Created slice 2-category.

]]>At the very least the pages all curently feel circular when trying to learn what exactly a 2-functor is, with [[pseudofunctor]] the only page really providing any accessible definitions (but they are very nice and explicit).

I have the various definitions and diagrams latexed up in my own notes (using xypic syntax) and would copy-paste them in and rewrite/merge the pages myself (assuming there are no objections) but I'm not sure which levers to pull on the nLab to merge/delete pages, or if a lowly browser like me even has such powers. ]]>

This is a different approach - more restrictive, I believe - than Mike's work at [[michaelshulman:exact completion of a 2-category]], where he talks about anafunctors in a 2-category (weak, by default, for him). There he talks about anafunctors in the 2-category of 2-congruences in a 2-site, and thinking of them more along the lines as in Cat(S). But I'm very interested in the relation between the two, especially if one could be derived from the other.

One spin-off of this is that I would like to provide another model for the localisation of a 2-category. Here J needs to be weakly cofinal in the class W one wants to invert. One point of my anafunctors paper was to show that the localisation of a 2-category of internal categories had a better model that the default one constructed by Pronk, and this theorem should go through, namely K[W^-1] ~ K_ana. Note that this is (2,2)-category localisation, not (2,1)-category localisation. (As an aside, the approach to localisation via bibundles, which is even simpler to describe, wouldn't work here because that assumes one is in a (2,1)-category.)

The one point which is a bit restrictive is that one needs covers to be an [[ff morphism]] in order to define the bicategory K_ana of anafunctors in K. (This reminds me somewhat of talking about S-local maps in a model theoretic setup, at least when the pretopology J is morally like a cover by open balls or affine schemes. But I haven't thought about this too much yet.)

One direction this may go is if the whole game can be phrased in a suitably 2-categorical way, then perhaps similar techniques could be used to talk about localisation of higher categories (say simplicial categories), at least in special cases. For example, defining weak maps between strict higher categories or something. This is complete speculation, and not a short-term goal by any means.

Thoughts? ]]>

In Cartan and Eilenberg there are several instances of squares which “anticommute”, that is, we have $h \circ f = - k \circ g$ instead of $h \circ f = k \circ g$. I was wondering if we could make this into an instance of a square commuting “up to a specified 2-morphism” and it turned out the answer was yes.

Let $\mathbf{C}$ be a (small) category. We attach to every parallel pair of 1-morphisms $f, g : X \to Y$ the set of all natural transformations $\alpha : id_\mathbf{C} \Rightarrow id_\mathbf{C}$ such that $g = \alpha_Y \circ f$. The vertical composition is obvious, and if we have another parallel pair $h, k : Y \to Z$ and a 2-morphism $\beta : h \Rightarrow k$, the horizontal composition of $\alpha$ and $\beta$ is just $\beta \circ \alpha$, since $k \circ g = (\beta_Z \circ h) \circ (\alpha_Y \circ f) = (\beta_Z \circ \alpha_Z) \circ (h \circ f)$, by naturality of $\alpha$. This yields a (strict) 2-category structure on $\mathbf{C}$. Note that we have to remember which natural transformation is needed to make the triangle commute in order to have a well-defined horizontal composition.

In the specific case of $\mathbf{C} = R\text{-Mod}$, the set (class?) of natural transformations $id_\mathbf{C} \Rightarrow id_\mathbf{C}$ include the scalar action of $R$, so in particular the anticommutative squares of Cartan and Eilenberg can be regarded as a square commuting up to a 2-morphism.

My question now: Is there a name for this construction?

]]>Created F-category and rigged limit.

]]>I have created lax morphism, with general definitions and a list of examples. It would be great to have more examples.

]]>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.

]]>People may have noticed my question on MO and on the categories mailing list, so I’m not going to labour the point too much. This thread is really to record my thoughts and welcome input from the peanut gallery.

Note $Cat$ is the 1-/2-category of small categories.

The functor $Obj:Cat \to Set$ is a fibration (amongst other things), because one can perform ’change of base’ given a category $C$ and a function $f:D \to C_0$, we have a cartesian arrow $F:C[f] \to C$. Given a natural isomorphism between $F,G$ arising from $f,g:D\to C_0$, we have a canonical isomorphism between $C[f]$ and $C[g]$ over $C$. I think that there is a more general behaviour for non-invertible 2-arrows, and for 2-arrows from and/or to $F$, but I will think about this later (busy, busy at work).

Now there is a notion of 2-fibration given by Hermida, and one given by Bakovic. I’ll examine whether $Obj$ is a 2-fibration by either of these definitions.

]]>