On the cartesian closed T-spaces Good morning to everybody. I have a question concerning exercise 27I (pag. 450) of Adamek's "The Joy of Cats" (that you can find here: http://katmat.math.uni-bremen.de/acc/acc.pdf)
Such an exercise characterize the $T$-spaces (where $T$ is a set-functor) which are cartesian closed. More in detail, I do not well understand the meaning of the sentence "T weakly preserves pullbacks, i.e., for each 2-sink $(f,g)$, the factorizing morphism of the $T$-image of the pullback of (f,g) through the pullback of $(Tf, Tg)$ is a retraction.
If I have well understood, I must consider the $2$-sink $(f,g)$ and next to compute their pullback. Call it $(P,h_1,h_2)$. Secondly, I must consider the image of the previous pullback square, namely the square with arrows $T(h_1),T(h_2),Tf,Tg$ and with vertices $T(P),T(Dom(f)),T(Cod(f)),T(Dom(g))$. In general this is not a pullback square. Therefore, if I consider the pullback $(X,k_1,k_2)$ of $(T(f),T(g))$, then there exists a unique morphism $s: T(P) \to X$ (the factorizing morphism of the $T$-image of the pullback of $(f,g)$ through the pullback of $(Tf,Tg)$) such that $s \circ k_1=T(h_1)$ and $s \circ k_2=T(h_2)$. I must only show that $s$ is a retraction. Is it my interpretation correct?
I have also a second question: in the paper "J. Adamek, V. Koubek, Cartesian closed functor-structured categories, Commentationes Mathematicae Universitatis Carolinae, Vol. 21 (1980), No. 3, 573-590." the authors characterize cartesian closed T-spaces by requiring that $T$ covers each non-empty pullback. A pullback is said to be covered by a functor if this functor maps it on a square, through which all commuting squares factorize but not necessarily uniquely.
Well, my question is: it seems that T covers non-empty pullbacks is equivalent to that of weakly preserve pullbacks. But why (without using their equivalence with cartesian closedness)?
Using the definition of retraction, it is easy to check that weakly preserves pullbacks implies covers non-empty pullbacks!
 A: Regarding the first question, your interpretation is correct. Regarding the second, one can interpret "covering" as sending a pullback to what is known as a weak pullback. One can then show weak pullbacks are exactly retractions onto pullbacks (hence "cover" being interpreted as retraction onto).
In detail, given a diagram $J\colon D\to C$, a cone $h_j\colon d\to Jj$ is called a weak limit of $J\colon D\to C$ if every cone $g_j\colon d'\to Jj$ factors as $h_j\circ g'\colon d'\to d\to Jj$ for some (not necessarily unique) morphism $g'\colon d'\to d$.
If there is a limiting cone $\pi_j\colon \lim J\to Jj$, then the cone $h_j\colon d\to Jj$ factors as $\pi_j\circ h\colon d\to\lim J\to Jj$ for a unique morphism $h\colon d\to\lim J$. Since any other cone $g_j\colon d'\to Jj$ factors uniquely as $\pi_j\circ g\colon d'\to\lim J\to Jj$, we see that $h\colon d\to\lim J$ is a weak limit if and only if every morphism $g\colon d'\to\lim J$ factors as $h\circ g'\colon d'\to d\to\lim J$ for some morphism $g'\colon d'\to d$. In particular, it is necessary that $\mathrm{id}_{\lim J}\colon\lim J\to\lim J$ factor as $h\circ s\colon\lim J\to d\to\lim J$ for some morphism $s\colon\lim J\to d$, i.e. that $h\colon d\to\lim J$ be a retraction. Conversely, given such a factorization, any $g\colon d'\to\lim J$ factors as $(h\circ s)\circ g=h\circ(s\circ g)\colon d'\to d\to\lim J$.
