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I don't think I've seen a definition of a space-like notion phrased only in terms of paths, but you could certainly write one down, perhaps as an example of concrete sheaves. It seems related to the …

Especially when doing topos theory, one sometimes uses the Sierpinski space (the two-point space with one open point) as a sort of "directed interval." This is convenient because "Sierpinski homotopi …

Actually, I think the statement that you quoted from the nLab is wrong. It was copied from v1 of the Bergner-Rezk paper, but v2 corrected the statement to be only about simplicial presheaves on $R$ ( …

One property that one sometimes wants, but which is not satisfied by most of the "usual" convenient categories, is that of being locally cartesian closed, or equivalently that each pullback functor $f …

Suppose $X$ and $Y$ are pointed Kan complexes. Is their smash product $X\wedge Y$ also a Kan complex? I expect the answer is probably no, but it would be nice to see a counterexample.

In many places (on MO, elsewhere on the Internet, and perhaps even in some textbooks) one finds a statement of the classical Brown representability theorem that looks something like this: If $F$ i …

The category of simplices $\Delta$ has a terminal object $[0]$, hence its nerve is contractible. What can be said about the nerve of its subcategory $\Delta_{\mathrm{mono}}$ which contains only the c …

I'm not sure whether you'll like this, but my natural response to "how should I think about delooping?" is to invoke (higher) category theory. You may know that a homotopy 1-type, i.e. a space (proba …

You may also be interested in this paper.

If I'm not mistaken, here is an "even worse" counterexample than Charles'. Let $R$ be the walking isomorphism $(0\cong 1)$, let $R_+$ be $(0\to 1)$, and let $R_-$ be $(1\to 0)$. The conditions on de …

Here's a partial answer to a related question. In classical stable homotopy theory, a spectrum is equivalent to a "strict" one (arising from a chain complex) if and only if it admits the structure of …

A very concrete example of a cospan having no pullback in the homotopy category, which does not require any knowledge of cohomology or Moore spaces, can be found here. It's phrased in terms of the ho …

If "space" means the same thing in the two cases, as seems to be implied, then there is no difference, at least not at that level of precision. There are many different models of $(\infty,1)$-categor …

As Ryan points out, if Y is allowed to be disconnected, then there is no hope, since the loop-space construction sees only the connected component of the basepoint. But even if Y is assumed to be con …

A different perspective is that of Elmendorf's theorem: the homotopy theory of G-equivariant spaces (in the sense usually meant) is equivalent to the homotopy theory of diagrams on the orbit category …