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?)

]]>Hi,

I am working on a theory of physics that is intended to allow for variability over categories. By this I mean, a science that allows the user to reason over categories and even evolve his theory according to an evolution over categories. I intend for this theory of physics to allow for something called the “approximation and idealization of structure” and this is meant to allow a scientist to “have” an approximation to a structure which represents only the information which he has had access to up to some instant. The physics would allow the scientist to evolve his approximation and refine it given new information. This kind of mathematical method, I believe, would be relevant when considering a physics that is true at all stages in the history of an observer’s universe. For instance, in an early universe, when there are few or no events, the mathematical structure that is assumed to be relevant, only fits the data seen thus far. As a simple example, in a universe with only a single event, a theory should not presume more structure than would be exhibited by a system whose type is in a category containing only one morphism.

The example I want to talk about here is the approximation to a popular toy quantum category, $FREL$, the category of finite sets and relations. I believe this category is interesting to some modern researchers for two reasons. First, it is a toy quantum theory intended to piece apart quantumness by allowing only some quantum properties. Second, the category deals with finite sets and this has a flavour of quantum gravity to an extent. Regardless, $FREL$, along with $REL$ (sets and relations) are important toy quantum categories for present day researchers. In the spirit of the theory I am working on, we would reject either FREL and REL because it calls on all sets in their construction and neither the category of Sets or finite sets makes sense in a universe that has only a finite number of events. The solution to the problem is to have an approximation to $FREL$ that can evolve.

To approximate FREL we choose an ambient category, $Cat$, the category of small categories. $Cat$ is locally finitely presentable (which helps us). Next, we understand that the compact objects in Cat are the finite graphs. A finite set is a discrete category in $Cat$ and all finite sets are in $Cat$, namely the discrete (finite) categories. A relation between two finite sets is a finite graph. Take a first approximation to $FREL$, call it $APPR$, as a set of objects $ob(APPR) \in ob(Cat)$ and a set of morphisms $mor(APPR) \in ob(Cat)$. Next, consider adjoining an object and some morphisms to $APPR$ by finding the disjoint union of categories in $Cat$. Coproducts in $Cat$ are the disjoint union of categories. Here is a defintion from nlab and there is an example there for $Cat$.

Ultimately, we want to find $FREL$ as a colimit over all such categories in $Cat$.

How do we develop colimits in $Cat$ from coproducts and coequalizers? I have been told by a researcher at Oxford that coequalizers are very difficult in $Cat$, thus making colimits difficult. I am wondering if anyone can walk through this calculation with me?

]]>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?

]]>The nLab article retract has Corollary 1, which sketches a proof of the fact that retracts of homotopy (co)limit diagrams are again homotopy (co)limit diagrams.

What is the original reference (if any) for this statement?

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