# Colimits, limits, and mapping spaces

It is true that in the category of topological spaces $\mathrm{Map}(\underset{i\in I}{\mathrm{colim}}\, X_i, Y)\cong \underset{i\in I}{\mathrm{lim}}\,\mathrm{Map}(X_i,Y)$ ? Here mapping spaces are endowed with the compact-open topology. One has a bijective map from the left-hand side to the right-hand side, but is this map a homeomorphism? For example, if $X_1\subset X_2$, is the inclusion $\mathrm{Map}(X_2/X_1,Y)\subset \mathrm{Map}(X_2,Y)$ a homeomorphism on its image? If not, what if this inclusion is a cofibration?

(Compare with my previous question Mapping space from a quotient space.)

• I just realized that the first question has a negative answer. One can easily cook up a pushout counterexample from the counterexample in: mathoverflow.net/questions/274994/…. But the second and third questions still need an answer. – Victor Jan 3 '18 at 17:21

This is true if, instead of topological spaces, you work in a convenient category of topological spaces, in the sense of Steenrod. These are the place you want to do homotopy theory in (assuming you want to do it using topological spaces and not, for example, Kan complexes). In this case, the functor: $$\mathrm{Map}(-,Y):C^{op}\to C$$ is right adjoint to the functor $$\mathrm{Map}(-Y):C\to C^{op}$$ (careful: adjunctions with the opposite category are confusing :)).

Indeed, if we denote by $C(X,Y)$ the set of continuous maps from $X$ to $Y$ $$C(Z, \mathrm{Map}(X,Y))=C(Z\times X,Y)=C(X,\mathrm{Map}(Z,Y))$$

(in this case $\mathrm{Map}(X,Y)$ will in general be different from the compact open topology, for example if we work in compactly generated spaces it will have the Kelleyfication of the compact open topology).

• Denis, thank you! I think that's exactly what I need. Please allow me some time to read the literature on this question. Indeed, I need a model category of topological spaces in which I would have the above property, also the exponential property $Map(X,Map(Y,Z))\cong Map(X\times Y,Z)$ and also $Map(X,\prod_i Y_i)\cong \prod_i Map(X,Y_i)$. By the way, can you suggest any good reference to read? – Victor Jan 3 '18 at 17:27
• @Victor The nlab page I linked in the answer is a good guide for the literature on this. The starting point is probably Steenrod's classical paper titled A convenient category of spaces. I don't know the subtleties very well (I prefer to work with simplicial methods) but I think it is pretty much irrelevant which category you choose, provided it is cartesian closed, locally presentable and contains all CW complexes. – Denis Nardin Jan 3 '18 at 18:31
• Denis, thanks again. I was a bit slow, but I finally understood what you meant: since Map(-,Y) has a right/left adjoint it preserves colimits/limits, which is exactly my question. I still need to figure out whether in a convenient category one has $Map(X,\prod_i Y_i)\cong \prod_iMap(X,Y_i)$, but I have a feeling I am on a right track! – Victor Jan 4 '18 at 1:41
• @Victor In a cartesian closed category this too is automatic: $\operatorname{Map}(X,-)$ preserves all existing limits, being the right adjoint to $X\times-$. – მამუკა ჯიბლაძე Jan 4 '18 at 5:07
• @DenisNardin None of the categories presented in ncatlab.org/nlab/show/convenient+category+of+topological+spaces are locally presentable. However, there does exist a locally presentable convenient category of topological spaces. – Philippe Gaucher Jan 9 '18 at 12:57