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The snarky response would be "the opposite category of any of the categories you could name on the spot". The less-snarky response is to observe that some of these are quite natural. For example, $\ma …

Take any simplicial set $X$ which is not a Kan complex. Let $K$ be a Kan replacement of $X$, and let $L$ be a Kan replacement of the pushout $K\amalg_X K$. Then the two maps $K\to L$ are levelwise inj …

Shouldn't it just be $\operatorname{Spec}(\Lambda)/\mathbb{G}_m$, where $\Lambda = \mathbb{Z}[d]/d^2$ and the $\mathbb{G}_m$ action encodes the grading with $d$ in degree $-1$? This is just the observ …

Yes, you have to include the case $i=j$. Just look at what happens in the case of a single $U_1$, in order for this to boil down to the concept of a single effective epimorphism (introduced in the pre …

Here's a partial answer constraining $\alpha_{S^1}$. First of all, I think my comment shows that (at least if we take $\operatorname{hTop}$ to be the homotopy category of CW complexes), it is possible …

Yes: Since $\mathcal{C}$ is stable, $\operatorname{Fun}(\mathcal{C},\mathcal{C})$ is stable, too. In particular, it has finite colimits, so $\operatorname{Ind}\operatorname{Fun}(\mathcal{C},\mathcal{C …

You can always reduce an arbitrary "zigzag" as in the first definition, to one of length two, as in the second definition, by applying the following trick: Whenever you encounter morphisms $E_{j-1}\ …

As Qiaochu has explained in the comments, strict higher categories are less general: You can pass from strict to weak higher categories, but many important $\infty$-categories (for example the fundame …

This does give you a vector bundle, and it comes with a flat (i. e. path homotopy invariant) parallel transport. You cannot get all vector bundles, but you can get exactly the ones with such a paralle …

I don't think this is true as written, for example there are anima $X$ which are not loop spaces of anything (e.g. $X=S^2$). One can also directly see that $\mathrm{ev}_n$ is corepresented by $\Delta^ …

A discussion of the relation between dependent coproducts and cartesian products is found at https://ncatlab.org/nlab/show/dependent+sum