A Category Theoretic Thinking Perspective on:

Studying Emergence in TeamsUsing Multiple Methods

A SymposiumAcademy of Management Annual Meeting

Vancouver BC, August 10, 2015

James K. HazyAdelphi University

Founder, Leadership Science, LLC

Organizing Organization TheoryCategory Theory

Systems Thinking

Complex Adaptive SystemsManagementTheory

Psychology

IO psych

Social Psych

Neuroscience

Natural Sciences

Traditional Economics

Evolutionary Economics

Resource-Based View

Dynamic Capabilities

Behavior Economics

Operations/Logistics

Decision ScienceSoft Systems Method

Chaos Theory/ Dynamical Systems

Linguistics/Semantics

Logic

MathematicalFields

Computer/DB Science

ITC systems

Social media

Social Physics

Neuroscience

Cybernetics

Statistical mechanics

(Adapted from Spivak, 2014; however, errors are mine alone).

Mathematical Category Theory

• Objects (Categories can be Objects)

• Relationships• Identity • Composition

• Associativity

“sets” or “collections”

“arrows” or “maps” object maps to itself f: A B, g: B C implies g ᵒ

f: A Ch ᵒ (g ᵒ f )= (h ᵒ g) ᵒ f

Think (but not exactly)Abstraction

Abstract Representations of Objects & Relationships that preserve Structure or Order (i.e. store “information”)

(Adapted from Spivak, 2014; however, errors are mine alone).

g ᵒ f gA B

C

f

g ᵒ f

A

Group of members, mi

has a

Team“limit”

has a

Project

hasa

has a

Cooperative Structure“colimit”

P X G P

G

“Pullback” or “meet”- ConstructOrdering in Space of Possibilities

Define: Observed “Emergence of a Team Cooperative Structure” as a “Fact” The above statement is “True” iff the diagram commutes, that is, iff …

P G

Π

“Pushout”or “join”Construct Disjoint Union for Clarity

of Role Interfaces & Map of Activities

Team Project Cooperative Structure Team Group Cooperative Structure

Construct broadestachievable Project given Group for planning purposes

Construct simplestProject roadmap &Group structureto enable action

Construction must be rigorous at increasing levels of detail

Universal property is not decomposable; component objects

“reconstructed” as functors, i.e, “integrated- selves.”

Inclusion

Inclusion

(Adapted from Spivak, 2014; however, errors are mine alone).

Projection

Projection

Hypothesis I: The above diagram commutes, iff… Benefit Maximized (PXG) less Cost Minimized (P G) = Value >

Hypothesis II: Autocatalysis occurs iff catalytic Leadership Practices are present

Hypothesis III: The presence of Residual Complexity between• External Complexity in the ecosystem (context) and the • Internal Complexity within the organizing system (category structure) enables the emergence of a new Cooperative Structure (storing additional information) that enhances the system’s predictability.

Π

Implications

OpportunitySeeking

Value Discovering

StrategicAmbiguityDivergence

Fine-GrainInteractions

(FGI)Autonomy

EntrepreneurialOrganizing

Value Gathering

InterpretativeDominanceConvergence

Coarse-GrainProperties

(CGP)Integrative

ChangingEcosystems

Dynamical Systems &Attractors

Information/ Entropy

Computational/Algorithmic Complexity

Identity/Entity

Structural & RelationalComplexity

Many (Plurality) One (Unity)

Abstract Framework

Level 3: Ecosystem-Level Dynamic search on fitness landscapes

Level 2: Event-LevelPredicting future state underuncertainty

Level 1: Identity-LevelDetermining “Logical” Categories of unity, relations & identity.

Human Interaction Dynamics (HID)Difference – Dualities – Sameness Analytical

RepresentationExploration Exploitation

Let’s look at implications of these layers from the bottom up

(Hazy & Backström, 2013b)

7

Research Questions & Methods

RQ1: What category theoretic frameworks can be developed & how can real-time data be gathered and processed to efficaciously solve organizing problems?

RQ2: Which leadership practices are necessary for catalyzing organizing and at what frequency is each required to sustain each requisite leadership function under various circumstances?

RQ3: What mathematical and computational models can be developed to simulate the potential field effects on organizing under various environmental and internal interaction conditions?

Methods Development: Data collection/analysis techniques and algorithms, a Leadership Practices Data Collection instrument, and mathematical & computational simulation models.

Category Theory (Very Partial) Toolkit

(Adapted from Spivak, 2014; however, errors are mine alone).

• Categories of abstract objects or types & inter-relationships – Sets (e.g., roles, workgroups, firms), Graphs (e.g., social networks), Functors, etc. – finite & infinite; discrete & indiscrete; continuous

• Recursively “layer up” or “drill down” (e.g., Power Laws?) – “Categories of Categories” & Sub-categories as Objects

• Database Schemas are Categories as are networks– Cognitive schema?

• Universal objects & properties might signal emergence– initial & terminal objects “constructed ” over indexed Functor Categories– colimit or “join” least upper bound; “simplest” disjunction into components (e.g.

a subway map has Letter-named and Color-named lines representing “train-equivalence”) – limit or “meet” or greatest lower bound; “broadest” still included in all objects

(e.g., the “ordered pair,” cross-product, i.e., multiplication)

• Left & right adjoints translate among categories – “Free” & “Forgetful” Functors between categories and generalized definitions

References1.Goldstein, J. (2015). Reimagining emergence, Part 3: Uncomputability, transformation, and self-transcending constructions. E:CO ,

16(2): 116-176.

2. Goldstein, J., Hazy, J. K., & Lichtenstein, B. (2010). Complexity and the Nexus of Leadership: Leveraging Nonlinear Science to Create

Ecologies of Innovation. Palgrave Macmillan: Englewood Cliffs.

3.Hazy, J. K., & Ashley, A. (2011). Unfolding the Future: Bifurcation in Organizing Form and Emergence in Social Systems. Emergence:

Complexity and Organization, ISSN: 1521-3250, 13(3), 58-80.

4.Hazy, J. K., & Backström, T. (2013a). Human Interaction Dynamics (HID): Foundations, Definitions, and Directions. Emergence:

Complexity and Organization, 15(4).

5.Hazy, J. K., & Backström, T. (2013b). Editorial: Human Interaction Dynamics (HID)—An Emerging Paradigm for Management Research.

Emergence: Complexity and Organization, (15)4: i-ix

6.Hazy, J. K. & Boyatzis, R. E. (in press). Emotional Contagion and Proto-Organizing in Human Interaction Dynamics. Frontiers in

Psychology Organizational Psychology.

7.Hazy, J. K., Millhiser, W., & Solow, D. (2007). Mathematical and computational models of leadership: Past and future. In J. K. Hazy, J.

Goldstein & B. B. Lichtenstein (eds.). Complex Systems Leadership Theory (pp. 345-370). Mansfield, MA: ISCE Publishing.

8.Hazy, J. K. & Wolenski, P. R. (in press). Phase transition and complex social contagion as enabling mechanism s for coordinated action

in populations: A mathematical framework. In E. Mitleton-Kelly, Prof. A. Paraskevas, C. Day (Eds). Edward Elgar Handbook of Research

Methods in Complexity Science: Theory & Application.

9.Holland, J. H. (1995). Hidden order: How adaptation builds complexity. Reading, Massachusetts: Perseus Books.

10.Mac Lane, S. (1978). Category Theory for the working mathematician, 2nd edition. New York: Springer.

11.Spivak, D.I. (2014). Category theory for the sciences. Cambridge, Massachusetts: MIT Press.

Questions?

Back-Up Slides

New Management Imperatives

• Context – Use information to parameterize internal and external complexity and map among categorical representations to enable efficacious decisions and actions

• Catalysis - Engage predictability (cycles) & uncertainty (events) by accelerating information gathering and using by testing ideas through action – “Leadership Practices”

• Categories - Determine and communicate a common understanding of who is acting and what is being done.– Mission: Construct largest Category supporting a Project that can be

accomplished by these People: “limit” as greatest lower bound (GLB) – Roadmap: Construct smallest Category reflecting: connected activities

connected people: “co-limit” as least upper bound (LUB)

ΠTraditional leaders often claim to embody these Categories. By doing

so, they become a personified structural attractor. Because these categories are “universal properties” transcending any individuals, those who seem to embody them may appear to have “charisma”.

A better solution is a rigorous mathematical treatment using diverse input. This enables additional levels of complexity to be layered onto the

mathematical representation that is understood & used by the team.

Five Leadership Functions

• Gathering Information for situational awareness• Generative activities to create options that

accelerate information producing events• Using information by implementing the structures

for resource and information flows• Administering the execution & control of activities,

resources & information• Building community & evolving collective identity

14–

Frequency of LeadershipPractices

Organizing/ Coordinating Mechanism

System Function Performed

Collective-LevelSituational Awareness

Emergence

Ratcheting/Timely & Effective Decision-Making

Entrainment

Shared Identity

Info Gathering/ Learning Culture

Generative/ Creating Alternatives & Options

Info Using/Accountability Culture

Administration/Efficient Executing

Sense of Community/Employee Engagement

Illustrative Practice:- Ask questions of diverse parties

- Bias to“ try something” with others and process feedback

- Promote people & activities for success not politics

- Offer incentives to meet objectives

- Find & promote a common cause or purpose

Culture of Leadership

Among Team Members

Complex Adaptive

System

OrganizationalProperties Enabling

Performance & Adaptation

Social Field Potential

• Action Vectors, ai Persons Directed To Goal• Order Parameters, Higher-scale relevant

observable among ai

• External Complexity, Opportunity/Threat• Internal Complexity, Organizing Potential • Singularities Discontinuous Change

ξ

cint

cext

Proposed definition:Value is created when Information is gathered and used

such that through agency the residual uncertainty that remains between internal and external complexity is reduced.