# nLab Beck-Chevalley condition

Contents

category theory

## Applications

#### 2-category theory

2-category theory

Definitions

Transfors between 2-categories

Morphisms in 2-categories

Structures in 2-categories

Limits in 2-categories

Structures on 2-categories

# Contents

## Idea

The BeckChevalley condition, also sometimes called just the Beck condition or the Chevalley condition, is a “commutation of adjoints” property that holds in many “change of base” situations.

## Motivation

The Beck-Chevalley condition may be understood as a natural compatibility condition for

### From integral transforms

From the point of view of geometry, in contexts such as of integral transforms one considers correspondences between spaces $A$, $B$ given by spans of maps between them:

Via composition of such correspondences by fiber product of adjacent legs, they form the 1-morphisms in a 2-category Span.

Assuming some base change adjoint pair $f^\ast \vdash f_\ast$, thought of as

• push-forward $f_\ast \,\colon\, \mathcal{D}(X) \longrightarrow \mathcal{D}(A)$

and

• pull-back $f^\ast \,\colon\, \mathcal{D}(A) \longrightarrow \mathcal{D}(X)$

is functorially associated with maps $f$ (i.e. such that there are natural isomorphisms $(f_2 f_1)_\ast \,\simeq\, (f_2)_\ast (f_1)_\ast$ and dually), the integral transform encoded by any span as above is the “pull-push” operation given by $g_* f^*$.

Now the Beck-Chevalley condition (on such assignment of base change adjoints to maps) essentially says that this pull-push construction is (2-)functorial under composition of spans:

Concretely, given a pair of composable spans:

and their composite in Span formed by the fiber product of the adjacent legs

then functoriality of pull-push means that the two different ways to pull-push through the diagram coincide:

Pull-pushing through the spans separately results in

$(k_\ast h^\ast) \circ (g_\ast f^\ast) \;=\; k_\ast \, \underbrace{h^\ast \, g_\ast} \, f^\ast \,,$

while pull-pushing through the composite span results in

$(k p)_* (f q)^* = k_* \, \underbrace{p_* \, q^*} \, f^* \,.$

For these two operations to coincide, up to isomorphism, hence for pull-push to respect composition of spans, we need an isomorphism between the operations shown over braces, hence of the form

(1)$h^\ast \, g_\ast \;\simeq\; p_* \, q^*$

between the two ways of push-pulling along the sides of any pullback square

This (1) is the Beck-Chevalley condition: A natural necessary condition to ensure that integral transforms by pull-push base change through correspondences is functorial under composition of correspondences by fiber product of adjacent legs.

More motivation along these lines also be found at dependent linear type theory. As a formulation of propagation in cohomological quantum field theory this is Sc14 there.

### From logic and type theory

From the point of view of formal logic and dependent type theory, the three items $f_! \dashv f^\ast \vdash f_\ast$ in a base change adjoint triple constitute the categorical semantics of quantification and context extension (existential quantification/dependent pair type $\dashv$ context extension $\dashv$ universal quantification/dependent product type), and so the Beck-Chevalley condition says that these are compatible with each other: concretely that substitution of free variables commutes with quantification — a condition which syntactically is “self-evident”, at least it is evidently desirable.

Original articles with this logical perspective on the BC condition: Lawvere (1970), p. 8; Seely (1983), p. 511; Pavlović (1991), §1; Pavlović (1996), p. 164.

For more on this logical aspect see below.

## Definition

Suppose given a commutative square (up to isomorphism) of functors:

$\array{ & \overset{f^*}{\to} & \\ ^{g^*}\downarrow && \downarrow^{k^*}\\ & \underset{h^*}{\to} & }$

in which $f^*$ and $h^*$ have left adjoints $f_!$ and $h_!$, respectively. (The classical example is a Wirthmüller context.) Then the natural isomorphism that makes the square commute

$k^* f^* \to h^* g^*$

has a mate

$h_! k^* \to g^* f_!$

defined as the composite

$h_! k^* \overset{\eta}{\to} h_! k^* f^* f_! \overset{\cong}{\to} h_! h^* g^* f_! \overset{\epsilon}{\to} g^* f_! \,.$

We say the original square satisfies the Beck–Chevalley condition if this mate is an isomorphism.

More generally, it is clear that for this to make sense, we only need a transformation $k^* f^* \to h^* g^*$; it doesn’t need to be an isomorphism. We also use the term Beck–Chevalley condition in this case,

### Left and right Beck–Chevalley condition

Of course, if $g^*$ and $k^*$ also have left adjoints, there is also a Beck–Chevalley condition stating that the corresponding mate $k_! h^* \to f^* g_!$ is an isomorphism, and this is not equivalent in general. Context is usually sufficient to disambiguate, although some people speak of the “left” and “right” Beck–Chevalley conditions.

Note that if $k^* f^* \to h^* g^*$ is not an isomorphism, then there is only one possible Beck-Chevalley condition.

### Dual Beck–Chevalley condition

If $f^*$ and $h^*$ have right adjoints $f_*$ and $h_*$, there is also a dual Beck–Chevalley condition saying that the mate $g^* f_* \to h_* k^*$ is an isomorphism. By adjointness, if $f^*$ and $h^*$ have right adjoints and $g^*$ and $k^*$ have left adjoints, then $g^* f_* \to h_* k^*$ is an isomorphism if and only if $k_! h^* \to f^* g_!$ is.

### For bifibrations

Originally, the Beck-Chevalley condition was introduced in (Bénabou-Roubaud, 1970) for bifibrations over a base category with pullbacks. In loc.cit. they call this condition Chevalley condition because he introduced it in his 1964 seminar.

A bifibration $\mathbf{X} \to \mathbf{B}$ where $\mathbf{B}$ has pullbacks satisfies the Chevalley condition iff for every commuting square

$\array{ & \overset{\psi^\prime}{\rightarrow} & \\ \downarrow^{\varphi^\prime} && \downarrow^{\varphi}\\ & \underset{\psi}{\rightarrow} & }$

in $\mathbf{X}$ over a pullback square in the base $\mathbf{B}$ where $\varphi$ is cartesian and $\psi$ is cocartesian it holds that $\varphi^\prime$ is cartesian iff $\psi^\prime$ is cocartesian. Actually, it suffices to postulate one direction because the other one follows. The nice thing about this formulation is that there is no mention of “canonical” morphisms and no mention of cleavages.

A fibration $P$ has products satisfying the Chevalley condition iff the opposite fibration $P^{op}$ is a bifibration satisfying the Chevalley condition in the above sense.

According to the Benabou–Roubaud theorem, the Chevalley condition is crucial for establishing the connection between the descent in the sense of fibered categories and the monadic descent.

### “Local” Beck–Chevalley condition

Suppose that $f^*$ and $h^*$ do not have entire left adjoints, but that for a particular object $x$ the left adjoint $f_!(x)$ exists. This means that we have an object “$f_! x$” and a morphism $\eta_x\colon x \to f^* f_! x$ which is initial in the comma category $(x / f^*)$. Then we have $k^*(\eta) \colon k^* x \to k^* f^* f_! x \to h^* g^* f_! x$, and we say that the square satisfies the local Beck-Chevalley condition at $x$ if $k^*(\eta)$ is initial in the comma category $(k^* x / h^*)$, and hence exhibits $g^* f_! x$ as “$h_! k^* x$” (although we have not assumed that the entire functor $h_!$ exists).

If the functors $f_!$ and $h_!$ do exist, then the square satisfies the (global) Beck-Chevalley condition as defined above if and only if it satisfies the local one defined here at every object.

## In logic / type theory

If the functors in the formulation of the Beck-Chevalley condition are base change functors in the categorical semantics of some dependent type theory (or just of a hyperdoctrine) then the BC condition is equivalently stated in terms of logic as follows.

$\array{ D &\stackrel{h}{\to}& C \\ {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{g}} \\ A &\stackrel{f}{\to}& B }$

is interpreted as a morphism of contexts. The pullback (of slice categories or of fibers in a hyperdoctrine) $h^*$ and $f^*$ is interpreted as the substitution of variables in these contexts. And the left adjoint $\sum_k \coloneqq k_!$ and $\sum_q \coloneqq g_!$, the dependent sum is interpreted (up to (-1)-truncation, possibly) as existential quantification.

In terms of this the Beck-Chevalley condition says that if the above diagram is a pullback, then substitution commutes with existential quantification

$\sum_k h^* \phi \stackrel{\simeq}{\to} f^* \sum_g \phi \,.$
###### Example

Consider the diagram of contexts

$\array{ [\Gamma, x : X] &\stackrel{}{\to}& [\Gamma, x : X, y : Y] \\ \downarrow && \downarrow \\ \Gamma &\to& [\Gamma, y : Y] } \;\;\; \simeq \;\;\; \array{ \Gamma \times X &\stackrel{(p_1,p_2,t)}{\to}& \Gamma \times X \times Y \\ {}^{\mathllap{p_1}}\downarrow && \downarrow^{\mathrlap{(p_1,p_3)}} \\ \Gamma &\stackrel{(id,t)}{\to} & \Gamma \times Y } \,,$

with the horizontal morphism coming from a term $t : \Gamma \to Y$ in context $\Gamma$ and the vertical morphisms being the evident projection, then the condition says that we may in a proposition $\phi$ substitute $t$ for $y$ before or after quantifying over $x$:

$\sum_{x : X} \phi(x,t) \simeq (\sum_{x : X} \phi(x,y))[t/y] \,.$

## Examples

### Various

• The codomain fibration of any category with pullbacks is a bifibration, and satisfies the Beck–Chevalley condition at every pullback square. In particular, locally cartesian closed categories always satisfy the Beck–Chevalley condition.

• If $C$ is a regular category (such as a topos), the bifibration $Sub(C) \to C$ of subobjects satisfies the Beck–Chevalley condition at every pullback square.

• The family fibration $Fam(C)\to Set$ of any category $C$ with small sums satisfies the Beck–Chevalley condition at every pullback square in $Set$.

• Mackey’s restriction formula for group representations.

### Images and pre-images

###### Example

(Beck-Chevalley for images and pre-images of sets)
Given a set $S \,\in\, Set$, write $\mathcal{P}(S)$ for its power set regarded as a poset, meaning that

1. $\mathcal{P}(S)$ is the set of subsets $A \subset S$,

2. regarded as a category by declaring that there is a morphisms $A \to B$ precisely if $A \subset B$ as subsets of $S$.

Then for $f \,\colon\, S \longrightarrow T$ a function between sets, we obtain the followin three functors between their power sets:

1. $f_! \,\coloneqq\, f(-) \,\colon\, \mathcal{P}(S) \longrightarrow \mathcal{P}(T)$

forms images under $f$

2. $f^\ast \,\coloneqq\, f^{-1} \,\colon\, \mathcal{P}(T) \longrightarrow \mathcal{P}(S)$

forms preimages under $f$,

3. $f_ast \,\coloneqq\, T \setminus f\big(S \setminus (-)\big) \,\colon\, \mathcal{P}(S) \longrightarrow \mathcal{P}(T)$

forms the complement of images of complements.

These three functors constitute an adjoint triple

$f(-) \;\dashv\; f^{-1} \;\dashv\; f_\ast$

as is readily verified and also discussed at some length in the entry interactions of images and pre-images with unions and intersections.

In particular, the familiar/evident fact that forming images and preimages preserves unions may be understood as a special case of the fact that left adjoints preserve colimits.

Now given a pullback diagram in Set

(2)$\array{ D' &\overset{\beta}{\longrightarrow}& D \\ \mathllap{{}^{\psi}}\big\downarrow &{}^{{}_{(pb)}}& \big\downarrow\mathrlap{{}^{\phi}} \\ C' &\underset{\alpha}{\longrightarrow}& C }$

then these operations satisfy the left Beck-Chevalley condition in the sense that the following diagram of poset-homomorphisms (functors) commutes:

(3)$\array{ \mathcal{P}(D') &\overset{\beta_!}{\longrightarrow}& \mathcal{P}(D) \\ \mathllap{{}^{\psi^{-1}}}\big\uparrow && \big\uparrow\mathrlap{{}^{\phi^{-1}}} \\ \mathcal{P}(C') &\underset{\alpha_!}{\longrightarrow}& \mathcal{P}(C) }$

To see this it is sufficient to check commutativity on singleton-subsets (because every other subset is a union of singletons and, as just remarked, images and preimages preserve unions) and given a singleton set $\{c'\} \subset C'$ on an element $c' \,\in\, C'$, we directly compute as follows:

(4)$\begin{array}{l} \beta \circ \psi^{-1}\big(\{c'\}\big) \\ \;\simeq\; \beta\Big( \big\{ (c',d) \,\big\vert\, d \,\in\, D \,, \phi(d) = \alpha(c') \big\} \Big) \\ \;=\; \big\{ d \,\in\, D \,\big\vert\, \phi(d) = \alpha(c') \big\} \\ \;=\; \phi^{-1}\big( \{\alpha(c')\} \big) \\ \;=\; \phi^{-1} \circ \alpha\big(\{c'\}\big) \,. \end{array}$

Here it is the first two steps which make use of the assumption that (2) is a pullback square, namely which means that $D'$ is compatibly bijective to the fiber product of $C'$ with $D$ over $C$:

$\array{ D' &\simeq& \big\{ (c', d) \in C' \times D \,\big\vert\, \phi(d) \,=\, \alpha(c') \big\} \\ \mathllap{{}^{\beta}} \big\downarrow && \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \big\downarrow\mathrlap{{}^{ (c',d) \mapsto d }} \\ D &=& \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! D }$

This elementary example actually constitutes half of the proof of the archetypical example of Prop. below.

### Systems of functor categories

###### Proposition

Given

1. $\phi \colon D \to C$ is an opfibration of small strict categories

2. a pullback diagram in the 1-category Cat of small strict categories and functors of the form

(5)$\array{ D' &\overset{\beta}{\longrightarrow}& D \\ \mathllap{{}^{\psi}}\big\downarrow &{}^{{}_{(pb)}}& \big\downarrow\mathrlap{{}^{\phi}} \\ C' &\underset{\alpha}{\longrightarrow}& C }$

(hence presenting a 2-pullback/homotopy pullback, see Rem. below),

3. $\mathcal{C}$ any category with all small colimits

then the induced diagram of functor categories

$\array{ [D', \mathcal{C}] &\overset{\beta^*}{\longleftarrow}& [D, \mathcal{C}] \\ \Big\downarrow\mathrlap{{}^{\psi_!}} && \Big\downarrow\mathrlap{{}^{\phi_!}} \\ [C', \mathcal{C}] &\underset{\alpha^*}{\longleftarrow}& [C,\mathcal{C}] }$

(for $(-)^\ast$ denoting precomposition and $(-)_!$ denoting left Kan extension)

satisfies the Beck-Chevalley condition, in that there is a natural isomorphism

$\psi_! \circ \beta^* \;\simeq\; \alpha^* \circ \phi_! \,.$

For this statement in the more general context of quasicategories see Joyal (2008), prop. 11.6.
###### Proof

The proof relies on two observations:

1. Since $\phi$ is opfibered, for every object $c \in C$ the embedding of the fiber $\phi^{-1}(c)$ into the comma category $\phi/c$ is a final functor. Therefore the pointwise formula for the left Kan extension $\phi_!$ is equivalently given by taking the colimit over the fiber, instead of over the comma category:
(6)$\phi_!(X)_c \;\simeq\; \underset {\underset{\phi^{-1}\big(\{c\}\big)}{\longrightarrow}} {\lim} \, X \,.$
1. Since (5) is a (strict) pullback of (strict) categories (i.e. an ordinary pullback of sets of objects and of sets of morphisms), the operations of taking images and preimages $(-)^{-1}$ (fibers) of the object-functions of the given functors satisfy the posetal Beck-Chevalley condition from Exp. :

(7)$\begin{array}{l} \beta \circ \psi^{-1}\big(\{c'\}\big) \\ \;\simeq\; \beta\Big( \big\{ (c',d) \,\big\vert\, d \,\in\, D \,, \phi(d) = \alpha(c') \big\} \Big) \\ \;=\; \big\{ d \,\in\, D \,\big\vert\, \phi(d) = \alpha(c') \big\} \\ \;=\; \phi^{-1}\big( \{\alpha(c')\} \big) \\ \;=\; \phi^{-1} \circ \alpha\big(\{c'\}\big) \,. \end{array}$

for all objects $c' \,\in\, Obj(C')$.

Now using first (6) and then (7) and then again (6) yields the following sequence of isomorphisms

\begin{aligned} \big(\psi_! \beta^* (X)\big)(c') & \;\simeq\; \underset {\underset{ \beta \circ \psi^{-1}\big(\{c'\}\big) }{\longrightarrow}} {\lim} \, X \\ & \;\simeq\; \underset {\underset{ \phi^{-1} \circ \alpha\big(\{c'\}\big) }{\longrightarrow}} {\lim} \, X \\ & \;\simeq\; \big(\alpha^* \phi_! (X)\big)(c') \,, \end{aligned}

all of them natural in $c' \,\in\, Obj(C')$.

###### Remark

As a counter-example showing that in Prop. the condition for $\phi \colon D \to C$ to be an opfibration is necessary for the statement to hold, consider

• $C \coloneqq \{0\to 1\}$ the interval category,

• $D \coloneqq \ast,\;\;C' \coloneqq \ast$ the terminal category,

• $\phi\coloneqq const_0,\;\;\alpha \coloneqq const_1$.

Then the pullback $D' = \varnothing$ is the empty category. Therefore

• $[D',\mathcal{C}] = [\varnothing, \mathcal{C}]$ is the terminal category, so that $\psi_!\circ \beta^\ast$ is a constant functor

(concretely, $\psi_! \colon \ast \simeq [\varnothing, \mathcal{C}] \to [C',\mathcal{C}]$ is left adjoint to the projection functor to the terminal category and hence picks the initial object of $[C',\mathcal{C}]$),

• while $\alpha^* \circ \phi_!$ is the identity functor

(to see this observe first that $[C,\mathcal{C}]$ now may be identified with the arrow category of $\mathcal{C}$, and second that, under this identification, $\phi_! \colon X \mapsto id_X$ sends any object of $\mathcal{C}$ to its identity morphism, as is verified by observing that this is clearly left adjoint to the domain-restriction $\phi^\ast$).

###### Remark

In Prop. , the pullback (5) of an opfibration of strict categories formed in the 1-category of strict categories may be understood as modelling the 2-pullback (cf. there) of the corresponding cospan in the actual 2-category Cat.

In particular, if the strict categories involved in the pullback diagram (5) are all groupoids, then a functor such as $\phi$ being a Kan fibration is equivalent to its nerve being a Kan fibration (see at right Kan fibration, this and this Prop), in which case the pullback diagram (5) models the homotopy pullback (by this Prop.).

### Enriched functor categories

Consider

• $(\mathbf{V}, \otimes, \mathbb{1})$ a cocomplete symmetric monoidal category, regarded as a cosmos for enrichment,

• $\mathbf{C}$ a cocomplete $\mathbf{V}$-enriched category which is also tensored over $\mathbf{V}$, denoted:

$(-)\cdot(-) \;\colon\; \mathbf{V} \times \mathbf{C} \longrightarrow \mathbf{C}$

and write

• for any small $\mathbf{V}$-enriched category $\mathcal{X} \,\in\, \mathbf{V}Cat^{small}$:

$\mathbf{V}Func(\mathcal{X},\,\mathbf{C}) \;\in\; Cat$

for the $\mathbf{V}$-enriched-functor category from $\mathcal{X}$ to $\mathbf{C}$,

• for any $\mathbf{V}$-enriched functor $f \,\colon\, \mathcal{X} \longrightarrow \mathcal{Y}$:

$\mathbf{V}Func(\mathcal{X},\,\mathbf{C}) \underoverset {\underset{f^\ast}{\longleftarrow}} {\overset{f_!}{\longrightarrow}} {\;\;\; \bot \;\;\;} \mathbf{V}Func(\mathcal{Y},\,\mathbf{C})$

for the pair of adjoint functors given by precomposition and by left Kan extension, the latter being expressible by the following coend-formula (see there):

(8)$\mathscr{V}_{(-)} \;\in\; \mathbf{V}Func(\mathcal{X},\,\mathbf{C}) \;\;\;\;\;\;\;\; \vdash \;\;\;\;\;\;\;\; (f_! \mathscr{V})_y \;\simeq\; \int^{x \in \mathcal{X}} \mathcal{X}\big(f(x),\,y\big) \cdot \mathscr{V}_{x} \,.$

The following is a rather simplistic but maybe instructive example:

###### Example

In the case of a cartesian closed cosmos $(\mathbf{V}, \times, \ast)$ consider for $\mathcal{X}, \mathcal{Y}, \mathcal{X}' \,\in\, \mathbf{V}Cat^{small}$ a commuting diagram of $\mathbf{V}$-enriched functors of the following form:

$\array{ \mathcal{X} \times \mathcal{X}' &\overset{f \times id}{\longrightarrow}& \mathcal{Y} \times \mathcal{X}' \\ \mathllap{{}^{pr_{\mathcal{X}}}} \big\downarrow &{}^{{}_{(pb)}}& \big\downarrow \mathrlap{{}^{pr_{\mathcal{Y}}}} \\ \mathcal{X} &\underset{\;\;\; f \;\;\;}{\longrightarrow}& \mathcal{Y} }$

(which is a pullback square in the 1-category of strict $\mathbf{V}$-enriched categories).

Then the Beck-Chevalley condition on $\mathbf{C}$-valued $\mathbf{V}$-functors holds, in that the following diagram commutes:

$\array{ \mathbf{V}Func( \mathcal{X} \times \mathcal{X}' ,\, \mathbf{C} \big) &\overset{(f \times id)_!}{\longrightarrow}& \mathbf{V}Func( \mathcal{Y} \times \mathcal{X}' ,\, \mathbf{C} ) \\ \mathllap{{}^{(pr_{\mathcal{X}})^\ast}} \big\uparrow && \big\uparrow \mathrlap{{}^{(pr_{\mathcal{Y}})^\ast}} \\ \mathbf{V}Func(\mathcal{X},\,\mathbf{C}) &\underset{\;\;\; f_! \;\;\;}{\longrightarrow}& \mathbf{V}Func(\mathcal{Y},\,\mathbf{C}) }$

To see this, we may check object-wise as follows:

$\begin{array}{l} \big( (f \times id)_! (pr_{\mathcal{X}})^\ast \mathscr{V} \big)_{(y,x')} \\ \;=\; \int^{(x,x'_0)} \mathcal{Y}\big(f(x),y\big) \times \mathcal{X}'\big(x'_0,x'\big) \cdot \big( (pr_{\mathcal{X}})^\ast\mathscr{V}_{x,x'_0} \big) \\ \;=\; \int^{(x,x'_0)} \mathcal{Y}\big(f(x),y\big) \times \mathcal{X}'\big(x'_0,x'\big) \cdot \mathscr{V}_{x} \\ \;=\; \int^{x'_0} \int^{x} \mathcal{Y}\big(f(x),y\big) \times \mathcal{X}'\big(x'_0,x'\big) \cdot \mathscr{V}_{x} \\ \;\simeq\; \int^{x'_0} \mathcal{X}'\big(x'_0,x'\big) \cdot \int^{x} \mathcal{Y}\big(f(x),y\big) \cdot \mathscr{V}_{x} \\ \;\simeq\; \int^{x} \mathcal{Y}\big(f(x), y\big) \cdot \mathscr{V}_{x} \\ \;\simeq\; \big( f_! \mathscr{V} \big)_y \\ \;\simeq\; \big( (pr_{\mathcal{Y}})^\ast ( f_! \mathscr{V} ) \big)_{y,x'} \,, \end{array}$

where we used (apart from the definition of precomposition), in order of appearance:

1. the coend-expression (8) for the left Kan extension,

2. the Fubini theorem for coends (here),

3. the fact that the closed tensor $(-) \times (-)$ preserves colimits and hence coends in each variable,

4. the co-Yoneda lemma in the coend-form here

and finally the expression (8) again.

### Proper base change in étale cohomology

For coefficients of torsion group, étale cohomology satisfies Beck-Chevalley along proper morphisms. This is the statement of the proper base change theorem. See there for more details.

### Grothendieck’s yoga of six operations

A Beck-Chevalley condition is part of the axioms of Grothendieck's yoga of six operations, e.g. Cisinski & Déglise (2009), §A.5, Gallauer (2021), p. 16, Scholze (2022), p. 11.

The Beck-Chevalley condition has arisen in the theory of descent - as developed from Grothendieck 1959. Jon Beck and Claude Chevalley studied it independently from each another. […] It is conspicuous that neither of them ever published anything on it. [Pavlović 1991, §14]

Original articles:

expanded on in

Review:

(in view of linear type theory)

Discussion for subobject lattices:

Review in the context of Grothendieck's yoga of six operations:

Discussion for derived functors with focus on the example of base change of retractive spaces and homotopy categories of parameterized spectra:

Discussion in the generality of enriched category theory:

Discussion in the generality of $\infty$-categories:

Discussion for presheaf categories in the context of quasicategories ($\infty$-categories of $\infty$-presheaves):

Discussion for Goursat categories: