August 23, 2016

Roald Koudenburg,

*Hypervirtual double categories*

**Abstract:**Hypervirtual double categories (a.k.a. fc-bl categories) are like virtual double categories but where there are multi-cells with empty codomain as well as the usual ones whose codomains are single arrows. The prototype involves (small) profunctors on large categories. In this way smallness conditions, of the sort needed for Yoneda structures, are coded into the double category structure.

September 13, 2016

Mark Johnson (Penn State Altoona),

*A concrete approach to higher homotopy operations*

**Abstract:**While computationally important, especially in studying the stable homotopy groups of spheres, higher homotopy operations are notoriously difficult to define explicitly. We will discuss an approach considering HHOs as the possible obstructions to realizing homotopy commutative diagrams. The explicit nature of this approach, using grids of pullback diagrams, will allow us to recover the familiar Toda brackets and other related constructions as examples. Joint with David Blanc (Haifa) and James Turner (Calvin).

September 20, 2016

Richard Wood (Dalhousie),

*On an idea of Freyd and Street*

**Abstract:**If a category C is equivalent to a small category then both C and set^{C^{op}} are locally small. Showing that the hom sets of set^{C^{op}} are small when C is small is an important exercise in introductory category theory but the converse to the first sentence is a rather strange result that was proved by Freyd and Street in the 1970's and independently by Folz. In the first volume of TAC, Freyd and Street published their proof (many years after the appearance of Folz's proof). Freyd and Street also conjecture, for a site (C,J), that if C and shv(C,J) are locally small then shv(C,J) is a Grothendieck topos. Apparently, this is still an open question. The speaker and Street, in proposed joint work, hope to show that, for C and shv(C,J) locally small, shv(C,J) is a lex total category. We also hope to show that, for a locally small taxon T with i-mod(1,T) locally small, i-mod(1,T) is totally distributive. It should be noted that our strategy bi-passes the question of whether C and T above are set-theoretically small and begs adoption of defining smallness in terms of local smallness as suggested in Street and Walters on Yoneda structures. The talk will begin with Freyd and Street's proof and elaborate on our strategy for the new problems.

September 27, 2016

Bob Paré (Dalhousie),

*The secret double life of graphs*

**Abstract:**By graphs I mean, as do most category theorists, what graph theorists might call directed multigraphs. A category is such a graph with an appropriate multiplication on its edges. In fact, a precise way to say this is that the forgetful functor from the category of (small) categories to graphs, which has a left adjoint, is monadic. This is of course well-known. What is perhaps less well-known is that the left adjoint, the paths functor, is comonadic. So one could say that graphs are categories with costructure. Here we have a nice comonad on

**Cat**which is not a 2-comonad, and this is a bit troubling. However,

**Cat**is not a mere 2-category but a double category, and the paths comonad lifts to a lax comonad on that. Considering the coalgebras on this gives a notion of "proarrow of graphs" thus giving a double category of graphs over which the double category of categories is monadic. We will study some of the properties of this double category.

October 4, 2016

Jeff Egger (Sackville, NB),

*Groupoid uniformities*

**Abstract:**During my talk at CT, I mentioned localic groupoids as one potential source of examples for fibred uniformities (that is, uniform locales internal to a localic topos). In this talk, I will show that they are; I will also attempt to describe the imagined obstacle which stopped me from trying to work out this class of examples earlier.

October 11, 2016

Bob Paré (Dalhousie),

*The secret double life of graphs, revealed*

**Abstract:**This is the second part of my September 27 talk.

October 18, 2016

Evangelia Aleiferi (Dalhousie),

*Cartesian double categories*

**Abstract:**A double category is said to be Cartesian if the diagonal double functor and the unique double functor to the terminal double category have right adjoints. We will give examples of such and we will compare with Cartesianess in the bicategorical sense. We will especially talk about fibrant Cartesian double categories and some of their properties.

November 1, 2016

Geoff Cruttwell (Mount Allison),

*Differential equations in tangent categories I*

**Abstract:**Tangent categories provide an axiomatic approach to the tangent bundle, one of the central objects in differential geometry. Working in this axiomatization, a number of further important concepts from differential geometry can be defined. In particular, vector fields, the Lie bracket, vector bundles, connections, and de Rham cohomology are all examples of concepts that can be defined within the axiomatic setting of a tangent category.

However, a number of other important concepts in differential geometry (for example, parallel transport, geodesics, and integrability theorems) rely on being able to solve certain (ordinary) differential equations. It is clear that the tangent category axioms alone are not powerful enough to allow one to "solve differential equations" in this setting. The purpose of this talk is to look at additional axioms to add to a tangent category which would allow one to do this, and to see some of the consequences these axioms would entail.

(joint work with Robin Cockett and Rory Lucyshyn-Wright)

November 8, 2016

Geoff Cruttwell (Mount Allison),

*Differential equations in tangent categories II*

**Abstract:**Continuation of last week's talk.

November 15, 2016

Darien DeWolf (Dalhousie),

*Restriction monads and algebras*

**Abstract:**A restriction monad in a double category $\mathbb{D}$ is a monad equipped with structure reminiscent of that in a restriction category. Indeed, the fact that monads in the double category $\mathrm{Span}(\mathbf{Set})$ are small categories generalizes immediately to restriction monads in $\mathrm{Span}(\mathbf{Set})$ being small restriction categories. We introduce also a notion of restriction algebra for a monad. If $X$ is a restriction category and $R(X)$ its corresponding restriction monad in $\mathrm{Span}(\mathbf{Set}),$ then the restriction algebras for $R(X)$ are right $X$-modules. Moreover, the modules arising in this way can be naturally given a suitable restriction structure. We will then organize these structures in a double category $\mathrm{rMod}(\mathbb{D})$ of restriction monads, which gives an example of a so-called double restriction category, all of whose vertical arrows are total.

November 29, 2016

Marzieh Bayeh (Dalhousie),

*Orbit Class and Invariant Topological complexity*

**Abstract:**Let G be a compact, Hausdorff, topological group acting on a Hausdorff topological space X. In this case X is called a G-space. It has always been interesting to develop equivariant versions of topological invariants for G-spaces; these would be invariants that are compatible with the G-action and preserve the orbit structures. One of these invariants is topological complexity. The topological complexity (TC) of the configuration space of a mechanical system was introduced by M. Farber, to estimate the complexity of motion planning algorithms. W. Lubawski and W. Marzantowicz developed an equivariant version of TC, called the invariant topological complexity.

In this talk we first define a new concept, called the orbit class, to study G-spaces. Then using this new tool, we study the invariant topological complexity. In particular, we introduce condition that ensures that the invariant topological complexity is finite.

January 10, 2017

Dorette Pronk (Dalhousie),

*Bicategories of Fractions Revisited*

**Abstract:**Motivated by questions about mapping spaces for orbifolds, we revisit the the bicategories of fractions conditions and construction. (I will give both in detail.) In particular, in this talk we will present a weaker set of conditions on the class of arrows to be inverted, and we will present a canonical form for the 2-cell diagrams using the pseudo pullbacks. This presentation makes horizontal composition much simpler than in the general case.

January 17, 2017

Dorette Pronk (Dalhousie),

*Bicategories of Fractions Revisited (continued)*

January 24, 2017

Michael Lambert (Dalhousie),

*A Categorical Approach to Wild and Undecidable Theories of Modules*

**Abstract:**Guided by definitions in the representation theory of associative algebras over an algebraically closed field, a representation embedding between categories of models of algebraic theories can be defined to be a functor between the respective categories of models that preserves indecomposibility and projectivity and that reflects certain epics. The main result is that such a representation embedding preserves undecidability of theories; that is, if there is a representation embedding of algebraic theories T and T' in this sense, then if T is undecidable, so is T'. This result is applied to obtain an affirmative resolution of a reformulation of a conjecture of M. Prest that every "wild" associative algebra over an algebraically closed field has an undecidable theory of modules.

January 31, 2017

Bob Paré (Dalhousie),

*The secret double life of graphs, exposed*

**Abstract:**I will introdue a notion of proarrow for graphs, simpler than the one hinted at in the previous talks and give some justification for it. Then I will look at the properties of the double category of graphs this gives.

February 7, 2017

Bob Paré (Dalhousie),

*Some properties of the double category of graphs*

**Abstract:**I will recall the notions of companion, conjoint, tabulator, and map in a double category and study how these pertain to the double category of graphs introduced in the last talk.

February 28, 2017

Rory Lucyshyn-Wright (Mount Allison),

*Introduction to commutants for algebraic theories*

**Abstract:**In 1963, Lawvere introduced an elegant approach to universal algebra in terms of the notion of algebraic theory, or Lawvere theory. In a 1968 article, Lawvere emphasized that algebraic theories generalize rings, so that certain notions in the theory of rings and modules admit generalizations to universal algebra. In this spirit, Wraith's 1969 lecture notes define a notion of commutant of a set of morphisms in an infinitary algebraic theory. Given instead a morphism of Lawvere theories A : T --> U, one can form an associated Lawvere theory, called the centralizer or commutant of T with respect to A. As a special case, one can take the commutant of a Lawvere theory T with respect to any given T-algebra A. In this talk we will discuss several examples of commutants of Lawvere theories, including theories of modules, affine spaces, convex spaces, and semilattices. Time permitting, we will also comment on generalizations to the realm of enriched algebraic theories and monads.

References:

F. W. Lawvere, Functorial semantics of algebraic theories, Dissertation, Columbia University, New York, 1963. Available in: Repr. Theory Appl. Categ. 5 (2004).

F. W. Lawvere, Some algebraic problems in the context of functorial semantics of algebraic theories, Reports of the Midwest Category Seminar. II, Springer, 1968, pp. 41-61. Available in: Repr. Theory Appl. Categ. 5 (2004).

R. B. B. Lucyshyn-Wright, Convex spaces, affine spaces, and commutants for algebraic theories. Preprint. arXiv:1603.03351 (2016).

R. B. B. Lucyshyn-Wright, Commutants for enriched algebraic theories and monads. Preprint. arXiv:1604.08569 (2016).

G. C. Wraith, Algebraic theories, Lectures Autumn 1969. Lecture Notes Series, No. 22, Matematisk Institut, Aarhus Universitet, Aarhus, 1970 (Revised version 1975).

March 21, 2017

Bob Rosebrugh (Mount Allison),

*Universal updates for symmetric lenses*

**Abstract:** "Asymmetric" lenses provide a strategy to lift updates in a model
domain along a morphism of model domains. A "symmetric" lens between two model
domains has state synchronization data and resynchronization operations. When the
model domains are categories we speak of delta-lenses (or d-lenses). In previous
work we showed that (certain equivalence classes of) spans of asymmetric d-lenses
represent symmetric d-lenses. Asymmetric c-lenses are a special case of asymmetric
d-lenses whose updates satisfy a universal property which can be construed as
"least-change''. So it was natural to hope that symmetric c-lenses characterize the
symmetric d-lenses which satisfy a natural universal property.

Instead, we'll explain why we do not expect all symmetric c-lenses (viewed as
equivalence classes of spans of c-lenses) to be central to developing universal
properties for symmetric d-lenses. We consider instead cospans of c-lenses and
show that they generate symmetric c-lenses with a universal property. That property
is further analyzed towards obtaining universal, least-change, properties for
symmetric d-lenses. We also explore how to characterize those symmetric d-lenses
that arise from cospans of c-lenses.

(Joint work with Michael Johnson)

March 28, 2017

Gabor Lukacs (Halifax),

*On group-valued continuous functions*

**Abstract:**For a space X and topological group A, one denotes by C(X,A) the set of continuous maps on X with values in A, equipped with pointwise operations and the compact-open topology.

If X is compact, then C(X,A) carries the uniform topology, and for suitable choices of X, one obtains groups that resemble l^\infty and c_0, which are of interest in their own right.

For an abelian group G, let G^ denote the group of all continuous characters of G equipped with the compact-open topology. The group G is *reflexive* if the evaluation map G --> G^^ is a topological isomorphism.

In this talk, we present results concerning reflexivity of C(X,A).

April 4, 2017

Bob Raphael (Concordia),

*On regrouping series to obtain absolute convergence*

**Abstract:**The talk is based on concepts from undergraduate analysis and requires no other background. Different kinds of convergence, regrouping, and rearranging are briefly reviewed.

A convergent series has an absolutely convergent regrouping. A discussion of how, and where, this can be generalized to series of functions ensues. Functions which have Taylor series representations, and functions satisfying Holder continuity conditions appear.

This is a report on joint work in progress with Emilia Alvarez of Concordia University