nLab Diary on Higher Category Theory and its Applications

Why these Notes

Unlike notes where one usually incorporates structure and some level of self-containment, the following write-up could be seen more like a diary and a means to guide one’s learning journey towards some more sophisticated references. In the cases where good references are lacking, we will aim to provide full detail or accumulate material that was hard to find (or stuff that is never spelled out in other references). In that way, we hope this will be of value to someone in the future.

Straightening & Unstraightening

For other Nlab entries see: (infinity,1)-Grothendieck construction, straightening functor, Grothendieck construction, …

Idea

Mathematical objects very often come in families. One quick example is the following: Let $\text{Ab}$ denote the category of abelian groups, and fix a commutative ring $R$. Then the there is a well defined category

$$\text{Mod}_R \coloneqq \text{Mod}_R(\text{Ab})$$

of $R$-modules (see also module object). Now every homomorphism of commutative rings $\varphi \colon R \to S$ induces two functors

$$\varphi^\star \colon \text{Mod}_S \to \text{Mod}_R, \qquad \varphi_! \colon \text{Mod}_R \to \text{Mod}_S$$

which are referred to as restriction of scalars and extension of scalars, respectively. It turns out that both these assignments are functorial: We have two functors

It is often more convenient to encode the functoriality of the (covariant or contravariant) construction $R \mapsto \text{Mod}_R$ in a different way. In fact, to every functor $U \colon \mathcal{C} \to \text{Cat}$, one can associate a new category $\text{Un}(U)$, referred to as the category of elements, the Grothendieck construction, or the unstraightening? of the functor $U$, which comes endowed with a “bundle map”

$$\text{Un}(U) \to \mathcal{C}$$

More generally, for an $\infty$-category $\mathcal{C}$ there is an (adjoint) equivalence of $\infty$-categories

Here by “nice” bundle maps we really mean cocartesian fibrations (in particular, there is another equivalence which encodes contravariant functors with domain $\mathcal{C}$ and target $\infty\text{Cat}$). This equivalence is referred to as the Straightening/Unstraightening equivalence and it is certainly one of the most foundational and important tools (maybe second only to the yoneda lemma) used to define functors in higher category theory. The idea is that, usually when we want to define a functor with values in $\infty$-categories we would have to specify an infinite amount of coherences which might be completely unfeasible; thus instead we go unstraightened and find us a nice bundle map (i.e. cococartesian fibration?) that encodes the functor we are after. Some applications for this fibrational approach are e.g. the theory of stacksthe definition of monoidal $\infty$-categories (or more generally for $\infty$-operads), operad algebras, Covering spaces, the calculation of $\infty$-(co)limits, …

We shall discover that when one seeks to pursue algebra in a homotopy coherent manner, $\infty$-operads present the ideal framework. In fact, all of higher algebra is written in the language of (infinity,1)-operads. But the thing that keeps the machine of $\infty$-operads well oiled, or makes riding it even possible is the theory of cocartesian fibrations and in turn therefore the Straightening/Unstraightening Equivalence.

(Co)Cartesian Fibrations - Warm up

See also Cartesian morphism.

Functoriality of Fibers

Given a functor $U \colon \mathcal{E} \to \mathcal{C}$ of $\infty$-categories, a natural construction is to look at the fibers of $U$, that is, for $c \colon \mathcal{C}$ an object, the fiber of $U$ at $c$, denoted by $\mathcal{E}_c$ is given by the pullback diagram:

Now the question is: When do the fibers have contra- or covariant dependence with regards to the category $\mathcal{C}$. It is certainly false that every functor $U \colon \mathcal{E} \to \mathcal{C}$ induces a “straightened” functor:

that retains all the information that was present in the original functor $U \colon \mathcal{E} \to \mathcal{C}$. In fact, an easy counter example is the following.

Example

Let $\mathcal{E}$ be the category with three objects $e_1, e_2, e_3$ such that there exists precisely one arrow from $e_1$ to $e_2$ and one from $e_1$ to $e_3$. On the other hand, let $\mathcal{C}$ be the category with two objects $c_1, c_2$ with precisely one arrow from $c_1$ to $c_2$ (i.e. $\mathcal{C} \simeq \Delta^1$). Define the functor $U \colon \mathcal{E} \to \mathcal{C}$ by the assignment

$$Ue_1 = c_1,\; Ue_2= Ue_3 = c_2,\; U(e_1 \to e_2) = (c_1 \to c_2),\; U(e_1 \to e_3) = (c_1 \to c_3)$$

Put graphically, we have

where the colors indicate where objects and morphisms are mapped to. Now the fibers $\mathcal{E}_{c_1}$ and $\mathcal{E}_{c_2}$ are given by the terminal category $\Delta^0 \simeq \{e_1\}$ and the disjoint union of two terminal categories $\Delta^0 \coprod \Delta^0 \simeq \{e_2,e_3\}$, respectively. There is now two (different) choices for a functor

$$\mathcal{C} \simeq \Delta^1 \to \text{Cat} \subset \infty\text{Cat}$$

which is really just a functor $\mathcal{E}_{c_1} \simeq \{e_1\} \to \{e_2,e_3\} \simeq \mathcal{E}_{c_2}$. Thus the choices are either $e_1 \mapsto e_2$ or $e_1 \mapsto e_3$, and both these functors are far from ever being able to recover all the information of the original functor $U \colon \mathcal{E} \to \mathcal{C}$.

Slick Definition

We shall first give slick, quick and abstract model-independent? definitions, and then after we shall unravel these definitions (in the $1$-categorical setting) to give some intuition.

Definition

Aaron Mazel-Gee, 2015 Let $U \colon \mathcal{E} \to \mathcal{C}$ be a functor of $\infty$-categories, and let $\varphi \colon e \to \tilde{e}$ be a morhpism in $\mathcal{E}$.

• $\varphi$ is called $U$-cocartesian morphism if it induces a pullback square:

• $\varphi$ is called $U$-cartesian morphism if it induces a pullback square:

Definition

Let $U \colon \mathcal{E} \to \mathcal{C}$ be a functor of $\infty$-categories, and denote by $U\text{-CoCart}$ (resp. $U\text{-Cart}$) the full subcategory of the arrow category $\text{Ar}\mathcal{E} \coloneqq \mathcal{E}^{\Delta^1}$ spanned by the $U$-cocartesian morphisms (resp. cartesian morphisms).

• $U$ is called cocartesian fibration, if the (induced) dashed arrow in the diagram below is an equivalence of $\infty$-categories:

where the functors $s$ denote the source projections, respectively.

• $U$ is called cartesian fibration, if the (induced) dashed arrow in the diagram below is an equivalence of $\infty$-categories:

where the functors $t$ denote the target projections, respectively.

Unraveling in $1$-Category-land

On the definitions

Let us unravel both these definitions. For $\varphi \colon e \to \tilde{e}$ to be a $U$-cocartesian morphism really boils down to saying that for each object $\overline{e} \colon \mathcal{E}$ we have a pullback square:

Let us suppose now that $\mathcal{C}, \mathcal{E}$ are 1-categories, then the above equivalence of morphism spaces is really just a bijection of hom-sets. In particular, for this morphism to be a bijection is equivalent to the following: For every $\psi \colon e \to \overline{e}$, and for every $\omega \colon U\tilde{e} \to U\overline{e}$ such that $\omega U\varphi = U\psi$, there exists a unique lift $w\colon \tilde{e}\to \overline{e}$ of $\omega$, i.e.

$$Uw = \omega$$

such that

$$\psi = w \circ \varphi$$

Writing this down by means of a diagram yields:

The above graphical representation thus encodes what it means for a morphism $\varphi \colon e \to \widetilde{e}$ to be $U$-cocartesian (for $U \colon \mathcal{E} \to \mathcal{C}$ a functor between $1$-categories. Staying in the $1$-categorical setting, let us look at the second condition, namely that we have an equivalence of categories

Here the important bit is really only that the dotted arrow is essentially surjective: The objects of $P$ are really just arrows $f \colon c \to \tilde{c}$ such that there exists an $e \colon \mathcal{E}$ so that $c = Ue$. Hence essential surjectivity tells us that for any arrow $f\colon Ue \to \tilde{c}$, there exists a $U$-cocartesian lift

$$\text{Lift}_{e}^{f} \colon e \to t(\text{Lift}_{e}^{f})$$

of $f$.

Remark

Let $U \colon \mathcal{E} \to \mathcal{C}$ be a functor of $1$-categories. Then a morphism $\varphi \colon e\to \overline{e}$ is * $U$-cocartesian if and only if * $U$-cartesian if and only if

Remark

There is another reformulation for $\varphi$ to be a $U$-(co)cartesian morphism: $\varphi$ is $U$-cocartesian if and only if the induced functor

$$\mathcal{E}_{\varphi/} \overset{\sim}{\to} \mathcal{E}_{\overline{e}/} \times_{\mathcal{C}_{U\overline{e}}} \mathcal{C}_{U\varphi/}$$

is a (surjective) equivalence (see Cartesian morphism Proposition 2.4).

Functoriality

Now assume $U\colon \mathcal{E} \to \mathcal{C}$, a functor of $1$-categories, is a cocartesian fibration.

We define the pseudofunctor $\text{St}(U) \colon \mathcal{C} \to \text{Cat}$ with values in the $2$-category of small categories as follows:

which on objects $c \colon \mathcal{C}$ assigns the fibers $\mathcal{E}_c$, while for a morphism $f \colon c \to \overline{c}$, the functor

$$f_! \coloneqq\text{St}(U)(f) \colon \mathcal{E}_c \to \mathcal{E}_{\overline{c}}$$

is defined as follows:

• for any object $e \colon \mathcal{E}_c$ (meaning $Ue = c$), by assumption, there exists a cocartesian lift
  $$\text{Lift}_e^{f} \colon e \to t(\text{Lift}_e^f)$$

  This yields a family of lifts $(\text{Lift}_e^f)_{e \colon \mathcal{E}}$ which we fix. Now, define the value of the functor $f_!$ on the object $e$ to be the target of the lift associated to $e$ and $f$:

  $$f_!(e) \coloneqq t(\text{Lift}_e^f)$$
• for a morphism $z \colon e \to \overline{e}$ in $\mathcal{E}_c$, we define $f_!(z)$ by means of the unique lift induced by the universal property of the cocartesian morphism:

• $f_!$ thus defined is a functor $\mathcal{E}_c \to \mathcal{E}_{\tilde{c}}$ by uniqueness of lifts:

implying $f_!(z')f_!(z) = f_!(z'z)$ (and thus functoriality).

Next up, we want to verify that the construction of $\text{St}(U)$ is actually a pseudofunctor. In order to verify this, let us prove a quick lemma:

Lemma

For $\varphi, \tilde{\varphi}$ composable morphisms of $\mathcal{E}$ with $\varphi'$ $U$-cocartesian and $\varphi' \coloneqq \tilde{\varphi}\varphi$, we have that $\varphi$ is $U$-cocartesian if and only if $\varphi'$ is $U$-cocartesian. In particular, $U$-cocartesian morphisms are closed under composition.

Proof

For $U$-cocartesian morphisms $e \overset{\varphi}{\to} \overline{e} \overset{\tilde{\varphi}}{\to} \tilde{e}$ we have a pasting of pullback squares which by pasting law for pullbacks yields the desired claim:

Theorem

$\text{St}(U)$ is a pseudofunctor.

Proof

We have specified what $\text{St}(U)$ does on objects and what it does on morphisms in $\mathcal{C}$. All that remains is to prove that this assignment is pseudo-functorial. For this, let $c \overset{f}{\to} \overline{c} \overset{g}{\to} \tilde{c}$ be a composable pair of morphisms in $\mathcal{C}$ and consider for an object $e \colon \mathcal{E}_c$ the diagram:

where the (unique) dotted arrows are induced by the lifting property of the $U$-cocartesian morphisms $\text{Lift}^{gf}_{e}$ and $\text{Lift}_{t(\text{Lift}^{g}_{e^f})}\text{Lift}_e^f$, respectively (justified by the above Lemma). In fact, again by uniqueness the dotted arrows must be mutually inverse isomorphisms - these will be the components of our natural isomorphism witnessing functoriality.

Moreover, by uniqueness yet again, we have a commutative diagram

which proves that we have a natural isomorphism

$$g_! f_! \cong (gf)_!$$

An analogous argument shows the remaining axioms of pseudo-functoriality.

Remark

One can now define a category of cocartesian fibrations with codomain $\mathcal{C}$, denoted $\text{CoCart}(\mathcal{C})$, whose objects are cocartesian fibrations and whose morphisms are “bundle” maps which preserve cocartesian morphisms: Now the construction $U \mapsto \text{St}(U)$ extends to a functor

$$\text{CoCart}(\mathcal{C}) \to \text{PseudoFun}(\mathcal{C}, \text{Cat})$$

with values in the category of pseudofunctors from $\mathcal{C}$ to $text{Cat}$. It then turns out that this construction yields an equivalence of categories (see Grothendieck construction). We will spell this out for the more general case of $\infty$-categories over the course of the next few chapters.

(Co)Cartesian fibrations of simplicial sets

Lurie‘s definition of (co)cartesian fibrations in the setting of quasicategories (or more generally simplicial sets) is the following:

Definition

(see Kerodon, Definition 5.1.1.1)

Let $U \colon \mathcal{E} \to \mathcal{C}$ be a morphism of simplicial sets, and let $\varphi$ be an edge of $\mathcal{E}$.

• We say that $\varphi$ is $U$-cartesian if every lifting problem

admits a solution, provided that $n \geq 2$.

• We say that $\varphi$ is $U$-cocartesian if every lifting problem

admits a solution, provided that $n \geq 2$.

The definition for a (co)cartesian fibration of simplicial sets is spelled out in Definition 5.1.4.1.. The general theory of fibrations of $\infty$-categories is explained in Kerodon Chapter 5. In this Nlab section, we will primarily concern ourselves with Kerodon Section 5.5 and Section 5.6 and we will stick to the notation used in Kerodon. However, a very short recap on the most important constructions in the previous sections of chapter 5 in Kerodon cannot hurt:

(Parametrized) Covariant/Contravariant Transport

Let $U \colon \mathcal{E} \to \mathcal{C}$ be a cocartesian fibration of $\infty$-categories. To every morphism $f\colon C \to D$ in the $\infty$-category $\mathcal{C}$, we want to associate a functor $f_! \colon \mathcal{E}_C \to \mathcal{E}_D$ (uniquely determined up to isomorphism by Proposition 5.2.2.8), which Lurie calls covariant transport functor?. We will skip a few steps and instead of proving existence of such a covariant transport functor $f_!$ for a single $f$, we shall immediately construct a functorial assignment $f \mapsto f_!$ for $f \colon \text{Hom}_\mathcal{C}(C,D)$. This is referred to as parametrized covariant transport? in Lurie‘s Kerodon.

Definition

Definition 5.2.8.1 Let $U \colon \mathcal{E} \to \mathcal{C}$ be a cocartesian fibration of simplicial sets and let $C,D$ be vertices of $\mathcal{C}$. We will say that a morphism

$$F \colon \text{Hom}_\mathcal{C}(C,D) \times \mathcal{E}_C \to \mathcal{E}_D$$

is given by parametrized covariant transport? if there exists a morphism

$$\widetilde{F} \colon \Delta^1 \times \text{Hom}_\mathcal{C}(C,D) \times \mathcal{E}_C \to \mathcal{E}$$

satisfying the following conditions:

• We have a commutative diagram

where the lower horizontal map is induced by the inclusion $\text{Hom}_\mathcal{C}(C,D) \hookrightarrow \text{Fun}(\Delta^1, \mathcal{C})$ by currying.

• The restriction $\widetilde{F}|_{0\times\text{Hom}_\mathcal{C}(C,D)\times\mathcal{E}_C}$ is given by projection onto $\mathcal{E}_C$, and the restriction $\widetilde{F}|_{1\times\text{Hom}_\mathcal{C}(C,D)\times \mathcal{E}_C}$ is equal to $F$.

• For every edge $f \colon C \to D$ of $\mathcal{C}$ and every object $X \colon \mathcal{E}_C$, the composite map

$$\Delta^1 \times \{f\} \times \{X\} \hookrightarrow \Delta^1 \times \text{Hom}_\mathcal{C}(C,D) \times \mathcal{E}_C \overset{\widetilde{F}}{\to} \mathcal{E}$$

is a $U$-cocartesian edge of $\mathcal{E}$.

Remark

Let us unravel these conditions:

• The first condition (the commutative diagram) really just says that $U\widetilde{F}$ evaluated at $(0 \to 1, f)$ is equal to the constant diagram $\mathcal{E}_C \to \text{Ar}\mathcal{C}$ with value $f$. In other words, plugging in $f$ into $\widetilde{F}$ yields a lift for $f$.

• The second condition says that $\widetilde{F}$, interpreted as a natural transformation between $\infty$-functors has domain and target as depicted:

• Finally, the third condition just states that the diagram $\widetilde{F}$ encodes, for every morphism $f \colon \text{Hom}_\mathcal{C}(C,D)$ and every object $X \colon \mathcal{E}_C$, a $U$-cocartesian edge $\widetilde{f}_X\colon X \to f_!(X)$ in $\mathcal{E}$, with $f_!(X) \coloneqq t(\widetilde{f}_X)$.

Remark

As already hinted at in the previous remark: For every edge $f \colon C \to D$, the composite map

$$\{f\}\times \mathcal{E}_C \hookrightarrow \text{Hom}_\mathcal{C}(C,D) \times \mathcal{E}_C \overset{F}{\to} \mathcal{E}_D$$

is given by covariant transport along $f$ (see also Definition 5.2.2.4).

Example

(see Example 5.2.8.3.) Let $\text{Set}_*$ denote the category of pointed sets, and let $V \colon \text{Set}_* \to \text{Set}$ denote the forgetful functor $(X, x \colon X) \mapsto X$. Then $V$ is a cocartesian fibration (in fact, it is a left covering map), whose fiber over an object $X \colon Set$ may be identified with $\{(X,x)\}_{x\colon X} \cong X$. For every pair of sets $X,Y$, the evaluation map

$$\text{Hom}_\text{Set}(X,Y) \times X \to Y, \qquad (f,x) \mapsto f(x)$$

is given by parametrized covariant transport.

Theorem

(see Proposition 5.2.8.4) Let $U \colon \mathcal{E}\to \mathcal{C}$ be a cocartesian fibration of simplicial sets, and let $C,D$ be vertices of $\mathcal{C}$. Then we have the following: 1. There exists a morphism $F \colon \text{Hom}_\mathcal{C}(C,D) \times \mathcal{E}_C \to \mathcal{E}_D$ which is given by parametrized covariant transport?. 2. An arbitrary diagram $F' \colon \text{Hom}_\mathcal{C}(C,D) \times \mathcal{E}_C \to \mathcal{E}_D$ is given by parametrized covariant transport if and only if it is isomorphic to $F$ (as an object of the $\infty$-category $\text{Fun}(\text{Hom}_\mathcal{C}(C,D) \times \mathcal{E}_C, \mathcal{E}_D)$).

In order to prove this, we need a lemma

Lemma

Let $U \colon \mathcal{E} \to \mathcal{C}$ be a cocartesian fibration of simplicial sets, let $K$ be a simplicial set, and suppose we are given a lifting problem

Then we have: 1. The lifting problem admits a solution $H \colon \Delta^1 \times K \to \mathcal{E}$ which is a $U$-cocartesian lift of $\overline{H}$. 2. Let $F$ be any object of the $\infty$-category $\text{Fun}_{/\mathcal{C}}(\{1\} \times K, \mathcal{E})$. Then $F$ is isomorphic to $H|_{1\times K}$ (as an object of $\text{Fun}_{/\mathcal{C}}(\{1\} \times K, \mathcal{E})$) if and only if $F = H'|_{1\times K}$, where $H'$ is yet another $U$-cocartesian lift of $\overline{H}$ which solves the above lifting problem.

Proof

This follows by means of the currying isomorphism and the fact that postcompositon with a cocartesian fibration is itself a cocartesian fibration. Therefore, the lifting problem reduces to $K = \Delta^0$. In this case, $U$ being a cocartesian fibration readily implies the first part, while Remark 5.1.3.8 does the second part.

Theorem now follows as a special case of the above lemma (see Proposition 5.2.8.4.).

The $\infty$-categories $\mathcal{S}$ and $\mathcal{QC}$

This is a summary of Section 5.5 in Kerodon.

Given a locally Kan simplicial category? $\mathcal{C}$, there is a canonical slice and coslice construction $\mathcal{C}_{/X}$ and $\mathcal{C}_{X/}$ for an object $X \colon \mathcal{C}$. The main question is now does this behave with regards to the homotopy coherent nerve operation?

Before adressing this, one has yet another natural defintion: For $\mathcal{C}$ locally Kan as above, one can define

$$\mathcal{C}^{\triangleleft}, \qquad \mathcal{C}^{\triangleright}$$

locally Kan enriched left and right cone categories. For those we have canonical isomorphisms:

$$N_\bullet^{\text{hc}}(\mathcal{C}^{\triangleleft}) \cong N_\bullet^{\text{hc}}(\mathcal{C})^{\triangleleft}, \qquad N_\bullet^{\text{hc}}(\mathcal{C}^{\triangleright}) \cong N_\bullet^{\text{hc}}(\mathcal{C})^{\triangleright}$$

Definition

Let $\mathcal{C}$ be a simplicially enriched category and let $Y \colon \mathcal{C}$. We have a functor $(\mathcal{C}_{/Y})^\triangleright \overset{V}{\to} \mathcal{C}$ which sends objects $(C, f) \colon \mathcal{C}_{/Y}$ to $C$, and carries the cone point $\infty$ to the object $Y$. If $(C,f)$ and $(D,g)$ are objects of $\mathcal{C}_{/Y}$, then the induced map of simplicial sets

$$\text{Hom}_{(\mathcal{C}_{/Y})^\triangleright}((C,f), (D,g)) \hookrightarrow \text{Hom}_\mathcal{C}(V(C,f), V(D,g)) = \text{Hom}_\mathcal{C}(C, D)$$

is simply given by the inclusion map. If $(C,f)$ is an object of $\mathcal{C}_{/Y}$, then the induced map

$$\Delta^0 = \text{Hom}_{(\mathcal{C}_{/Y})^\triangleright}((C,f), \infty) \to \text{Hom}_\mathcal{C}(V(C,f), V(\infty)) = \text{Hom}_\mathcal{C}(C, Y)$$

is equal to $\Delta^0 \overset{f}{\to} \text{Hom}_\mathcal{C}(C,Y)$. This is referred to as the right cone contraction functor?.

Analogously we have a functor $V \colon (\mathcal{C}_{X/})^\triangleleft \to \mathcal{C}$, which is referred as the left cone contraction functor?. We then have the following useful result:

Proposition

Propositon 5.5.2.16 We have a bijection

induced by postcomposition with the right cone contraction functor?. The dual result holds too of course.

Now we realize that there is a map

$$N_\bullet^{\text{hc}}(\mathcal{C}_{/X})^\triangleright \cong N_\bullet^{\text{hc}}(\mathcal{C}_{/X}^\triangleright) \to N_\bullet^{\text{hc}}(\mathcal{C})$$

carrying the cone point to the vertex $X$. Therefore by the above bijection we obtain the slice comparison morphism

$$c \colon N_\bullet^{\text{hc}}(\mathcal{C}_{/X}) \to N_\bullet^\text{hc}(\mathcal{C})_{/X}$$

This map is a monomorphism, but it is generally not an isomorphism, nor is it a homotopy equivalence in all cases. However, in good situation it will be a homotopy equivalence.

Theorem

Theorem 5.5.2.21 Let $\mathcal{C}$ be locally Kan and let $X$ be an object of $\mathcal{C}$ with the following property: For every morphism $f \colon X \to Y$ and every object $Z \colon \mathcal{C}$, the morphism of simplicial sets

$$\text{Hom}_\mathcal{C}(Y,Z)_\bullet \overset{f^*}{\to} \text{Hom}_\mathcal{C}(X,Z)_\bullet$$

is a Kan fibration. Then the coslice comparison morphism $c' \colon N_\bullet^\text{hc}(\mathcal{C}_{X/}) \to N_\bullet^{\text{hc}}(\mathcal{C})_{X/}$ is an equivalence of $\infty$-categories.

This result is useful for the following reason: Later on in Chapter 5.5 Lurie defines a chain of $\infty$-categories: where the first morphism is an equivalence by Proposition 5.5.6.6 (this is really a corollary of the above theorem).

We can describe the above categories informally as follows: All the above categories have the same collection of objects. For the morphisms we have quite different situations however: Morphisms in $N_\bullet^{\text{hc}}(\text{QCat}_*)$ are really just (strictly) commutative triangles: Morphisms in $\mathcal{QC}_\star$ are triangles that commute up to isomorphism in the $\infty$-category $\mathcal{D}$: Morphisms in $\mathcal{QC}_\text{Obj}$ are diagrams with the comparison morphism not necessarily being an isomorphism. Finally, the last play in the game $\mathcal{QC}_{\text{Ob}}^{(\infty,2)}$ is not an $(\infty,1)$-category but an $(\infty,2)$-category, with the same objects and morphisms as $\mathcal{QC}_\text{Obj}$, but with non-invertible natural transformations between functors.

Derived Categories

Homological Algebra

For amazing Nlab lecture notes see An Introduction to Homological Algebra.

Differential-Graded Categories

Derived $\infty$-categories of (Grothendieck) abelian Categories

Derived Categories of Rings

Derived Categories of (quasicoherent) Sheaves

Stable Homotopy Theory

Motivation from Classical homotopy Theory: Suspension-Loop Adjunction

Amazing Nlab Lecture notes at Introduction to Stable Homotopy Theory.

Freudenthal-Suspension Theorem

Stable $\infty$-Categories

For now see stable (infinity,1)-category.

The Stable Yoneda Lemma

How to Stabilize - All about Spectrum Objects

Brown Representability

A universal Example - Spectra

Let there be algebra - Operadic Yoga

$\mathbb{E}_n$-operads

Module-Operads

Operad-Algebras

Monoidal Categories

Definition and Intuition

Algebra Objects

Derived Categories

Spectra

Brave new Algebra

Ring Spectra, modules, algebras

Presentable and Accessible Categories

Why size matters

Cocompletions

Presentability and the Adjoint Functor Theorem

Lurie Tensor Product

Lax Stuff via (Co)Ends

Enriched $\infty$-Categories

$k$-linear presentable stable $\infty$-Categories

Derived Representation Theory

Recollements

Generalized BGP Reflection Functors

Ladkani’s Theorem in the context of Ring Spectra

References

• Aaron Mazel-Gee, A user’s guide to co/cartesian fibrations, arXiv:1510.02402 (2015).