Can conservativity depend on the universe?

Question

Question 1: Let $F: C \to D$ be a conservative, $\kappa$-cocontinuous functor between small, $\kappa$-cocomplete categories. Is the induced functor $Ind_\kappa(F): Ind_\kappa(C) \to Ind_\kappa(D)$ also conservative?

Terminology: I think this is pretty self-explanatory, but to be clear:

• $\kappa$ is a regular cardinal. Things are perhaps most familiar when $\kappa = \aleph_0$.

• A $\kappa$-cocomplete category is a category with $\kappa$-small colimits, i.e. colimits indexed by categories with fewer than $\kappa$-many morphisms.

• A $\kappa$-cocontinuous functor is a functor preserving $\kappa$-small colimits.

• $Ind_\kappa(C)$ is obtained from $C$ be freely adjoining $\kappa$-filtered colimits, or by the formula $Ind_\kappa(C) = Fun_\kappa(C^{op},Set)$, where $Fun_\kappa(A,B)$ is the category of $\kappa$-continuous functors from $A$ to $B$.

• The induced functor $Ind_\kappa(F)$ is defined by left Kan extension along the Yoneda embedding.

I'm a little worried that Question 1 is too much to ask for, so here's an even milder variant.

• Let $\kappa$ be inaccessible.

• Let $Pr^L(\kappa)$ be the category (really a $(2,1)$-category) of categories which are locally presentable with respect to the universe $V_\kappa$. That is, $Pr^L(\kappa)$ consists of categories of the form $Ind_\lambda^\kappa(C)$ where $C$ is a $\kappa$-small and $\lambda$-cocomplete category with $\lambda < \kappa$; here $Ind_\lambda^\kappa$ is the free cocompletion under $\kappa$-small, $\lambda$-filtered colimits. The morphisms are left adjoint functors.

• Similarly, let $Pr^L$ be the $(2,1)$-category of locally presentable categories and left adjoint functors.

• Then we have a functor $Ind_\kappa: Pr^L(\kappa) \to Pr^L$.

Question 2: Does the functor $Ind_\kappa: Pr^L(\kappa) \to Pr^L$ preserve conservative functors?

Question 2 is asking whether the property of a left adjoint between locally presentable categories being conservative depends on which universe we work in.

In the setting of either question, it's clear that if $Ind_\kappa(F)$ is conservative, then $F$ is conservative. So in either case, if the answer to the question is affirmative, then we have "$F$ conservative $\Leftrightarrow$ $Ind_\kappa(F)$ conservative", which would be reassuring.

Answer 1

Probably not the best one can do, and what follows might be a bit 'overkill', but it answer the question about dependency on universe, and it is a nice argument.

Also if you know how the proof of the left transfer results I will use below works, it might give some idea on how to prove more general case of the result.

Theorem: Let $\kappa$ be an uncountable regular cardinal. Let $F: \mathcal{C} \to \mathcal{D}$ be a strongly $\kappa$-accessible left adjoint functor between locally $\kappa$-presentable categories. Then $F$ is conservative if and only if the restriction of $F$ to the full subcategory of $\kappa$-presentable objects is conservative.

(Here strongly $\kappa$-accessible = sends $\kappa$-presentable objects of $\mathcal{C}$ to $\kappa$-presentable objects of $\mathcal{D}$)

Proof: The weak factorization system (isomorphisms, all maps) on $\mathcal{D}$ is ($\kappa$-)combinatorial: it is generated by the empty set of maps. Hence the left transfered weak factorization system on $\mathcal{C}$ along $F$ exists and is also $\kappa$-combinatorial (I mean by this that is is cofibrantly generated by maps between $\kappa$-presentable objects). By definition, the left class of this weak factorization system is the set of maps such that $F(i)$ is an isomorphism. But if the restriction of $F$ to $\kappa$-presentable objects is conservative it means that all generators are isomorphisms hence, the left class only contains isomorphisms, so $F$ is conservative.

Note: the finer version of the left transfer theorem that I'm using which specify the presentability rank can be found as Theorem B.8.(4) in this paper of mine, but at the end of day it mostly follows from an analysis of the proof of the existence of left transfer available in the litterature (for e.g. in Makkai and Rosicky Cellular categories.)

Answer 2

Here's a proof of a more general statement, which unfortunately still fails when $\kappa = \aleph_0$. The key tool is Fodor's lemma:

Proposition: Let $\kappa$ be an uncountable regular cardinal, and let $F: C \to D$ be a filtered-colimit-preserving functor between $\kappa$-accessible categories with filtered colimits, which preserves $\kappa$-presentable objects. Then $F$ is conservative iff its restriction to the $\kappa$-presentable objects is conservative.

Proof: Because each category has filtered colimits, it suffices by induction on presentability rank to show that if $(f_i: c_i \to c_i')_{i < \kappa}$ is a map of smooth $\kappa$-chains between $\kappa$-presentable objects with colimit $f: c \to c'$, and if $Ff: F(c) \to F(c')$ is an isomorphism, then $f: c \to c'$ is an isomorphism. (Here "smooth" $\kappa$-chain means that $c^{(')}_i = \varinjlim_{j < i} c^{(')}_j$ for $i < \kappa$ a limit ordinal. Because $F$ preserves filtered colimits, the image chains $(F(c_i^{(')}))_{i < \kappa}$ are also smooth.) Let $g: F(c') \to F(c)$ be inverse to $F(f)$, and write $\gamma_{ij}^{(')}: c_i^{(')} \to c_j^{(')}$ for the linking maps.

Define $\phi: \kappa \to \kappa$ by taking $\phi(j) = \sup S_j$ where $S_j$ is the set of $i< \kappa$ such that $g\gamma_{i\kappa}$ factors through $F(c'_j)$. If $\phi(j) < j$ on a stationary set, then by Fodor's lemma $\phi$ is constant on a stationary set; it follows that $\phi(j)$ is bounded for sufficiently large $j$, which is impossible because every $i$ lies in some $S_j$. Therefore we have $\phi(j) \geq j$ on a club set $S$ of $j$'s. By restricting to the limit ordinals in $S$ and using smoothness of our chains, we may assume that $g: F(c') \to F(c)$ is induced "levelwise": there is a map of chains $g_i: F(c_i') \to F(c_i)$ such that $g = \varinjlim_i g_i$.

A similar argument shows that we may assume that $g_i$ is a levelwise retract of $F(f_i)$, and similarly that $g_i$ is a levelwise section of $F(f_i)$. Thus $F(f_i)$ is invertible. Because $F$ is conservative on $\kappa$-presentable objects, $f_i$ is invertible, and so $f$ is invertible as well.

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Note that we needed $\kappa$ to be uncountable and regular in order to apply Fodor's lemma. We also needed this to pass to the limit ordinals in $S$.

My suspicion is that a more general version of Fodor's lemma working on more general $\kappa$-directed posets ought to allow one to take $C,D,F$ to be arbitary $\kappa$-accessible categories and functors, but I think $\kappa$ will still need to be uncountable (and regular).