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- Anders Kock
- 2007

We present here the equational two-dimensional categorical algebra which describes the process of freely completing a category under some class of limits or colimits. It is crystallized out of the authors 1967 dissertation [6] (revised form [7]). I presented a purely equational aspect of that already in 1973 [9], [10] , and the present note is in some sense… (More)

- ANDERS KOCK
- 1996

We analyze the Bianchi Identity as an instance of a basic fact of combinatorial groupoid theory, related to the Homotopy Addition Lemma. Here it becomes formulated in terms of 2-forms with values in the gauge group bundle of a groupoid, and leads in particular to the (Chern-Weil) construction of characteristic classes. The method is that of synthetic… (More)

- ANDERS KOCK
- 2005

In [4] we proved that a commutative monad on a symmetric monoidal closed category carries the structure of a symmetric monoidal monad ([4], Theorem 3.2). We here prove the converse, so that, taken together, we have: there is a 1-1 correspondence between commutative monads and symmetric monoidal monads (Theorem 2.3 below). The main computational work needed… (More)

The notion of commutative monad was denned by the author in [4]. The content of the present paper may briefly be stated: The category of algebras for a commutative monad can in a canonical way be made into a closed category, the two adjoint functors connecting the category of algebras with the base category are in a canonical way closed functors, and the… (More)

- ANDERS KOCK
- 2012

It is shown how the theory of commutative monads provides an axiomatic framework for several aspects of distribution theory in a broad sense, including probability distributions, physical extensive quantities, and Schwartz distributions of compact support. Among the particular aspects considered here are the notions of convolution, density, expectation, and… (More)

- Anders Kock, Max Kelly
- 1991

a ≤ b iff a is the supremum of the subset {a} ∩ {b}. On any elementary topos E , one has the functor which to an object A associates the object TA = Ã which classifies partial maps into A (cf e.g. [J] 1.2). This functor T carries a monad structure T= (T, η, μ) ; it is a submonad of the power ”set” monad P= (P, η, μ), as described in, say, [AL],[Mi], or [J]… (More)

- Anders Kock
- 1989

If A is a small category and E a Grothendieck topos, the Kan extension LanF of a flat functor F : A → E along any functor A → D preserves whatever finite limits may exist in D; this is a well known fundamental result in topos theory. We shall present a metamathematical argument to derive out of this some other left exactness results, for Kan extensions with… (More)

- Anders Kock
- 1999

In this paper, I intend to put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid, and groupoid action in the focus for fibre bundle theory in general, and for connection theory in particular. In so far as connection theory… (More)

- Anders Kock
- 2005

We prove that the forgetful functor from groupoids to pregroupoids has a left adjoint, with the front adjunction injective. Thus we get an enveloping groupoid for any pregroupoid. We prove that the category of torsors is equivalent to that of pregroupoids. Hence we also get enveloping groupoids for torsors, and for principal fibre bundles.

- ANDERS KOCK
- 2000

In the context of synthetic differential geometry, we study the Laplace operator an a Riemannian manifold. The main new aspect is a neighbourhood of the diagonal, smaller than the second neighbourhood usually required as support for second order differential operators. The new neighbourhood has the property that a function is affine on it if and only if it… (More)