Created complete small category, and moved the proof of Freyd’s theorem to there from adjoint functor theorem.

]]>Are there results about the existence of lax and oplax limits and colimits in locally posetal categories (such as Pos and Rel)? I would for example imagine that all those limits exist in Pos. But what about, for example, the category of locales or the one of topological spaces, with their canonical 2-cells?

Also, are there lax/oplax analogues of the facts that right adjoint preserve colimits, that monadic functors create them, and so on? Which results carry over?

If this has been done, does anyone of you know where I can find this? (Is all of this obvious?)

]]>Can we view an integral somehow as transfinite composition of “small” arrows, in a rigorous (but possibly nonstandard) way?

For example, if we view a real 1-form as a functor with values in $B\mathbb{R}$, is its integration along a curve a transfinite composition of its values at all the tangent vectors of the curve?

I don’t know if the question is clear enough, in case it isn’t, feel free to ask. (It may also be that none of this makes any sense.)

]]>I started a new page dedicated to ind-pro-objects, more general pro-ind-objects and their iterations.

]]>Homotopy colimits of simplicial diagrams and homotopy limits of cosimplicial diagrams have their own special names: realization and totalization.

Is there a special name for homotopy limits of simplicial diagrams? In general, are there any examples in the literature where such homotopy limits are computed?

]]>(Homotopy) sifted colimits commute with finite (homotopy) products in the category of sets (respectively spaces).

Is it possible to point out a bigger class of categories for which this is true?

Jacob Lurie points out in a comment on MathOverflow (http://mathoverflow.net/questions/181188/commutation-of-simplicial-homotopy-colimits-and-homotopy-products-in-spaces) that this is false for arbitrary presentable ∞-categories.

On the other hand, it seems like this might be true for cartesian closed presentable ∞-categories, because the argument for sets seems to go through in this case.

Also, could it be true for algebras over a finitely accessible ∞-monad? The forgetful functor from algebras to spaces creates limits and sifted colimits, so commutativity should follow from commutativity in spaces.

In general, is it possible to describe a more general class of categories that covers the above examples?

]]>I noticed in passing that the entry *fiber* had been in a sad state all along. I threw in a few more keystrokes to help it a bit. And also created *cofiber*.

I am wondering, if we are working in Choice-free foundations, about the bifurcation of the sorts of finite (co)limits we can have. To save typing, I will only consider limits, and also because they seem to be what I need to think about for what I’m working on.

We have the standard result in usual foundations that terminal object, binary products and equalisers implies all finite limits, but in the case where we have the existence of non-finite D-finite sets, do we still get this? In particular, might it be the case that really we would like the existence of the smaller class $tbe$ of limits generated by terminal object, binary products and equalisers? For the sort of simple reasoning I’m envisaging, these are certainly all I’d want. It seems to be only theorems do we usually get given an arbitrary finite diagram of which we need to take the limit. But could these really be referring to $tbe$-limits? (Hammering the point, I know, but I’m genuinely curious.) Of course, I’ve probably just hidden the non-finite, D-finite sets in the definition of ’generated’, but can we adjust that definition so it is something along the lines of ’generated by a finite sequence involving $t$, $b$ and $e$, which is in bijection with a standard numeral’?

]]>I created inserter and equifier, and added a bit to inverter to bring them all parallel. Then I created a stub for PIE-limit but ran out of steam.

]]>