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    • CommentRowNumber1.
    • CommentAuthorPeter Heinig
    • CommentTimeJun 14th 2017
    • (edited Jun 21st 2017)

    What do you consider a usual technical term for the following kind of functor PP? Or rather, for the pair of functors (F,P)(F,P) described by the list below?

    Can you point to parts of the literature where such a situation is studied more or less in general, for its own sake? (Although we do not expect there to be much to say about it in this generality, the literature might hold surprises.)

    • 22 is the discrete category with two objects,

    • C\mathsf{C} is any (infinite) category,

    • P:C2P\colon \mathsf{C}\rightarrow 2 is any functor,

    • F:CCF\colon\mathsf{C}\rightarrow\mathsf{C} is an endofunctor such that

      • for each i2i\in 2 and each object OO of C\mathsf{C} with P(O)=iP(O)=i we have (PF)(O)=1i(P\circ F)(O)=1-i.

    The latter property explains our (temporary) working terminology that PP is an FF-wise switchable property functor: since F(O)F(O) is an object of C\mathsf{C} again, we can switch between the two values of PP as often as we like, functorially, by applying FF over and over. (This implies obvious things, such as that FF cannot have any “fixed point”, since the OO in the property after the black square is arbitrary).

    We also know that our endofunctor FF has the property that

      • FF is not involutive, also not in weakened senses; the objects OO, F(O)F(O), F(F(O))F(F(O)) are quite “far” from each other, though thery are all related, because of:
      • for each object OO of C\mathsf{C} there is at least one morphism OfF(O)O\overset{f}{\rightarrow}F(O) in C\mathsf{C}.

    (In various contexts there are other terms for such functors FF, such as inflationary operator, closure operator etc. This question is more about PP and the joint property it satisfies w.r.t. to FF than about this latter property that FF happens to have into the bargain.)

    Comments, more or less irrelevant to the question:

    • This question is partly motivated by ongoing graph-theoretical work with colleagues, applying methods from to prove a non-axiomatizability-theorem for a large number of natural classes of vertex-reconstructible graphs. We make essential use of an “operation” on the category of all countable forests, with isometric embeddings as the morphisms. It is essential that this “operation” actually is an endofunctor, and switches a property functor P:C2P\colon\mathsf{C}\rightarrow2 important to us.

    • It is easy to make up small artificial examples of such a situation. In a sense, the above situation occurs all over the place. (Though we are not aware of it being singled out as such, named and studied conceptually.) There also appear to be rather natural examples with C\mathsf{C} being some category of topological spaces and FF being a suspension-endofunctor, but using such examples in this question more than briefly touching upon them in this comment would be irrelevant and distracting.

    • The concrete category C\mathsf{C} that we are using is rather special from a categorical point of view, in particular, it is a left cancellative category, but it is not at all finite. Spelling out these properties seems irrelevant to the question.

    • Needless to say, writing “functor P:C2P\colon\mathsf{C}\rightarrow 2” immediately implies that PP formalizes the usual notion of an isomorphism-invariant property: if there is an iso O 0fO 1O_0\overset{f}{\rightarrow}O_1 in C\mathsf{C}, then of course P(O 0)P(f)P(O 1)P(O_0)\overset{P(f)}{\rightarrow}P(O_1) is an iso in 22, so already its being a morphism at all, together with 22 being discrete, implies P(O 0)=P(O 1)P(O_0)=P(O_1). One could say that any functor into the discrete category with two objects formalizes the concept of a morphism-invariant property: if there is a morphism between two objects, then they must either both have the property or both not have it.

    • CommentRowNumber2.
    • CommentAuthorTodd_Trimble
    • CommentTimeJun 14th 2017
    • (edited Jun 14th 2017)

    Putting Ob(2)={0,1}Ob(2) = \{0, 1\}, it seems to be tantamount to a pair of functors G:C 0C 1,H:C 1C 0G: C_0 \to C_1, H: C_1 \to C_0. Given such a (P,F)(P, F), let C 0C_0 be the fiber over 00 and C 1C_1 the fiber over 11, etc. In the other direction, given the pair G,HG, H, let C=C 0+C 1C = C_0 + C_1, and define F:CCF: C \to C by GG on C 0C_0 and HH on C 1C_1, etc. Of course you know this already.

    I can’t think of a name from the literature, but a whimsical name might be “pisces-functor”. It amounts to a quiver map (ab)Cat(a \leftrightarrows b) \to Cat where a,ba, b are the “two fish”.