# Cohomology vs. Reduced Cohomology in homotopy type theory

I just read Mike Shulman's blog post the other night, discussing the definition of cohomology in Homotopy Type Theory. I really enjoyed the new (to me) perspective, but I had a few questions on technicalities.

Given types $$X$$ and $$Y$$, Mike defines

$$H^n(X; Y) :\equiv \| X \to \Omega^{-n} Y \|_0.$$

This agrees with the definition given in the nLab article on cohomology. Here is where my first question comes up:

1. Although it's not stated in the article, I assume that for $$n \neq 0$$ we need $$Y$$ to be a pointed type, since (if I understand correctly) looping/delooping requires a choice of basepoint? Or does it not matter?

(Mike does explain in the article that for $$n>0$$ we need to specify choices of deloopings, since deloopings might not exist and might not be unique, and that we usually just take $$Y$$ to be a spectrum so that this works.)

However, in section 1.1 of this paper on Eilenberg-MacLane spaces in HoTT by Daniel Licata and Eric Finster they seem to be saying that the definition above actually produces reduced cohomology, since they write:

On the other hand, in his blog post Mike writes defines reduced cohomology for pointed types $$X$$ and $$Y$$ by $$\tilde{H}^n(X; Y) :\equiv \| \operatorname{Map}_*(X, \Omega^{-n} Y) \|_0.$$ (He technically defines it only in the case that $$Y$$ is a spectrum, but I'm extrapolating here.) So this leads me to my second question:

1. What is the correct definition for reduced cohomology in HoTT?

1. In order for the notation $$\Omega^{-n} Y$$ to make sense for $$n \ge 1$$, $$Y$$ must be a grouplike $$E_n$$-space, which in particular means it has a basepoint (the identity of the $$E_n$$ multiplication). For $$n \le -1$$ we also need a basepoint to define the based loop space. However note that $$X$$ does not need to have a basepoint.