotopy) I think that I know how to do that but for 3-5 especially in homotopy I dont have any idea ...

This gives rise to my questions :

1. how to give motivation infinite category or with more generallity homotopy theory\higher category theory but from a homological point of view I read somewhere a maybe good idea :

For an abelian category $\mathcal{A}$

the derived category $\mathcal{D(A)}$ is not defined by a universal property

I read somewhere that in some sense higher category theory resolve the problem,Okay but why ? and do we need quasi categories or model categories will be sufficient for doing that?

2. and if someone have some idea to organize this text I open to any suggestion

I will be grateful if someone could give me some clues for doing this self exercise .